The Magic Of Mathematics-slicer.pdf

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Discovering the Spell of othematics

M*L«*

lJ

,z

author of The Mathematics Calendars The Joy of Mathematics

THE

MAGIC Discovering the Spell of Mathematics

THEONI PAPPAS WIDE WORLD PUBLISHING/TETRA

Copyright © 1994 by Theoni Pappas. rights reserved. No part of this work may be reproduced or copied in any form or by any means without written permission

All

from Wide World

Publishing/Tetra.

Portions of this book have

appeared

in

previously published

works, but were too essential to not be included.

Wide World

Publishing/Tetra

P.O. Box 476 San Carlos, CA 94070

Printed in the United States of America. Second Printing, October 1994.

Library of Congress Cataloging-in-Publication

Data

Pappas, Theoni, The

magic of mathematics Theoni Pappas.

:

discovering

the

spell of mathematics /

cm.

p.

Includes

bibliographical

references and index

,

ISBN 0-933174-99-3 1.

Mathematics--Popular works.

QA93

,

P368

510-dc20

I. Title.

1994

94-11653 CIP

This book is dedicated to mathematicians

who have created and are

the magic

creating

of mathematics.

CONTENTS

PREFACE

1

MATHEMATICS IN EVERYDAY THINGS

3

MAGICAL MATHEMATICAL WORLDS

33

MATH EMATICS & ART

63

THE MAO IC OF NUMBERS

97

MATHEMATICAL MAGIC IN NATURE

119

MATHEMATICAL MAGIC FROM THE PAST

143

MATHEMATICS PLAYS ITS MUSIC

173

THE REVOLUTION OF COMPUTERS

189

MATHEMATICS & THE MYSTERIES OF LIFE

223

MATHEMATICS AND ARCHITECTURE

243

THE SPELL OF LOGIC, RECREATION & GAMES

265

SOLUTIONS

311

BIBLIOGRAPHY

315

INDEX

321

THE MAGiCOf MATHEMATICS

1

PREFACE You don't have to solve

be

problems

or

mathematician to discover the

a

magic of mathematics. This book with not

underlying

an

expect

Is

collection of Ideas

a

mathematical theme. It is not

to become

exhausted. The Magic

proficient of

in

topic

a

ideas

—

textbook. Do

a

find

or

an

idea

delves into the world of

Mathematics

ideas, explores the spell mathematics casts

on our

lives, and helps

you discover mathematics where you least expect it.

Many

think of mathematics

as a

fixed curriculum.

rigid

could be further from the truth. The human mind mathematical ideas and

creates

independent

of

world

our

world almost

as

which

from

new

objects

point

can

If

worlds

new

these ideas connect to

our

magic wand had been waved. The way

one

always

operate, equations solves

a

fascinating

and presto

—

Nothing

continually

dimension

can

disappear

be found between any two

into another,

in a

points, numbers

solved, graphs produce pictures, infinity

are

problems, formulas

generated

are

—

all

seem

to possess

a

magical quality. Mathematical ideas

are

in alien worlds and its

A

creativity. world,

while

perfect our

figments of the imagination. Its ideas exist objects are produced by sheer logic and

square

world

or

has

circle exists in

a

mathematical

of

only representations

things

mathematical. The topics and concepts which

by

no means

can

it

mentioned in each

easily cross over the arbitrary boundaries

were

possible,

idea to and

are

can

a

area.

Each topic

enjoyed independently.

I

stepping stone into mathematical worlds.

are

examples

of chapters. Even if

it would be undesirable to restrict

specific

be

chapter

confined to that section. On the contrary,

a

mathematical

essentially self-contained, hope this book will be a is

Print Gallery by M.C. Escher. © 1994 M.C. Escher/Cordon Art-Baarn-Holland, All

rights reserved.

MATHEMATICS IN EVERYDAY THINGS THE MATHEMATICS OF FLYING

THE MATHEMATICS OF A TELEPHONE CALL PARABOLIC REFLECTORS & YOUR HEADLIGHTS COMPLEXITY AND THE PRESENT MATHEMATICS & THE CAMERA

RECYCLING THE NUMBERS

BICYCLES, POOL TABLES & ELLIPSES THE RECYCLING NUMBERS

LOOKOUT FOR TESSELLATIONS

STAMPING OUT MATHEMATICS MOUSE'S TALE A MATHEMATICAL VISIT THE EQUATION OF TIME

WHY ARE MANHOLES ROUND?

4

THE MAGIC OF MATHEMATICS

There is

no

branch of mathematics, however abstract,

which may not someday be applied to of the real world.

So

many

things

routines have

from one

taking least

a

a

with which

we

come

mathematical basis

plane flight

expects,

one

to the

or

—

Nikolai

phenomena

Lobachevsky

into contact in

our

daily

connection. These range

shape of a manhole. Often when

finds mathematics is involved. Here is

random sampling of such

cases.

a


5

THE MATHEMATICS

OF FLYINC

The grace and

ease

human's desire to

of the

fly. Ancient

interest in various

realizes that the

just

a

Greek

flight

of birds have

always

tantalized

stories from many cultures attest to

creatures.

flying flight of Daedalus

myth. Today enormous

Viewing hang gliders,

and Icarus

was

probably

one

not

sized alrcrafts lift themselves

and their cargo into the domain of the bird. The historical steps to achieve

downs.

flight,

as we

Throughout

mathematicians and other

of flying and have

now

know It, has

had Its up and

literally

the years, scientists. Inventors, artists,

professions

have been

Intrigued by

developed designs, prototypes,

the Idea

and experiments

in efforts to be airborne.

Here Is

a

condensed outline of the

•Kites

were

invented

•Leonardo da Vinci

history of flying:

by the Chinese (400-300 B.C.).

scientifically studied

sketched various flying machines (1500).

the flight

of birds and

6


Italian mathematician Giovanni Borelli

•

muscles •

were

too weak to support flight

proved

that human

(1680).

Frenchmen Jean Pilatre de Rozier and

Marquis d! Arlandes

made the first hot air balloon ascent (1783). •

British inventor. Sir

George Cayley, designed the airfoil

(cross-section) of a wing, built andflew (1804) thefirst model

glider, andfounded the science of aerodynamics. •

Germany's

Otto Lilienthal devised

a

system to measure the lift

produced by experimental wings and made thefirst successful manned gliderflights between 1891-1896. •

In 1903 OrviUe and Wilbur

powered propeller

driven

with wind tunnels and

Wright made the first engine

airplane flights. They experimented

to measure the lift of designs. They perfected theirflying techniques and machines to the point that by 1905 their flights had reached 38

weighing systems

and drag

length covering a distance of20 miles!

minutes in

Here's how

we

In order to

fly,

get offthe ground: there

be balanced.

are

vertical and horizontal forces that must

Gravity (the downward vertical force) keeps

earthbound. To counteract the

pull of gravity, lift (

a

upward force) must be created. The shape of wings and the

airplanes

Is essential In

to

flight

is

always

a

think of air

medium,

as

airplane itself,

as a

never

or

that

as

the

speed of gas

or

seems

flight of birds,

of

almost

but without

of the components of

ground.

One does

since it is invisible. Yet air

an

slices the air

Swiss mathematician Daniel Bernoulli

Bernoulli's

It

have left the

substance,

The wing of

water.

divides

of the

key.

physical analyses

flying, today's airplanes would not

holds the

quantify the elegance

the mathematical and

design

creating lift. The study of nature's design

of wings and of birds In

sacrilegious

us

vertical

airplane, as

as

it passes

well

as

the

through

it.

(1700-1782) discovered

fluid increases its pressure decreases.

principle1 explains

the lift force. The top of the

how the

wing

shape of a wing

is curved. This

curve

creates increases


the

of air and

speed

passing curve,

speed of the

thus its air pressure is

wing

and thus lifts the

gravity)

air

passing under the wing

higher.

pushes toward

moves or

the air pressure of the air

Since the bottom of the wing does not have this

over it.

the

thereby decreases

7

plane

The

high

is slower and

air pressure beneath the

the low pressure above the wing,

The

into the air.

weight

(the pull of

is the vertical force that counteracts the lift of the

plane.

The wing's shape makes the distance over the top longer, which means air must travel over the top faster, making the pressure on the top of the wing lower than under the wing. The greater pressure below the wing pushes the

wing

up.

When the wing is at

a

steeper angle,

the distance

the top is

over

even

longer, thereby increasing the

liftingforce.

Drag and thrust

picture.

Thrust

backwards. relies

pushes

the

plane

A bird creates thrust on

its

another, i.e. be thrust and

greater the

the horizontal forces which enter the

flying

forward while

drag pushes

it

its

wings, while

a

by flapping For

to maintain

plane propellers jets. straight flight all the forces acting on it must equalize

plane and

are

zero.

or

a

The lift and gravity must be

drag must balance. During take off the

than the

drag,

but in

flight they

must be

zero,

a

level one

while the

thrust must be

equal, otherwise

plane's speed would be continually increasing.

Viewing birds swooping and diving reveals

two other

flying factors.

8

THE MACIC OF MATHEMATICS

When the will

also

speed of air over the top of the wing Increase.

approaching of the

wing

approximately bird

or

15

on

angle of attack,

be further Increased. If this or more

plane begins

it is called the

vortexes

By Increasing

air, called the

can

the

degrees,

the lift

can

Increased, the lift

wing's angle the

speed

angle

over

to

the

the

top

Increases to

stop abruptly and the

to fall Instead of rising. When this takes

angle of stall. The angle of stall

the

Is

place

makes the air form

top of the wing. These vibrate the wing causing the

lift to weaken and the force of gravity to overpower the lift force. Not

having

been endowed with the

flying equipment

humans have utilized mathematical and themselves and other

physical principles

to lift

off the ground. Engineering designs continually adapted to Improve an

things

and features2 have been aircraft's

of birds,

performance.

lLaws governing

the flow of air for

airplanes apply

to many other aspects certain computer

lives, such as skyscrapers, suspension bridges, disk drives, water and gas pumps, and turbines.

In

our

^The flaps and slots are changes adapted to the wing which enhance lift. The flap Is a hinged section that when engaged changes the curvature of the wing and adds to the lift force. Slots are openings In the wing that

delay

the stall for

a

few

degrees.


S

THE MATH EMATI C time

Every

telephone send

a

you

pick-up

the

place

call,

receiver to

fax,

a

OFATELEPHONI CALI

modem Information

or

you are entering a phenomenally complicated and enormous network. —

encompasses the

many calls How does

are a

globe

is

The communication net that

amazing. It

system

which is

call find its way to

day

over

"broken-up" by

different countries and bodies of water

phone

is difficult to

fielded and directed each

someone

varied

operate?

in your

Imagine how this network.

systems

How does

city,

state

or

a

of

single

another

country?

In the

early years

cranked the

of the

phone

to

telephone,

get

an

one

picked-up the

operator.

A local

the line from the local switch board and said and from there connected you with the reach.

Today

operator "number

party you

the process has mushroomed

as

receiver and

were

came on

please",

trying

to

have the various

10


methods used to convert and direct calls. Mathematics

Involving sophisticated types of linear programming, coupled with binary systems and codes, make

sense

out of a

potentially precarious

situation.

How does your voice travel? Your voice

converted In the receiver to electrical electrical

impulses

produces sounds which are

signals. Today

these

can

40

be carried and

converted In

3 ^l^.

J976

16'

a

variety of ways.

They may be changed

/*U*X~W*A~*-u^ft

to laser

light signals which

then

are

carried

along

fiber optics

cables1, they maybe N

converted to radio

signals

and transmitted over

radio

or

microwave links

from tower to tower

country,

or

kX**4-u4

u-

fcwvÛ^

-fri,

tf^.c-

4L~-~i

Tt-t.

e^4r*y**f

**7

they

may remain electrical

along

*****

across a

the

as

signals

4tt~ZE^ce.

i

?k7\ lf~«Zio-t<.

—

—

£•-*-+. fan*.

—

J

***~r-

£* 4,

phone

lines. Most of the calls

Pages from the notebook of Alexander Graham Bell

connected in the USA are done an

automatic

by

in

his first telephone message on his invention talking to his assistant, Mr. Watson. "I then shouted into M the following sentence: 'Mr. Watson —Come here—I want to see you.' Tb my delight he came and declared that he had heard and understood what J said." which he writes

of

MATHEMATICS IN EVERYPAYTHIKICS

switching system. Presently the fastest. Its

system has

information for all

a

aspects

telephones

available. Calls

can

frequencies

or

switching system

converted to

multiple conversations

program which contains the needed of

telephone operations

then encoded with

calls

can

thus travel

order until

When

a

keeping

digital signals.

to be transmitted

Either method enables

a

the

same

wires.

digital signals

which

along

binary number sequence.

"simultaneously" along

The Individual

the lines in

a

specified

they are decoded for their destinations.

call is placed, the system chooses the best path for the call

and sends

a

chain of commands to

entire process takes

a

fraction of

direct route to the other view

while

are

The most modern systems convert calls Into are

is the

being used and which paths are transmitted by electric currents at different

track of which

be

electronic

point

a

complete the circuitry. The

second.

party— that would

Ideally

it would take

a

be desirable from the

of the economics of distance and time. But if the direct

line is at

capacity servicing

other calls, the

new

call must be sent

along the best of the alternative routes. Here is where

linear

programming2 comes

the

Into

picture.

Visualize the

telephone routing problem as geometric possible having

solid with millions of facets.

solution. The

challenge

to calculate every

one.

Each vertex

a

complex

represents

a

is to find the best solution without

In 1947, mathematician

George

B.

11

12


Danzig developed the simplex method

of the solid,

the

edges along always heading

after another, while as

the solution to

programming problems. The simplex method.

linear

complex

essence, runs

to find

the number of

possibilities

checking

for the best solution. As

programming problems, such calls

telephone

chosen point

so

exactly

the

best

for

routes

the middle of the solid. After

point, the algorithm warps the

interior

that it

as

reshapes

the

Into the center.

problem

which

The next

step

brings

the

is to find

a

point In the direction of the best solution and to warp the

new

again, and bring the

structure

warping

is

transformations and lead

Today, double

rapidly

the old

are

point Into the

center.

time

based

is

an

on

concepts

of

give

These

illusion.

Unless the best

repeated

projective geometry

to the best solution.

telephone salutation "number please" takes

meaning.

telephone

new

done, the direction that appears to each

improvement

and

method that

distances. The Karmarkar algorithm

by going through

arbitrary

an

entire structure

the

long

over

short-cut

selecting

a

cuts down the time needed to solve very cumbersome

drastically

a

In 1984,

efficiently.

mathematician Narendra Karmarkar discovered

takes

long

is no more than 15,000 to 20,000,

this method manages to find the solution

linear

In

one corner

The

receiver and

once

simple

placing

a

process of

call,

now

picking

on a

up your

sets Into motion

a

vast

complicated network that relies on mathematics.

^Depending

the type of lines used, the number of "simultaneous" range from 96 to over 13000. Fiber optic systems can than the traditional copper/aluminum more information on

conversations

carry

even

can

cables.

^Linear programming techniques are used

to solve a variety of problems. the problems entail many conditions and variables. A simple case may be an agricultural problem: A farmer wants to decide how to most effectively use his/her land to maximize production and profit Conditions and variables would involve such things as considering different crops, how much land each crop requires, how much yield each produces per acre, and how much revenue each brings when sold. To solve such a problem, one writes linear Inequalities and/or equations for each condition and looks at a 2-dimensional graph of a polygonal region for the solution.

Usually


13

PARABOLK R E F L E CTO R S & YO UI

When you flick the switch of your

headlights

from

dim,

to

bright

HEADLIOHT!

mathematics is at work. To be

specific, the principles of a parabola do the trick. the In

The reflectors behind

headlights

shape.

are

In fact,

parabolic

they are

paraboloids (3dimensional formed

parabola1

parabolas

by rotating a about its axis of

symmetry). The bright beam is created by a light source located at the focal

point of the parabolic

flectors. Thus, the

light

axis of

When the

symmetry.

changes

light

location. It is

rays travel out

lights

longer at

no

rays do not travel

re

parallel

are

the focus, and

to the axis.

parallel

to the

parabola's

dimmed, the light as a

source

result the

The low beams

now

that

only

point down and up. Those pointing up

are

the downward low beams

shorter distance than the

are

reflected

a

shielded,

so

high beams. The

by

while he

was

the

that

a

parabola

discovered

involving

the

example,

it

trajectile's path uses

only

of BTU thermal units

a

ancient

to

that

was

Over

(1564-1642) who showed

parabolic. Today

highly

the cube.

and discoveries

have been made. For

Galileo

one

energy efficient

1000 watts but

as a

curve

duplicate

uses

new

parabola

was

was

an

trying

centuries,

hardware store and find heater which

is

Menaechmus (circa 375-325 B.C.)

on

go into

parabolic

produces the

heater that operates

can

same

a

electric

number

1500 watts.

1 Parabola is the set of all points in a plane which are equidistant from fixed point called its focus and a fixed line called its directrix.

a

14


COMPLEXITY & THE PRESENT "The hours

midnight the at

exactly

seven

Island that evening."

excerpt from

The Law

by Robert M,

Coates Illustrates, sometimes

things

just seem to take place with no apparent reason. Nor Is there a

%

warning

that

a

particular event Is

about to take

place.

We have

all experienced such events and

usually attributed them to

"coincidence", since there were no

apparent indicators to

predict otherwise

bridge.

a

seven

'til

normally quiet ...

o'clock, ...it just looked

Manhattan who owned

As this

from

are

nearly ones on

Beginning almost

as

if everybody

motorcar had decided to drive out

on

in

Long

MATHEMATICS IN EVERYDAY THIN6S

may hold

Complexity Is an emerging science which explanations to such questions as:

answers or

at

least

How is it that •

the universe

•

cells know which organs and

emerged out of the void ?

•

on

1994 Los

January 17,

to become and when?

parts

Angeles

suffered

an

earthquake of unexpected magnitude and destruction? the Soviet Union's

•

collapsed

in such

Yugoslavia

•

long reign

short time?

a

thrown

was

its satellite countries

over

into

suddenly

severe

internal

wars?

species that has

•

a

•

for

not

for millions of years

changed

suddenly experiences a mutation? or

no

apparent

reason

is that each

represents

underlying a

very

common

factor of these events

complex system.

A

system governed

of factors, which

number and

diversity delicately balanced, tittering between stability an

enormous

factors which act

such

on

changing. Consequently,

tug

of

war

changing

and

system

is

edge of chaos.

are an

by which

the

circumstances. Those

artificial

to

studying

constantly this

new

In

a

seems

state of to be

high

changing

science draw as

chaos

intelligence, fuzzy logic,

and mathematicians feel that other tools and

always

There

a

self-

part of a complex system. It

mathematical and scientific ideas, such

probability,

and chaos. The

system regains equilibrium by

Itself

adapting

essential

are

growing and

are ever

between order and chaos. Spontaneous

organizing dynamics is the means

a

complex system

a

potential chaos i.e. at the continual

upward

plunges downward?

The list is endless. The

by

the stock market surges

factors/

on a

host of

theory, fractals,

etc. These scientists

today's mathematics, along with are capable of creating a

tech innovations,

complexity framework that

can

impact major aspects of our global

world, especially economics, the environment, and politics.

15

16


MATHEMATICS & THE CAMERA Ever

wonder

about

the

f-stop

number of a camera? Where did It

get determined?

"f

Its

name?

How

stands for the mathematical term

brightness of the photographic image

on

film

Is

It

The

factor.

depends

on

the

aperture and focal

length of

the lens.

Photographers use

what Is

known

as

the

f-number

system to relate focal

length and aperture. The

f-stop Is calculated by

measuring the diameter of the

aperture and

dividing It into the focal

length of the lens. For

example, f4= 80mm lens/20mm aperture. fl6=80mm We

see

the lens

opening

f-stop number shutter

lens/5mm aperture.

speeds,

is smaller

increases.

you

can

(the aperture decreases)

Working manually

photograph you want in focus.

with

as

the

f-stop numbers and

decide how much of the


17

RECYCLINC THE NUMBER!

Here mathematical units and

used to get the point

symbols were across

about

recycling paper!

ton of

•A ton of

virgin

•A ton of

recycled paper uses 4102 kwh less energy.

•A ton of

recycled

paper

uses

recycled

paper

produces 60 pounds less air

recycled

paper

produces 3 cubic yards less solid

•A ton of

recycled

paper

uses

less tax money for landfill.

•A ton of

recycled

paper

uses

17 fewer

paper

* a

recycled paper

7000

gallons less water to

produce. •A ton of

pollution. •A ton of waste.

—the numbers behind •

37% of all landfill is

•

Only

logged trees.

recycling and landfill

—

comprised of paper.

29% of all newspapers

produced

are

recycled by the

consumer. •

165 million cubic

yards of landfill are needed for our paper

wastes per year.

•97% of the

virgin forests of the continental

cut down in the I Was Once A Tree

Alonzo Printing,

past 200 years.

...Newsletter, Spring 1990,

Hayward

CA.

USA have been

18


BICYCLES, POOL TABLES &

The

ELLIPSES

section curves,

ellipse, along with other

Greeks

as

early

was as

studied

the 3rd

conic

by

the

century

If this ball is hit through the location of the focus, marked with an X, it will bounce off the cushion and go to the other focus where the pocket is located.

B.C.. us

of

associate

the an

Most

ellipse with

angled

circle

or

the

orbital a

path of planet, but

elliptical shapes and properties also lend themselves to


An

ellipse

has two foci, and the

sum

of the distances

from the foci to any point of the ellipse always equals the length of its major axis. le. \PF1\ + \PF2\.= \AB\.

imagined

that

and

gears

applications. Who would have ellipse would find itself in the design of bicycle

nonscientific

contemporary

an

pool

manufactured with The

drawing,

on

tables? a

front

Today

elliptical gear leg

power and

a

property of the ellipse's

the previous page, the

elliptipool has

rear

been

gears.

design can

quick upward

Elliptipools, elliptical shaped pool tables,

utilize the reflection on

and circular

the previous page, illustrates how this

utilize the downward thrust of return.

have

bicycles

some

are

designed

to

two foci. As illustrated

one

pocket located

at

one

points of the ellipse. A ball hit so that it passes ellipse's non-pocket focus will bounce off the side of

of the two focus

through

the

the table and travel the reflected

focus).

path over to the pocket (the other

19

20


LOOKOUT FOR

TESSELLATIONS

u

„

This Escher-like transformation

Mark Slmonson Illustrates the of tessellations

as a

communication.

appeared

In The Utne Reader and

Metropolitan Transportation Reprinted courtesy

Minneapolis,

MN.

on

the

cover

Communication

of Mark Slmonson.

by

use

form of visual

This

graphic

of Transactions,

publication.

Bluesky Graphics,

a

MATHEMATICS IN EVERYDAYTHIN6S

21

STAMPING OU1 One

usually

doesn't

expect

encounter mathematical Ideas

trip

to the

post office, but here

MATHEMATICS

to

on a

are a

few of the stamps that have been

printed with have

These and many other Ideas

mathematical themes.

appeared

on

such

popular

Items

as

posters, television,

T-shirts, post-Its, mugs, bumper stickers, and stickers. 7*^**^V

US 10 FORMULAS MAT£MATKAS ODE CAMBIABOHIA FU DE The

The

Pythagorean

Pythagorean

theorem

LA TIERS*

—Nicaragua

theorem

—Greece

Bolyai—Rumania

Gauss—-Germany

Mathematical Formulas —Israel

22


THE MOUSE'S TALE 'Contrariwise,' continued Tweedledee. "if it

was so, it

might be;

and if it were so, it would be: but

'Fury a

as it isn 't, it

said to

ain 't, That's logic.' —Lewis Carrol

mouse, That he met in the

Alice in Wonderland

Charles

house,

Lutwidge Dodgson

(1832-

'

Let us both go to In w. / will

1898)

of mathematics,

proseouto

children's

Come, I '11

books1,

mathematics

no

puzzles

a

When

issues.

trial : For

his

roally

seems

morning

1 'vo

that

nothing

to do.'

was

social

Lewis Carroll.

Dodgson

be associated Lewis Carroll

Said tho mousoto

on

writing children's tales

pseudonym

this

of

children's

and essays

stories, poetry

"We must hnvc

It

did not want to

connected with his

or

identity.

In

fact,

he

would return mail addressed to Lewis

tha our, a

trial,

Carroll.

dflur sll', With

of

and author

photographer

denial;

'Such

creator

and games, renowned Victorian

you.— take

mathematician, lecturer

was a

When the Bodleian

Library

111)

cross-referenced

Jjrjr or Judge,

Dodgson

wuulil Uo

WMllnauur urdKtll. 1

and

Carroll, Dodgson took exception to the

I' II ll«

lil
connection.

IMIb* CMnolaf

M rnrri

had

Dodgson

U»»l»fe

games.

a

passion for word

teenagers from

In 1991, two

New Jersey discovered

(both visual and verbal a

in

nature)

Long Tale from Chapter Two

the tale

by a

mouse

to this tale and tail Maiden

in

a

poem

four-way pun

in the poem, A Caucus-Race and

in Alice in

shaped

Wonderland. Alice is told

like

a

long

tail. In addition

link, students Gary Graham and Jeffrey

discovered that when the poem

outline of

a

a mouse was

was

written in stanza form the

formed— each stanza had two short lines


'wttftktw.te hmxi ^tfy^Hiiln

*$ipf$4 'tfkljJmf, shir. ^2-

mtoptipi, Vti fa jury*

•

'""

'

*kk wtipte tm$4?'«tiA condemn $mt£&&

(the mouse's body) and

a

long third line (the

mouse's tall).

Lastly,

poetic structure defined by

a

pair

of rhyming lines followed by another line of different length. this intentionally? you think Lewis Carroll planned all

Do

they

found that

a

tall-rhyme Is

a

1 Euclid and His Modem Rivals, An Elementary Treatise on Determinants, Alice in Wonderland, The Hunting of the Snark, Phantasmagoria and Other Poems, Through the Looking Glass are of a few of Dodgson works.

23

24


A MATHEMATICAL

VISIT

Not quite

sure

what to expect, I rang

the doorbell A voice asked me to

please push thefirstfive terms of the Fibonacci sequence.

Fortunately, I had done some research after my magazine assigned me the story on the home of the renowned mathematician, Selath.

I pushed 1,1,2,3,5 and the door slowly

opened.

As I passed

through the doorway, I was struck by the catenary stone shaped archway independently suspended

at the entrance. After

minute, Selath entered saying, "May I offer you something

long

a

after your

drive?'

"TdreaRy appreciate a

glass ofcold water," I

replied. "Please

come

with me," he

said, leading the way.

As Ifollowed, I couldn't

kitchen, an

help noticing the many unique and unusual objects.

we came

to

a

In the

peculiar table with many legs. Selath pulled

equally unusual bottlefrom the refrigerator.

I must have had

a

quizzical expression, for Selath began, "While you drtnk your water, we

might as well start the tour here in the kitchen."

this table and bottle

are

As you noticed

not your everyday accessories. I use


tangram tables for dining because its

rearranged into

as

many

shapes

as

seven

the

components can be

tangram puzzle. Here in the

kitchen its made into a square shape today, while Tve one

in the

living room into a triangle,

The water container is whats known

for dinner.

arranged the

since I am expecting two guests as a

Klein

bottle— its inside and outside

are one.

If

you look at thefloor you'll notice only two

shapes of tiles are used." "Yes," I replied, to

seem

"

but the design doesn't

repeat anywhere."

"Very perceptive." Selath seemed pleased with my response. "These

are

Penrose

tiles. These two shapes can cover a

plane

in

a

non-repeating

fashion." "Please continue," I urged. most anxious to

see

"

Tm

all the

mathematical parts of your home."

"Well, actually almost all of my house is mathematical.

Anywhere you see wallpaper Tve designed special

tessellation patterns for walls room.

room

a

la Escher. Lets proceed to the Op

Every item in here is an optical illusion. is

an

illusion. Furniture, fvdures,

In fact,

photos, everything!

example, the couch is madefrom modulo cubes fabric stacked to give thefeeling

sculpture

in the middle

divergence,

was

of

an

designed

reality in this For

in black and white

oscillating

illusion. The

to show convergence and

while the two stackedfigures

are

exactly the same size.

25

26


This

lamp's base, viewedfrom this

location, makes the impossible tribar."

"Fascinating! I could spend hours discovering things

in this room," I

replied enthusiastically. "Since move

we are on a

tight schedule,

lets

to the next room," Selath said

as

he

led the way.

We entered a darkened

room.

"Watch your step here. Come

this way to the parabolic

screen," Selath directed.

As I peered into the disc a

moving this

a

scene

appeared.

"Is

video camera?" I asked.

"Oh, no,"Selathlaughed, "Icall it my

system. The lens above the hole captures rotates to project scenes outside my camera

would. It is called a

camera

antique surveillance

light in the day time and

home, much the same way obscura. I have

a

special

a

lens

for night viewing." I was

busily taking notes, realizing I would have much additional

research to do

before writing my article.

Glancing around I noted, 'Yourfluorescent clock seems

to be off."

My watch read 5:30 pm while his read 21:30. "No, its just that I have the 24 hours of the day arranged in base

MATHEMATICS IN EVERYDAYTHIN6S

eight because Tm working on eight hour cycles this week.

So 24:00

hows would be 30:00 hours, 8:00 would be 10:00, and

on."

so

Selath explained.

"Whatever works best for you," I replied,

a

bit confused.

"Now, let's go to the master bedroom."

And off we went, passing all sorts seen in a

of shapes and objects rd never

home before.

'The master bedroom has

a

semi-spherical skylight

in addition to

27

28


movable geodesic skylights.

They are designed to optimize the use

of solar energy." "Marvelous, but where is the bed?' I asked. "Just push the button on this wooden cube, and you will

unfold "What

see a

bed

with a head board and two end tables."

a

great way to make a bed," I replied.

"There are many more things to see, but time is short. Lets go in the bathroom

so

you can

see

the mirrors over the basin. Come this way.

Now lean forward." To my

surprise I saw

an

infinite number of images ofmyself

repeated. The mirrors were reflecting back andforth into one another ad infinitum. "Now turn around and notice this mirror. Whats

different about it?"

Selath asked.

"My part

is on the wrong

'To the contrary this seen

side," I replied.

mirror1 lets you see yourself as you are really

by others," Selath explained.

Just then the doorbell rang. The dinner guests had arrived.

don't you stay to dinner?' Selath asked.

living room yet, It

was

"Why

'You haven't seen the

and Fm sure you'll enjoy meeting my guests."

hard to conceal by enthusiasm. "But your table is set for

three," I blurted. "No problem.

parts

With the tangram table I canjust rearrange

and we'll have

a

a

few

rectangle.

1Made from two mirrors placed at right angles to each other. The right-angled mirrors are then positioned so that they will reflect your reflection.


29

THE EQUATION

OFTIME

If you have

sundial, noticed

ever

that

the

registered slightly your watch.

time

the sundial

differed

difference

a

you may have

on

on

used

from that

This

Is tied Into the

length

daylight during the year.

of

In the

15th century, Johannes Kepler formulated three

that

laws

planetary

how the

described

governed

motion.

Kepler Earth

around the Sun In an

travels

elliptical

orbit, and also explained that the line

Sun and

segment joining the

the Earth sweeps out

equal

(sectors)

In

equal

Intervals of

time

its orbit.

The Sun Is

along

located at

one

areas

of the foci of the

ellipse

thereby

sector's

area

making

equal for

a

time Interval and the arc

each fixed

lengths

of the sectors

unequal. Thus

Earth's orbit

speed

its

„

,

,

along

This accounts for the

path.

variations ,

varies

the

,

in ,

the

lengths ,

,,„.

year. Sundials

rely

on

of

A 10th century pocket sundial times of ^^ m six ^^ listed on each stlck te placed in. the hole of daylight, side-A the column with the current month.

daylight during different

,,

,

30


and

daylight depends

on

the time of the year and

geographic

location. On the other hand, the time intervals of our other clocks are

consistent.

ordinary

The difference between

clock is referred to

as

a

sundial's time and

the equation

of time.

an

The almanac

lists the equation of time chart, which indicates how many minutes fast

or

slow the sundial is from the

the chart may look like the

one

Equation

regular

clocks. For

example,

below.

of Time Chart

(The negative and positive numbers indicate the minutes the sundial is slower or faster than an ordinary clock. Naturally the table does not take into consideration daylight difference within time zones.) VARIATION

DATE 1

-3

15

-9

Jan

Feb

1 -13 15 -14

Mar

1

-3

15

-9

1

-4

April

15

May

0

1

+3

15

+4

// the sundial shows 11:50 on May 15, its time should be increased by 4 minutes to 11:54.


31

WHY ARE

MANHOLES ROUND?

Why Is the shape of a manhole

circular?

Why not a square, rectangular, hexagonal,

or

elliptical

shape? Is it because

There Is

a

a

circle's shape Is

more

mathematical reason.

What is your

explanation?

pleasing?

Metamorphosis © 1994 M.C.

W

by

M.C. Esclwr.

IDscher/Cordon Art-Baam-HoUand. All rights reserved.

MACICAL

MATHEMATICAL WORLDS HOW MATHEMATICAL WORLDS ARE FORMED GEOMETRIC WORLDS NUMBER WORLDS

THE WORLDS OF DIMENSIONS THE WORLDS OF INFINITIES

FRACTAL WORLDS MATHEMATICAL WORLDS IN LITERATURE

34


can it

How

be that mathematics,

product of human

a

thought independent of experience, is so admirably adapted to the objects of reality. —Albert Einstein

Mathematics

alien to

totally

can

and

of

things

exist

on

our

so

world.

many

in

so

our

world,

perfect,

so

complete mathematical or

composed

in the infinite set of

of

points, equations,

fractals,

knots,

so on.

A

things

strange,

pin point of a needle

the

One finds such worlds

numbers. curves,

by

in its own worlds—worlds so

yet delves world

is linked and used

Until

one

understands how mathematical worlds and

systems

formed,

some

of its

worlds may

seem

are

contradictory.

For example,

might ask how an world

can

exist

one

infinite

only on

tiny line segment, world be created

a

or a

using

three

points/This only chapter seeks to explore the

%

XT

magic of some of these

mathematical worlds and

_

—

£

delve into their domains. As discussed later, the counting numbers form a mathematical world in. themselves.

MAGICAL MATHEMATICAL WORLPS

HOW MATHEMATICAL WORLDS ARE FORMED

Why, sometimes I've believed many as six impossible things

as

before breakfast.

—Lewis Carroll Little did Euclid know in 300 B.C. when he geometric ideas into

a

to

began

that he

mathematical system

organize

developing

was

the first mathematical world. Mathematical worlds and their elements abound

—

here

find the world of arithmetic with its

we

elements the numbers, worlds of algebra with variables, the world of Euclidean such

objects

objects are

geometry with

that

squares and

continually change

interrelated with

universe,

yet

a

sets the

world. It new

is

All form the universe of

universe that describes and

ground

explains

objects,

These terms

can

mean, but

exists in

explains things all

exist which

can

mathematical system. The

a

they

objects are

be described,

A mathematical

governed. so

called

are

that

one

has

in its

words there

a

to

are no

system

undefined

defined.

definitions, and you have

beginning

objects

formed, how they generate

are

technically they cannot be

terms. For these

feeling Why?

begin

terms.

of what Because

with

some

other terms that

be used to define them.

The best way to understand such a

anything from our

rules for the existence of the

how its

and how

it takes terms to form

how

Independent worlds yet

are

exist without

can

composed of basic elements, which

they

with

us.

Every mathematical world system

all

—

another.

one

mathematics. A universe that

around

triangles, topology

the Mobius strip and networks, fractals with

as

a

system

finite mini mathematical world

this mini world's undefined terms to undefined terms,

a

35

are

is to look at

might

one.

Here's

take form. Assume

points and lines.

In addition

mathematical system also has axioms,

theorems, and definitions. Axioms (also called postulates) are ideas

36

TH E MAC IC O F MATH EMATICS

we we

being true without proof. Definitions are new terms describe/define using undefined terms or previously defined

accept

as

terms. Theorems

are

axioms, definitions

What

type

Ideas which must be proven

or

of definitions, theorems and axioms

have? Here

are some

by using existing

theorems.

that

can our

mini world

might evolve—

Undefined

terms: Points and lines.

Definition

1: A set

of points

is colltnear

if a

line contains

the set.

Definition

2: A set

ofpoints

is noncolltnear

if a line cannot

contain the set.

Axiom 1: Our mini world contains

only

3 distinct

points,

which do not lie on a line.

Axiom 2: Any two distinct points make

a

line. Theorem 1:

Only three distinct lines exists

In

can

this

world.

proof: Axiom 1 states that there are 3 distinct points in this world. Using Axiom 2 we

know that every pair of these points determines a line. Hence three lines are formed by the three points

of this world. This As

example Illustrates how new

Ideas

come

to

a

mind,

mathematical world one

adds

axioms, definitions, and theorems and The

more

might

evolve.

undefined terms,

thereby expands

the world.

following sections introduce you to some mathematical worlds

and their inhabitants.

MAC ICAL MATH EMATICAL WO RLDS

37

GEOMETRIC ...

The universe stands

continually open

WO RLDS

to our gaze,

but it cannot be understood unless

one

first learns to

language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are...geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.—Galileo comprehend the

Mathematics has many types of

geometries. These include Euclidean and

analytic

geometries and

a

host

of non-Euclidean

geometries. find

Here

we

hyperbolic,

elliptic, projective,

topological,

fractal

geometries.

Each

geometry forms a mathematical system with its

own

undefined terms,

axioms, theorems and definitions.

Although these geometric worlds may use

the

same names

This is an abstract design of Henri Poincare's (1854-1912) hyperbolic world. Here a circle is the

boundary of this world. The sizes of the inhabitants change in relation to their distance from the center. As they approach the center they grow, and as they move away from the center they shrink. Thus they will never reach the boundary, and for all purposes, their world is infinite to them.

38

THE MAC\C OF MATHEMATICS

for their elements characteristics.

straight parallel,

or

For

properties, their elements possess different

or

can

geometry

either intersect In

one

lines

are

point, be

be skew. But lines In

lines but great circles of a distinct lines

In Euclidean

example.

and two distinct lines

always

elliptic geometry are not straight sphere, and therefore any two of Its

Intersect in two

points.

Consider the word parallel. In Euclidean geometry parallel lines are

always equidistant and never so in elliptic or

Intersect. Not

hyperbolic geometry. Why? Because every great circle of a

sphere Intersects another. Thus, elliptic geometry has no parallel lines. In

lines

hyperbolic geometry parallel

never

Intersect, but

they do not

resemble Euclidean lines.

Hyperbolic

The above diagram shows two great circles, line 1 and 2 intersecting

at points A &B. closer and continually closer together, yet never Intersect. They are called asymptotic. Euclidean, hyperbolic, and elliptic

parallel

lines

geometries

create three

come

dramatically

different worlds with lines and

points, etc., but whose properties these worlds Is

applications

in

a

mathematical

our

are

system

universes

apart. Each of

unto Itself, and each has

universe.

L^hyperbolic geometry, lines M and N are both parallel to line L and pass through point P. M and N are asymptotic

In

to line L.

¦}


39

NUMBER WORLD! Numbers

be

can

considered

first

elements

Their

early symbols

marks

drawn

indicate

never

Stone

were

the

in

number of

a

entered the

scene

been the

the

same.

probably earth

things. simple

and

computations. irrational

But

tn La

mathematicians

since

counting

are

numbers

familiar with

has

integers,

Pdeta, Spain.

and

decimals,

ever

world of

use

these

their

for

daily

But number worlds also include the rational and

numbers,

non-repeating

to

Many people

Age number patterns found

fractions

the

mathematics.

of

the

decimals,

complex numbers, transcendental

the

numbers, and many many subsets of numbers that

specific properties, such or

perfect

transfinite

are

linked

by

numbers whose proper factors

polygonal numbers shapes of regular polygons,

total the number, connected to the

as

never-ending

numbers,

whose and

on

shapes and

on.

are

It is

40


Interesting to delve

Into the

they developed,

The

counting numbers date back

the

and

interrelationship

how

simple marks of the

Cave In southern

of numbers, surmise

explore their various properties.

Stone

prehistoric times. Consider Age number patterns from La Pileta

Spain, which

to

Inhabited

was

over

ago until the Bronze Age (1500 B.C.). The number over

three

thousand

ago,

years

calculations of a circle's

when

that fractional

glyph

for mouth, O

For example, ^^

1/3, ^^

|/2

0=

known

were

fascinating developed

by

the ancient mathematicians,

approximating their

values.

approxlmatle

Hexagrams

3

=

two

two

two

and their

101

100=

H=

two

12

5

4

binary eqiduaLents.

Over the centuries different civilizations

counting systems

were

1/10.

the ladder method to

10=

two

0

the

Babylonians used another method.

1=

two

Ancient civilizations

was

methods for

while the

in

to write their fractions.

In fact, the Greeks

the

known

used

quantities existed. The Egyptians used the ,

was

Irrational numbers who devised

was

and circumference, and later shown

area

to be irrational and transcendental. aware

It

25,000 years n was

developed symbols

for numbers, and In the 20th

century

the

and

binary

numbers and base two have been put to work with the computer Gottfried Wilhelm

revolution.

about the

(1679).

binary system

He

missionary that

the

I

numeration

corresponded in China. It

(1646-1716) first wrote

In his paper De

Progressione Dyadica

with Pere Joachim Bouvet,

was

through

Chlng hexagrams system.

Leibniz

were

a

Jesuit

Bouvet that Leibniz learned

connected

He noticed that If he

to

replaced

his

zero

binary for each


broken line and 1 for the unbroken line, the the

binary

upon the Sumerian

developed

and

develop

base 60 number

a

hexagrams illustrated Babylonians

numbers. Centuries prior to this, the

Improved

sexagimal system

But this section

system.

on

to

number

worlds Is not about number systems but about types of numbers.

Let's take

at the first

glimpse

a

type of

numbers

numbers. In the world of counting numbers terms

are

the numbers 1, 2, 3

order in which two sum

the

counting

find the undefined

with such axioms

—

counting numbers

we

—

are

as

added does not

the

affect

the

(a+b=b+a, called the commutative property for addition); the are multiplied together does (axb=bxa, called the commutative property for

order in which two counting numbers not

affect

the product

multiplication). even

—

and such theorems

number is also

numbers is numbers evolve

over

solution

always

were

not

an even an even

enough

the value

x

An

number.

even

an

of any two odd

counting problems that were to

Imagine tackling

for the

number plus

sum

But the world of

to solve all the

the years. Can you

was

as

number. And, The

equation

a

problem whose

x+5=3 and not

knowing

about

negative

numbers? What would have been

the

problem

Is defective, there is

no answer.

negative

numbers In

but most mathematicians of

—

Introduced

Europe,

some

reactions

Arab texts

the the 16th and 17th centuries

numbers.

Nicholas

were not willing to accept these (15th Chuquet century) and Michael Stidel

(16th century) referred to negative numbers Jerome Cardan (1501-1576) gave to

equations, he considered them

as

absurd.

negative numbers as

Impossible

as

Although solutions

answers.

Even

Blaise Pascal said "I have known those who could not understand that to take fourfrom zero there remains zero."

42

TH E MAC l< O F MATH EMATICS

history reveals required the Invention

Thus

the

explain

that the solutions of

new

meaning of y7-1

the creation of

or

numbers. to solve

certain

problems to

example, trying the equation x2 -1 led

to

=

numbers. And

imaginary

to

For

imaginary

mathematicians

to

numbers led the

expand

world of numbers to Include all iO, «, 12, t% I4»

real and Imaginary numbers and Here's where the

morel

?+S=3

'

street us can toe

complex

numbers (numbers of the form

'"

>pr«ttkta<

a+bl, where

l=y^-l) the 16th

century.

are a means

of

as

organization of

this line Is linked to

The

has

only one only one

Imaginary numbers

line. So 2i is located and the

Imaginary

complex

use

numbers

location

on

they

on

other

the ordered

Combining

we

get

a

the

complex number

number 4.

organize —

only complex number has

one

This coordinate

and picture the

(4,0)

system

complex

was

chapter

The Magic

plane

number

imaginary axis

1

*f.

an

+

21^

(-31/2,2 i)

<

-?2i

.—! i

i i*i

-5 -31/2

complex

number

equals

the

ingenious way to

numbers. The questions

ofNumbers has

0

¦"-.

¦f—l

So

4+Oi which

Are there any numbers which do not appear

bet! The

-1

-4

-31/2

are

is matched with

means

r-r-r-^

complex

In the

complex

real axis

-31/2

picture

that location.

point (-4, 3)

-4 +31.

the real

way to

origins. Every point

pair for the

Every point on

this line.

what Is called the

form Is associated with

no

—

at their

zero.

imaginary number

an

shown.

as

number lines,

The real

real number and vice

number plane. The real and Imaginary lines

perpendicular

during

all real numbers and

their respective distances and sizes relative to

-31/2

real and

complex.1

picturing mathematical objects.

number line shows the the

versa—so

are

introduced

With their Introduction, all numbers Invented

thus far could be classified

Graphs

and b

a

were

on

some

this

now

is

plane? You

examples.

Any real number can be thought of as a complex number whose imaginary part is 0, and any imaginary number is a complex number with real part 0.

r -1

r-i

plane^


43

THE WORLDS Ol

DIMENSION! • Let's look at the worlds which

created

are

by the Idea of dimensions. A

mathematical world

single point,

on a

exist

on a

single line,

on a

can

In space. In

plane.

(tesseract). Each

a

hypercube

higher dimension

encompasses those beneath It, each lower dimension In Itself.

your life

can

be

a

yet

world

Imagine your world and

on a

flat plane. You cannot

look up or down. Three dimensional creatures

without you

can

even

Invade your world

knowing by simply

entering your domain from above

or

below. Mathematicians, writers, and artists have used various Ideas to

to

try capture the essence of different dimensions In their works.

Dimensions

beyond the third have

alway been Intriguing. one

The cube

of the first 3-D

objects

was

to be

introduced Into the fourth dimension

hypercube. a

The

by becoming a

stages for arriving at

hypercube are illustrated.

Computer programs devised to derive fourth dimension

have

even

glimpses

been

of the

by picturing 3-D

perspectives of the various facets of the hypercube.

44


THE WORLDS OF

INFINITIES

To

see

the world in

a

grain of sand, And

a

heaven in

a

wildflower; Hold

And

has stimulated

Infinity an

Idea drawn upon

infinity eternity in

an

palm

of your hand,

hour. —William Blake

imaginations for thousands of years.

It is

by theologians, poets, artists, philosophers,

writers, scientists, mathematicians and

in the

—

an

idea that has

perplexed

idea that remains illusive.

Infinity has taken on different Identities In different fields of thought. In early times, the idea of Infinity was, rightly or wrongly, linked to large Intrigued

numbers.

by gazing Ancient

—

an

People of antiquity experienced at stars and

philosophers

planets and

or

at

argued

Aristotle

argued

the ideas that

feeling of the

grains of sand

mathematicians

Anaxagoras, Democrltus, Aristotle, and

a

on a

beach.

as

Zeno,

such

Archimedes

infinite

pondered, posed

infinity presented.

proposed the Ideas of potential and only potential infinity existed. 1

actual infinities. He

that

In The Sand Reckoner Archimedes

number of

determining

grains of sand a

method for

on

a

dispelled

beach

calculating

are

the idea that the

infinite

by actually

the number

on

all the

beaches of the earth.

Infinity has been the culprit In many paradoxes. of Achilles and the tortoise and the readers for centuries.

Galileo's

Zeno's

paradoxes

perplexed Dichotomy2 paradoxes3 dealing with have

segments, points, and infinite sets should also be noted.


This field of sunflowers in the Spanish

countryside gives

the illusion

The list of mathematicians with their discoveries and

of infinity.

uses

or

misuses of

300

Infinity extends through the centuries. Euclid (circa B.C.) showed that the prime numbers were Infinite by showing

there made

was no

by

last prime.

Headway in

the realm of the infinite

(1646-1716), and J.W.R. Dedeklnd (1831-1916). phenomenal work of

Georg Cantor (1845-1918)

major breakthrough. Building, creating found set.

a new

a

transflnlte numbers

Using

-

assigned

these topics

are

a

theory

equivalent

sets and same

by developing

cross

the realm of

countabillty,

number of

transflnlte number. His work and

ingenious.

was a

refining Ideas, Cantor by use of the notion of a

numbers that dared to

the idea of

them

But the set

way to compare Infinite sets

determined which Infinite sets had the and

on

and

way to organize mathematics

He determined

the finite.

was

Bernhard Bolzano (1781-1848), Gottfried W. Leibniz

he

objects

proofs

on

45

46


In addition to

teasing

our

mathematical tool. It has

minds, Infinity

played

a

mathematical discoveries. We find it used In: volumes

for

and

calculus

—

e

and

—

Indispensable

infinite sets

—

areas

and

calculating approximations numbers—trigonometryself-perpetuating geometric

and more. dynamic symmetry Other parts of this book explore various notions of Infinity, such Ideas

objects

—

limits

—

other irrational

half-lives

an

determining the

both in geometry and calculus

7i,

Is

crucial role In many

infinite

as

search for ad

—

series

—

—

generating fractals, the chaos theory, the continual a

larger prime number, transfinite

numbers and others

Infinitum.

^The counting

numbers are potentially infinite, since one can be added get the next, but the entire set cannot be actually

to any number to

attained.

Dichotomy Paradox Zeno argues that a traveler walking to a destination will never reach the destination because the traveler must first walk half the distance. Reaching this halfway point, the traveler then has to walk half the remaining distance. Then half of the Since there will always be half of the part that part that remains. remains to walk and an infinite number of halfway points to pass, the traveler will never reach the destination. In The

specific

3 In Galileo's 1634 work, Dialogues Concerning Two New Sciences, he discusses infinity in relation to the positive integers and the squares of the positive integers. He even deals with one-to-one correspondence between these two infinite sets. But he reaches the conclusion that the concepts of equality, greater than, and less than were only applicable to Galileo believed the principle that the whole is always finite sets. greater than its parts had to apply to both finite and infinite sets. Three hundred years later, Cantor showed this principle did not hold for infinite sets and used the idea of one-to-one correspondence to revise the traditional notions of equality, greater than, and less than when dealing with infinite sets. Cantor's modifications did away with many paradoxes involving infinite sets and the whole is always greater than its parts.


47

FRACTAL

WORLDS

/ coined fractal from the Latin

adjective fracfus. The

corresponding Latin verb fragere means 'to break': to create irregular fragments. how appropriate for our needs! that, in addition to 'fragmented' (as in fraction or refraction) fractus should also mean 'irregular1, both meanings preserved in fragment. —Benoit Mandelbrot ...

—

Fractals

are

shapes.

Ernesto Cesaro

magnificent objects

this about the strikes

me

which

come

In

Infinitely

geometric fractal, the Koch snowflake

above all about the

whole. To try to imagine it

as

curve

triangle

whole

an

reduced version

curve—

What

is that any part is similar to the

completely as possible,

realized that each small

shape reduced by

many

(Italian mathematician 1859-1906) wrote

it must be

in the construction contains the

appropriate factor. And this contains

of each small

triangle

a

which in turn contains the

II II

II II

ii ii

ii ii

llll llll

iiii mi

mi mi

mi nn

In 1883, Cantor constructed this fractal called the Cantor set Starting with the segment of length the unit interval on the number line, Cantor removed

the middle one third and got stage 1. Then to each remaining 3rds, removed the middle one-third, thereby creating the 2nd stage. Repeating the process ad infinitum, the infinite set of points that remains is called the Cantor set Here

are

the first stages

of the

Cantor set

48


whole

reduced

shape

further

and

Infinity...

It

so

is

even

to

this

self-

its

part,

all

in

similarity

on

the

curve

wondrous.

reality

a

€

however small, that makes so

seems

If it appeared

tr

in

it would not be

possible to

destroy

it without

removing it altogether, for otherwise it would

ceaselessly the

rise up

depths of

like the

itself.

its

life of the

universe

This is the

essence

regenerate itself.

by repeatedly applying

triangles

that portion retains the

have

The first three stages of the Peano Curve. The Peano curue voas mode in. the 1890's,

again from

generation to of fractals.

essence

So what Is

a

a

If

successive

segment. a

portion

of it remains,

of the fractal—which in turn

fractal?

Perhaps

purposely avoided giving a definition

can

mathematicians

to not restrict

or

inhibit

The first four stages of the Koch snowjlake. The Koch snowjlake is generated by starting with an equilateral triangle. Divide each side into thirds, delete the middle third, and construct a point off that length out from the deleted side.

the

creativity of fractal

this very as

new

creations and ideas that

field of mathematics. With this

fractional dimensions, iteration

theory,

are

new

formulating

In

field, Ideas such

turbulence

appllca-


self-similarity have evolved. Applications for tions,

?A A^\

fractals

astronomy to medicine, from

cartography

to economics, and

on

and

on.

Mathematically speaking, fractal is

as

triangle—

Divide it into Jour congruent triangles as shown and remove the middle one. Repeat this process to the smaller trianglesformed ad infinitum. The resultingfractal has infinite perimeter and zero areal

object

an

segment,

a

being

a

that is

altered

—such

point,

rule

can

by reapplying

be described

The previous

words.

a

constantly

rule ad infinitum.

a

a

form which

a

begins with

four stages of the Slerplnksi triangle. Begin with an equilateral triangle.

The first

by

to

cinematography

A^J^ /K?^

or

acid

rain to zeolites, from

^K

mathematical formula

from

range

The

by

a

diagrams

illustrate four of the earliest fractals made. One

can

think of

a

fractal

as an ever

fractal, you must really view it

developing. Today

we are

in

growing motion.

curve.

It

is

fortunate to have computers

generating fractals before

our

Benoit Mandelbrot, in the

same

eyes. It

spirit

was

of the

equally

To view

a

constantly capable of

fortunate that

early mathematicians,

almost

expanded the ideas and applications of fractals singlehandedly from 1951-75. In fact, he coined the word

fractal

How astonished the adventurous mathematicians * of the

studied and

century, who first dared to look at these ideas most monstrous2 and psychopathic, would be to see the

19th

considered wondrous

geometry of fractals

When

view

we

seeing of its

it for

an

one

growth.

links fractals

illustration

in motion.

or

moment in time

In

essence

photograph

—

it is frozen at

it is this idea of

dramatically

of

a a

growth

fractal,

we are

particular stage or

change

that

to nature. For what is there in nature

49

50


that Is not

Fractals

changing? Even a rock is changing on a molecular level. be designed to simulate almost any shape you can

can

imagine. Fractals

series of rules and own

fractal. Pick a

necessarily confined to one rule, but a stipulations can be the rule. Try creating your

are

not

simple object and design a rule to apply to It.

The firstjive stages

of a computer generated geometric fractal

1 Mathematicians Georg Cantor, Helge von Koch, Karl Welerstrass, Dubois Reymond, Gulseppe Peano, Waclaw Slerplnskl, Felix Haussdorff, A.S. Besicovitch (Haussdorff and Besicovitch worked on fractional dimensions), Gaston Julia Pierre Fatou (Julia and Fatou worked on Iteration theory), Lewis Richardson (worked on turbulence and self-similarity) spanning the years from 1860's to early 20th century—explored Ideas dealing with the "monsters". ,

—

2These

"monsters" were neither accepted or considered worth exploring conservative mathematicians of the time. It was felt that fractals contradicted accepted mathematics because some were continuous functions that were not dlfferentlable, some had finite areas and infinite

by

perimeters, and

some

could

completely

fill space.

MAC ICAL MATHEMATICAL WORLDS

THE PARABLE OF THE FRACTAL

'Wake-up Fractal!

You must

get

to

work," the voice prodded the

sleeping Fractal. "Not again and

so

early," pleaded Fractal "IJust got my dimensions

in order."

"Wake-up Fractal! Come downfrom that cloud sleeping Fractal.

'You're needed at the to be

you made," the voice prodded the

Geological Survey— another coastline needs

described," the voice continued.

"When will I get a break?' questioned Fractal 'YouVe had it easy for centuries voice

replied.

—

now

it's time to

get to work," the

51

52

TH E MAC \C O F MATH EMATICS

'Work, u>ork, work. Why don't they call on Square, Circle, Polygon, or

any other Euclideanfigure?

"First you

Whyme7' asked Fractal

and that they called you a understanding you, you want to finally they're

complained at being ignored,

monster. Now that

retire. Just be thankful you

"Popular is one thing,

but

are so

popular," the voice rebutted.

they won't let me rest.

Its

never

been the

since that Mandelbrot christened me and gave me my

same

replied Fractal.

"Mathematicians

were

debut,"

tediously struggling with

Tm sure my fractional dimensions threw them offfor a whUe.

me.

Those poor souls from the 19th century had

no

computers to help

them. Most mathematicians would not accept me, for I did

follow

their mathematical rules. But

stubborn. Now here I am, areas—

some

being designed and used

computers certainly

were a

notfit or

mathematicians in

so

were

many

boon. One moment the screen

displays afragment or beginning part of a fractal and the next moment the

screen

is

being filled with its generations—

ever

growing. They are now using me in almost everything— describe roots,

vegetables,

must say its is very

exciting to stretch my limits.

coastlines because it still

enclosed a region whose Fm

Icon

trees, popcorn, clouds, scenery... I I love to do

baffles many people to learn lean area

Is finite while my perimeter Is

serving for modeling many of the world's phenomena.

infinite.

For

example—population with Peano curves, fractal curvesfor creating scene

in movies, fractalsfor describing

economics,

ecology,

involved that since

they

astronomy, meteorology,

et cetera, et cetera, et cetera. Tm so

busy and

things are beginning to get a bit chaotic, especially

mixed me Into the chaos

theory," Fractal said sounding

very tired.

The voice started you

some

again. "Stop complaining! Chaos theory offers

variety. Without it you'd Just be continually repeating

the same rule and generating the

same

old shape

over

and over, at


The

beginning stages of a fractal cloud.

least when

different

some

can

"I suppose

slightly varied, something totally

evolve."

you're right," Fractal sighed.

"Of course I'm same

initial input is

right.

Just think how boring it would be to be the

shape forever, like a poor square or a circle," the voice

asserted.

"Well, at least there are no surprisesfor a square or circle." Fractal countered. "That's

precisely it.

Life is full of surprises; that's

why they are

calling on you so often. You are more like life." The voice seemed complementing Fractal. 'You

mean

Fm human?' Fractal asked.

"I wouldn't go that far. And besides all life isn't human. Let's say

you're just different, and you're non-Euclidean!" And with comment the voice drifted off.

that

53

54


FINDING THE AREA OF A SNOWFLAKE CURVE This beautiful

geometric fractal was created

Koch. To generate

Koch snowflake

a

curve,

equilateral triangle. Divide each side middle third, and construct deleted side.

a

in 1904

by Helge von

begin with

an

into thirds. Delete the

point of that length out from the

Repeat the process for each resulting point ad

infinitum. A

Two

fascinating properties, which

contradictory, •the the

area

are—

of the snowflake that

original triangle

curve

an

snowflake

informal curve

I. Assume the

is

area

Now concentrate

8/5 of the area of

curve

is infinite,

of equilateral AABC is k.

a, as on

congruent equilateral

shown. Thus k=9a.

determining the limit of the area

of one of the 6 initial points of the snowflake

triangles.

from it have

triangle

area

g

8/5 of its generating triangle.

We know the area of the

of the nine

j

proof that the area of the

II. Divide AABC into nine

triangles of area,

is

generates it; »the

perimeter of the snowflake Here is

seem

large point

curve.

is a, since its

one

*.,"

The next set of points

(a) (1 /9) each, Just like

had been divided into 9

generated the original

congruent triangles

it also is. In

fact, each successive point is broken down into nine congruent

triangles with two triangles springing from it. STEP III shows the summation of the various STEP IV: Now,

points plus

triangle,

get expression IV.

STEP IV is

of this

point.

by adding up the areas created by each of the 6 hexagon in the interior of the original generating

the

we

areas

changed to STEP V.

The

resulting series

in the

MAC ICAL MATHEMATICAL WORLDS

III.

,9*9

^•9»9, Notice there

Notice there are

8

of this

are

IV.

a +

2

+2

-

9) 2 H +

-

+

—Z-

+

brackets is

a

93

geometric

so we can

STEP VI.

9.9.9.9 n-2

-r-

+

6a

.

9l9

94

series with ratio

calculate its limit.

4/9 and 2/9

as

=

of the

6a

its initial

we

get

72a/5

Now we need to express the area

+

(2/9)/(l-(4/9))=2/5.

Substituting the 2/5 for the limit of the series,

[l+2/5]6a+6a

k, the

9.9.9

!•¥

5—+

92

of this

42 + 21

14 + 2

2*4Z

2*4

9

term,

19*9

32

stage -points.

stage points.

area

of the snowflake

original generating triangle.

a=k/9. Substituting this for

a

in

72a/S,

(8/5)k.

we

curve

in terms of

Since k=9a,

get get (72/S)(k/9) we

=


56

MONSTER CURVES

The stages of the Sierpinskt triangle. Suppose the area of the initial generating is 1

equilateral triangle

square unit. The sum of the areas of the black and white indicated

triangles are through the first

five generations. Suppose the black triangle represents removal of area. Notice how the value for the white triangles is

continually decreasing, meaning the white

area

is

approaching zero. Thus the area for the Sierpinskt triangle approaches 0, UJhlle its perimeter

approaches infinity. 64

64

XS6*

*56

Until Benolt Mandelbrot coined the term "fractal" In the late referred to

1970's, these

curves were

conservative

mathematicians

pathological.

They

neither

as

monsters.

19th

century

considered these monster

accepted

or

curves

considered them worth

exploring because they contradicted accepted mathematical ideas. For

example,

any

gaps)

some were

that

were

not

infinite perimeters, and

Sierpinski triangle perimeter and out how the

Earned

a

J

continuous functions

dlfferentlable, some

(also called

finite

area.

some

could

(functions without

had finite

areas

fill space.

completely Sierpinskt gasket)

has

an

and The

Infinite

The illustration above tries to point

Sierpinskt triangle's area is zero.

after mathematician Waclaw

Sierpinski (1882-1920).

MAGICALMATHEMATICAL WORLDS

MANDELBROT SET CONTROVERSY In the 17th

century

number of prominent

a

mathematicians (Galileo, Pascal, Torricelli, Descartes, Fermat, Wren, Wallis, Johann Bernoulli, Leibniz, intent

cycloid. Even at this

as

period

there

were

many discoveries

of time, there

arguments about

were

the properties of the

discovering

on

Huygens,

Newton)

were

also many

who had discovered what

first, accusations of plagiarism, and minimization of one another's work. As

the

cycloid

result,

apple of of geometry. 20th century mathematicians

discord and the Helen to have

now seem

a

has been labeled the

a new

Helen of geometry— the Mandelbrot set.

Who first discovered the Mandelbrot

question among present day

set1?

This is

mathematicians.

a

very heated

The contenders

are:

—

Benoit Mandelbrot is

work

on

fractals

the Mandelbrot set

as a

pioneer because of his initial

Mandelbrots work

showing

variants

of

December 26, 1980 in Annals of the of Sciences. His work on the actual Mandelbrot set

was

New York was

often described

in the 1970s.

published

Academy published in 1982.

Hubbard of Cornell University and Adrien Douady of the University of Paris named the set Mandelbrot in the 1980s while working on proofs of various aspects of the set In 1979, Hubbard says

—John H

he met with Mandelbrot, and showed Mandelbrot how to program a to plot iterative functions. Hubbard admits that Mandelbrot

computer later —

developed

Robert

Brooks

a

superior methodfor generating the images of the and

J.

Peter

Matelski

claim

they

discovered and described the set prior to Mandelbrot work was not published until 1981.

set

independently although their

—Pierre Fatou described Julia sets' unusual Gaston Julia's work

on

Julia sets

properties around 1906, and predates Fatou's. (Julia sets acted as

springboards for Mandelbrot sets.) Perhaps all.

Who gets the credit?

iThe illustration above is the most familiar fractal form from the Mandelbrot set. The Mandelbrot set is a treasure trove of fractals, which contains an infinite number of fractals. The set is generated by an iterative equation, e.g. z^+c, where z and c are complex numbers and c produces values than are confined to a certain boundary.

57

58


MATHEMATICAL WORLDS IN

There is

astonishing imagination even in

LITERATURE

the science of mathematics.

Is the tesseract the

only

an

figment of a mathematical imagination?

Is the

"real" dimension the 3rd dimension? We learn in Euclidean that

geometry

since it has

point only shows location, and dimension. Yet

of these invisible

composed such

a

zero

a

in

asymptotic lines of

geometry;

numbers.

transfinite

complex these

number

can

concepts

their

in are

our

world.

Many writers,

point thick.

our

of

pseudosphere

the

Madeleine

the

circle. One wonders

there

is no

mathematical

if

doubt of their

these

systems,

world.

artists and mathematicians have

as

a

hyperbolic

imaginary numbers,

ingeniously

these concepts to describe worlds where these ideas Such writers

What is

functions, infinities of

even a

Although

respective

only models in

one

exponential

Consider

plane, fractals, and

exist in

existence

only

world? Consider the

our

seen

segment

length, yet does What about a plane?

exist in the realm of our lives?

figure

it cannot be

line

A line is infinite in

points.

Infinite in two dimensions and

plane

we can see a

come

used

to life.

Dante, Italo Calvino, Jorge Luis Borges, and

L'Engle have

drawn

on

mathematics to enhance their

creations.

In the

19th

model of

a

century,

hyperbolic

Here to all

things

mathematician Henri Poincare created world contained in the interior of

and inhabitants,

never

was

everything would shrink

it moved away from the center of the circle, while

approached

a

circle.

their circular world

infinite. Unbeknownst to these creatures, as

a

the center. This meant the circle's

to be reached, and hence their world

growing

boundary

as

it

was

appeared infinite

to


Circle Limit IV (Heaven Si Hell) by M.C. Escher depicts a world reminscent of Henri. Poincare's hyperbolic wodd. © 1994 M.C. Escher/Cordon Art-Baam-HoEand. AR rights reserved.

them. In 1958, artist M.C. Escher created

a

series of woodcuts,

entitled Circle Limit I, II, III, IV which convey Poincare had described. Escher described

this

infinite

For her

a

a

world

feeling as

"the beauty

of

world-in-an-enclosed plane." 1

novel, A Wrinkle in Time, Madeleine L'Engle

ract and

of what

multiple

dimensions

as means

uses

the tesse-

of allowing her characters

"...for the 5th dimension you'd through square the fourth and add that to the otherfour dimensions and you to travel

can

outer space,

travel through space without having to go the long way

59

60


around...In other words

a

straight line

is not the shortest distance

between two points." Italo Cavino describes short

believe such we were

been? what

a

world that exists in

story AH At One Point.

"packed

did

we

have for time,

exists.

even

space to

pack

we

Looking back to the find Euclidean The cone's

a

us

of

one

"Naturally, we

have

Or time either:

packed in there like sardines? I say

like sardines," using

which was where

hell.

actually

In his

makes

Qfwfg said,—where else could

coincided with every point

we

single point

ingenious creativity

dimensional world

all there, —Old

a

Nobody knew then that there could be space.

use

wasn't

a zero

His

literary image:

into.

each

of

in

reality

there

Every point of each of the others in

a

us

single point,

all were."

Middle

Ages

and Dante's The Divine

geometric objects

shape

was

were

used to hold

Comedy,

the bases for Dante's

people

in

stages of hell.

Within it, Dante had nine circular cross-sections that acted

as

platforms which grouped people by sins committed.

From Dante's The Divine Comedy. The plan of concentric spheres, which shows the Etarth in the sphere (bearing the epicycle) of the Moon, and these are also enclosed in the sphere (bearing the epicycle) of Mercury.


In the 1900's

of Sand.

infinity was

featured in

Jorge

Luis

Here the main character acquires

"The number None is the

of pages

first

numbered in this

in this book is none

page,

arbitrary

Borges'

The Book

"marvelous" book.

no more or

less than infinite.

I don't know

the last.

way.

a

why they're

Perhaps to suggest the terms of an

infinite series admit any number."This book adversely changes his life and his outlook to

dispose

infinite

on

book

might

until he realizes he must find

things,

of it— "J thought

a

way

offire, but Ifeared that the burning of an

likewise prove

infinite

and

with smoke." What would your solution be?

suffocate You

the planet

might

want to

read the book to find how the hero resolved his dilemma. Science fiction writers have utilized

mathematical Ideas to

example,

Next Generation, the

being pulled by an "invisible" force Only when the ship's schematic monitor

toward

a

starship

black hole.

changes perspective

does the

in

an

episode of

help

create their worlds. For

Star Trek —The

is

crew

realize the unknown force is

a

2-dimensional world of minute life forms. Mathematics Is full of Ideas that make one's and wonder— Are

Mathematicians reside

—

they real?

are

perhaps

imagination churn

To mathematicians

they

are

real.

familiar with the worlds in which these Ideas

not within

our

realm, but real in their

nonetheless!

M.C.Escher, Harry N. Abrams.Inc, New York, 1983.

own

61

MATHEMATICS &ART ART,THE4TH DIMENSIONS. NON-PERIODIC TILING

MATHEMATICS & SCULPTURE MATHEMATICAL DESIGNS & ART MATHEMATICS & THE ART OF M.C.ESCHER

PROJECTIVE GEOMETRY & ART MIXING MATHEMATICS & ART OF ABRECHT PURER COMPUTER ART

64


The most beautiful It is the

mysterious.

thing

source

we can

experience is the

of all true art and science. —Albert Einstein

Linking mathematics and art may seem alien to many people. mathematical

worlds

of

computers have and

simplify,

provided perfect their work.

dimensions,

algebra,

geometries,

tools for artists to

But

explore, enhance,

Over the centuries, artists and

their works have been Influenced

mathematics. The used the

golden

Albrecht Dfirer

by the knowledge and use of ancient Greek sculptor, Phidias, Is said to have

mean In

the proportions of many of his works.

employed concepts from projective geometry

achieve perspective, and

geometric

fj^JNÂ WH*il»'rHHfi «{»$*«$*

constructions

played

a

to

vital

role In his

fc3h

typography of Roman letters.

Since

religious doctrine

¦&

prohibited the use

A sketch

from one of Leonardo da VincCs notebooks Illustrating lines converging to a vanishing point

Moslem art, Moslem artists had to avenue

for their artistic

wealth of tessellation

inquiry

can

rely

on

as

an

to create

a

Leonardo da Vinci felt "...no human

be called science unless it pursues its

mathematical exposition and demonstration." sculptures and

objects In

mathematics

expression, thus leading them

designs.

of animate

path through

Leonardo's

paintings illustrate his study of the golden rectangle,

proportions, and projective geometry, while his architectural

designs

show his work in geometric structures and

symmetry.

illustrating

The

topics

in

this

section

are

a

knowledge

few

of

examples

the connection between mathematics and art.

MATH EMATICS & ART

65

ART,THE4TH Mathematics takes

us

DIMENSIONS, NON-PERIODIC TILING

into the

region of absolute necessity, to which not only the actual world, but every possible world,

must conform. —Albert Einstein

On

a

the artist Is restricted to

canvas,

communicate the

two-dimensions to

feeling of other dimensions.

Icon artists of the

Byzantine period depicted three-dimensional religious only two-dimensions, giving the subject matter appearance.

a

scenes

in

mystical

During the Renaissance, artists using the concepts of

projective geometry transformed their flat canvas into the threedimensional world they wanted to convey. Today, mathematics

plays

an

active role In

mathematical

hypercube1,

the

adapted

dimensional

Salvador unfolded

Artists

higher dimensions. by artists to take

example, has been used

Into the fourth-dimension.

Bragdon

artist's Ideas.

an

Ideas to escape into

for

and tools for the

providing Inspiration

creation and communication of

hypercube

early

hypercube

in his

designs Dali3 delved

In the

use

The

step

a

1900's architect Claude

along

with

other

four-

work2. Intrigued by the hypercube,

Into mathematics

which is the focal

for his model of

point

in his

painting

an

The

Crucifixion4 (1954). Today, with

there

are a

number of artists pursuing art In connection

mathematical

Ideas

in

—

particular,

mathematics

non-periodic tiling, multi-dimensions and computer renditions. fact,

computer

renditions

of

the

hypercube,

created

of In

by

mathematician Thomas Brancroft and computer scientist Charles Strauss of Brown

University, produce visualization of the

hypercube moving in and out of

hypercube

a

in the 3-D world are

3-D space. Various

thereby captured

on

images of the the

computer

66


>¦*'

The unfolded hypercube was the Inspiration for Salvador Dall's The Crucifixion (1954). Metropolitan Museum of Art, Gift of the Chester Dale Collection, 1955. (55.5)

MATH EMATICS & ART

monitor. Introduced to this

part of mathematics,

bin has created 3-D

representations of the hypercube with

the

canvas

acting

which

have aided Robbln in

One moment

intertwined in

to the

pentagonal

non-periodic tiling unusual

an

Tony Rob-

creating fascinating structures,

views a series of

one

interlaced

position

combination of

artist

plane intersecting the hypercube. Nontiles, quasicrystal geometry and fivefold

change dramatically according

viewer. next

as a

Penrose

periodic tilings,

symmetry5,

canvas

perspective of the

triangles

stars

while In the The

appear.

of both 2 and 3-D forms

type of symmetry

create

almost

an

contradictory image.

* Also known 2

as

the tesseract— a 4-dimensional representation of a cube.

same time Bragdon used magic lines graphic designs of books and textiles.

At the

and

3 Dali contacted the mathematics

in architectural ornaments

department

at Brown

University for

further information. *

Jesus

Christ

5Non-periodic tiling designs which have n-Jold symmetry

nailed

is

fourth-dimensional

:

to

a

cross

represented by

the

unfolded

hypercube. tessellating

is no

If

with tiles

or

shapes

which

create

pattern.

a

pattern is preserved when rotated 360'/n, it is Therefore, a pattern has fivefold symmetry pattern.

said to have n-fold symmetry. if a rotation of 72' retains the

Quasicrystals are a newly discovered state of solid matter. Solid matter thought to exist only in two states, amorphorous or crystalline. In amorphorous the atoms (or molecules) are arranged randomly, while in crystalline the arrangement is the periodic repetition of a unit cell building block. The discovery of quasicrystals revealed a new state in which the arrangement of units is non-periodic and with an unusual symmetry, e.g. fivefold, not present in amorphorous or crystalline matter. was

67

68


MATHEMATICS & SCULPTURE Dimensions,

gravity,

geometric

symmetry, and

objects,

are

sets

complementary

all mathematical ideas

which when

role in

into

come

sculptor

a

Space plays a

play

creates.

prominent sculptor's works.

Some works

a

simply

occupy space in the we

of

center

space,

and other

same

way

living things

do. In these works the center of

gravity1

within

These

the are

the

to

For

Greek The Discobolus (circa 450 B.C.)

bronze, captures

a

a

example the

by artist,

the ancient

Myron,

or

Bufano's

St.

Horseback

all

Beniamino

by Myron,

in

we are

or

Michelangelo's David, Discobolus

are

ground

with which

comfortable

in

that

occupy space

manner

accustomed.

point

a

sculpture.

objects

anchored and

is

cast

moment in motion.

Francis

on

have their center of gravity within the

play with These

mass

use

as

uchi, the

Some modern art

sculptures

space and its three dimensions in unconventional ways. space

center of gravity mass,

of their sculpture.

as an can

integral part of the work. Consequently the

be

a

point of space rather than

such works

illustrated

by

Eclpse by

Charles

Perry,

as

Red Cube

a

by

point of the Isamu

Nog-

and the Vaillancourt Fountain

MATH EMATICS & ART

by

69

Louis Vaillancourt.

Other

depend

sculptures their

on

interaction with space. Here

the space around the artwork

(the

complementary

set of

points

of the mass) is

as,

or

equally, important the sculpture.

as

Consider

Zinc Zinc Plain

by

Carl Andre. This sculpture

is

room

works

other

The

in

staged

a

devoid of any

objects.

or

is created

plane

small

36

forming

by

squares

a

square

which lies flat

on

the San Francisco's controversial Vaillancourt Fountain

_-

„

tloor.

ine room repre-

has

as

its center of gravity

a

point of space.

sents space, the set of

all

points, and he describes his work

works

seem

to

defy gravity.

as

"a cut of

These include such

space".2

Some

sculptures

as

the

mobiles of Alexander Calder with their exquisite balance and

symmetry

and

mysteriously

on

Isamu's

Noguchi's

its vertex. There

the Earth itself as

an

integral part

are even

mysterious geometric

grass theorems

possible.

sculptures

balancing which

Crista, Carl Andre's Secant, and

by

physical nature

mathematical

Cube

use

of the art and its statement, e.g.

The Running Fence

Often the

Red

appearing

in

the

England.

of an artist's conceptional work requires

understanding and knowledge to make the work mathematically analyzed most of his

Leonardo da Vinci

creations before

mathematically

undertaking

a

work.

If M.C. Escher had not

dissected the ideas of tessellation and

optical

70


Illusions, his works would not have evolved with the which he

was

able to undertake them

once

ease

with

he understood the

mathematics of these ideas.

Today there are many examples of sculptors looking at mathematical ideas to

expand their art. Tony Robbin

uses

the

study of

quasicrystal geometry, 4th dimensional geometry, and computer science to develop and

expand his art. In his giant sculpture

Easter egg

Ronald Dale Resch had to

use

Intuition, ingenuity,

mathematics,and the computer as

This sketch by Leonardo shows his analysis of the horse's anatomy.

well

as

his hands to

complete his creation. And artist-

mathematician Helaman R.P.

Ferguson uses

traditional sculpting, the

computer and mathematical equations to create such works

as

Wide Sphere and Klein Bottle with Cross-cap & Vector. is not

Consequently it

surprising

to find

mathematical models

doubling as artistic models. Author In front of Continuum by Charles Perry. National Air & Space Museum,

Washington

D.C.

Among these we

find the cube, the

polycube,

the

sphere,

the

MATH EMATICS & ART

torus,

the

trefoil

hemisphere,

knot,

knots,

M6bius

squares,

Mathematical

prisms

the

objects

strip, polyhedrons, the

circles,

triangles,

from Euclidean

pyramids,

geometry and

topology have

played important roles in the

sculptures

of such

artists

Isamu

as

Noguchi,

David

Smith, Henry Moore, Sol LeWitt.

Regardless of the

sculpture, mathematics is inherent in it. It may

have been

conceived and created without

a

mathematical

thought, nevertheless An Alexander Calder mobile. East

National

Building Gallery of Art, Washington, D.C.

of the

mathematics

.

exists in that

work, just as it exists in natural creations.

^The

center

of gravity Is the point

on which an object can be balanced. the center of gravity or centrold of a triangle can be determined by drawing that triangle's medians. The point where the three medians Intersect happens to be the center of gravity.

For

example,

2Art & Physics,

Leonard Shlaln, William Morrow & Co. NY, 1981.

71

72


PUTTING MATH EMATICS INTO STONE Trefoil knots

spheres

—

—

—

torus

vectors

flow— movement

—

—

these

are some

of

the mathematical ideas inherent In the

sculptures of Helaman Ferguson. We have often heard

of artists

using

mathematical ideas to enhance their

work. Mathematician- artist

Helaman R.P. Ferguson

conveys the

beauty of mathematics in his Eine Kleine Rock Musik HI Photography by Ed Bernik. From Helaman Ferguson: Mathematics in Stone and Bronze by Claire Ferguson. Meridian Creative Group, Copyright © 1994.

phenomenal sculptures. art form and

believe it is feasible to communicate mathematics channels to the To create his

traditional

a

an

science...!

along

aesthetic

general audience."1

exquisite forms, Helaman utilizes methods from

sculpting,

the

His works bear such

Cross-cap

As he states

"Mathematics is both

computer, and mathematical equations. names as

Wild

Sphere;

Klein Bottle With

And Vector Field, Torus, Umbilical Torus With Vector

Field., WhaledreamU (horned sphere). 1 Ivars Peterson.

Equations

in Stone, Science News Vol. 138

September 8,

1990.

MATH EMATICS & ART

LAYING AN ECC MATHEMATICALLY When Ronald Dale Resch mission to

design for

sculpture

a

com

gigantic Easier egg he

Vegreville, Alberta,

discovered he would have

soon

develop

the

accepted

to

the mathematics for the task

virtually from scratch. Over the years Resch has refined

the art of

manipulating

into 3-D forms.

problems

2-D

objects

His work and the

he has solved

point

to

mathematics, yet he has had little formal mathematical training.

Working with sheets of such

as

various materials

aluminum,

or

paper

transforms them into works of art he has

he

by folding

He solves

developed. geometric problems using intuition, ingenuity, mathematics, the computer and his hands. techniques

His initial instincts about the

design

of the egg

were

that he could

ellipsoids for the ends and a bulging cylinder for the center. He quickly realized this would not work. Discovering that available mathematics for the egg was limited1, he realized he make two

would have to go it alone. His Easter egg

524

required 2,208 identical equilateral triangular tiles and

three-pointed

width varied

by varying

resulting design for the magnificent

stars

according

the

angle

tiles, which

were

to their location

on

equilateral

and whose

the egg. He found that

of placement of the tiles

ever so

slightly (from

less than 1" to 7'), the flat tiles gave the Impression of curving and the contour of the egg resulted.

long and 1

The final structure is 25.7 feet

18.3 feet wide, and weighs 5000 pounds.

Algebraic equations for egg-shaped curves were developed by French mathematician Rene Descartes (1596-1650). In 1842, as a youth, Scottish mathematician James Clerk Maxwell (1831-1879) devised a method for constructing an egg using a pencil, string and tacks.

73

74


MATHEMATICS DESIGN* & ART

The

following figures

can

and have

been used to create artistic and

graphic designs.

Their bases

mathematical, but that does take away from their beauty and simple

are

not

elegance.

MATHEMATICAL STARS The

illustrations

show

how

to

by using regular polygons

generate

stars

polygons.

For odd sided

simply join every other vertex of the polygon, the

and you will arrive back at

starting point.

For

even

pointed

stars, note that the star is made up of two rotated

polygons.

These will

have half the number of sides the

generating polygon.

as

MATH EMATICS & ART

75

MATHEMATICAL EMBROIDERY Math enthusiasts have been

mathematical It Is

curves over

to discover

always fascinating formed

curve

from

"embroidering"

the centuries

series

a

a

of

straight line segments. The "stitch es" (line segments) end up being
for the

One can

mathematically

"embroider" a

a

circle,

hyperbola,

an a

curve

they

ellipse,

cardioid,

a a

form.

parabola, limacon,

a

deltoid, etc. In the process of embroi

dering some

these

of their

Here is how

curves,

special

an

one

discovers

characteristics.

astroid

can

be made.

One way to make an astroid is to think of it as a sliding ladder. Circles, the radius of the ladder, are used to mark the base of the ladder's new location.

Concentric circles appear when the

diagonals of a 24-sided polygon are stacked.

Variations

ofparabolas.

76


This nest

of squares is the first

<0

step

STRAI6HT LINE SPIRALS

in

making spiral

the

below.

Spirals

are

mathematical

to convey motion.

of

curves

objects

They encompass

from two-dimensional

equiangular (logarithmic) and Archimedean spirals for examples

—

dimensional

that

such

a

seem

family

spirals

—

to three-

the conchoid

spirals Spirals touch many areas and many living objects. A few as

and helix spirals. in our lives

examples

are

—

antlers of certain

seedheads of flowers, many

growth of certain vines, art, graphic designs. Straight line spirals

are

goats,

shells, the

DNA, architecture,

formed

by making a

In which each

of

nest

sea

regular polygons polygon is formed by connecting the midpoints of its sides. Looking at a nest of polygons, the spiral is hard to see until it is shaded. Here are some exciting designs these straight line spirals produce.

reduced

There

are

many variations of these

spirals.

Some of these variations have been

connected" to pursuit four

spiral

is formed from

square root numbers using the Pythagorean theorem.

one

such

as

the

spiders problem.

Another famous This

problems,

straight

line

produced by constructing

numbers

using

the

spiral

is the

square roots

lythagorean theorem.

spiders start crawlingfrom the four comers of a square. Each spider is crawling toward the one on its rtoht moving toward the center at a constant rate Thus, the spiders are always located at four comers of a square. The curves formed by the spiders' path are equiangular spirals. The size of the initial square and speed of spiders determines how long it will take for the spiders to meet. Four

.

MATHEMATICS S. ART

LOOKING AT THE DUALS OF PLATONIC SOLIDS The Platonic solids— convex

convex

polygons—are only

polyhedra with

faces that

five In number. It Is

are

regular

interesting

to

discover the connection between the Platonic and their duals.

'\t^*

\^W^|^ off

a

tetrahedron & its dual

an

a

cube & Its dual

dodecahedron & Its dual

octahedron & its dual

a

To make the dual of a

polyhedron,

one

an

first finds the center of each

face and then joins the centers of faces that share The illustrations reveal that the dual for

tetrahedron, the dual for while an

an

a

icosahedron, and

an

a

a

an

edge. a

octahedron,

cube. The dodecahedron's dual is

icosahedron's is

the five Platonic solids' duals is

a common

tetrahedron is

cube (hexahedron) is

octahedron's dual is

icosahedron & Its dual

a

dodecahedron. None of

anything but a Platonic solid!

77

78

THE MACK OF MATHEMATICS

DEFLATED POLYHEDRA

—

SCHLEGEL DIAGRAMS

Each of these

represents

one

diagrams of the five

Platonic solids. called

tetrahedron

They are

Schlegel diagrams, after the German

mathematician Viktor

Schlegel, who invented particular diagrams

these

in 1883.

of 3-D hexahedron

(cube)

They remind one

objects which have

been deflated. In

however, the

essence,

polyhedron

is inscribed in

sphere projected through a point that is along the line a

and

of the north

sphere.

Its

pole of the

projection can

change according to the polyhedron's position in the sphere. These Schlegel diagrams have the special property that one face of the dodecahedron

polyhedron contains

all the

images of all the

other vertices. In addition, the

diagrams show all the polyhedron's vertices, edges and faces,

though

even

their size and

shape are distorted. icosahedron

MATH EMATICS & ART

79

MATHEMATICS The laws of

&THE ARTOF

mathematics

M.C. ESCHER

are

not

merely

human inventions

creations. They simply "are"; they quite independently of the human

or

exist

intellect. The most that any

man

with

a

keen intellect can do is to find out that

they

are

there and to take

cognizance

of

them. —M.C. Escher

And M.C. Escher

certainly

of mathematics.

It is

his works

Most of

did take

exciting

cognizance

to view many of

through mathematical eyes.

us

are

familiar with Escher's

creations of the tessellations of

a

magical

plane.

His

works far surpassed the traditional tessellations of

a

He

plane.

objects

gives

he tessellates,

famous

works

Water, Day and

as

motion and life to the as

illustrated in such

Night,

transforming the plane,

Besides

tessellated

objects undergo

themselves. In addition,

a

the

transformation

one sees

his

mastery

of

concepts of translations, rotations, and

reflections in

periodic tiling.

Escher also used

objects and

of topology. The Mobius his woodcuts Mobius Horseman.

and

Fish and Scales, and

Encounter.

the

Sky

Metamorphosis,

He

ideas from the field

strip plays

Strip

I and II

exquisitely produces

a

key role

in

and

trefoil knots in his

A section from Escher's work Metamorphosis in. © 1994 M.C. Escher/Cordon Art-Baarn-Holland. All rights reserved.

80


work Knots. And his Snakes Is theory. Print

even

though

Escher

probably

piece of art to Introduce

perfect Gallery and Balcony are a

wonderful

a

did not Intend It,

topic on knot

examples

of topolog-

M.C. Esher's Print Gallery Illustrates a topological distortion. © 1994 M.C. Escher/Cordon Art-Baam-Holland. AU. rights reserved.

leal distortion.

These

rubber sheets which

lithographs almost seem to be printed magically distorted via topology.

Manipulation and mixing dimensions

other mathematical

are

themes found in many of Escher's works. In

2-dimensional lizards dlmensional forms. Mirror and

Cycle.

on

are

eerily

come

Reptiles,

to life in realistic

Similar transformations take

His

use

of concepts from

Escher has

crawling

place

in

3-

Magic

projective geometry—

MATHEMATICS S, ART

illustrates Eschews use of tessellations as well as his mastery of moving between the worids of the 2nd and 3rd dimensions. © 1994 M.C. Escher/Cordon Art-Baam-Holland. All rights reserved.

Reptiles

perspective, traditional vanishing points, his

own

curved

vanishing points— are responsible for the feeling of depth and dimension in St. Peter's Rome, Tower ofBabel and

High and Low.

Circles, ellipses, spirals, polyhedra and other solids the

are some

of

geometric objects one finds in Escher's work. For example. Spheres creates a 3-dimensional Illusion of spheres,

Three

although find

a

it is

variety

composed entirely of circles and ellipses. of solids,

Including

tetrahedron is the focus in Tetrahedral Planetoid. And in

stellated dodecahedron is present.

In Stars we

the Platonic solids, while

a

Gravity the

81

82


Escher makes the concepts of needed to define it, his work

infinity

spirals

lead one's eyes

the

feeling

of infinite sequence

projected.

use

endless

in the field of

of such

help trick

to life.

meaning.

approaching

its

space

In

Whirpools

Square Limit

boundary

eyes and confuse

our

water is

Escher is as

We

get both the

phenomenal.

Waterfall

endless

endlessly climbing

descending

a

figures

a

staircase and

staircase all in are

also his

a

one

the other

for

Relativity.

and Convex, Escher is

a

us

loop.

means

illusions in Belvedere and

has

while in

loop, going upstream believing Ascending and Descending there are two sets of people— an

His

the Penrose tribar,

minds. His in

Concave

is

tessellating in Cubic Space Division.

optical illusion,

Impossible

are

ideal model for Henri Poin-

an

impossible geometric figures

our

No words

In

journey.

yet infinite non-Euclidean geometry.

concepts of infinity and

Lastly,

on an

And Circle Limit would be

care's finite

its

portrays

the

come

creating In

master of

oscillation illusion. Our eyes and mind

are

taken

back and forth in the interior and exterior of an

incredible structure and cast of

characters. For

dome is

a

example,

one

moment the vaulted

roof and then within seconds it is

a

ceiling. It is

satisfying

to

explore Escher's work

on

many different levels. This introduction is but

a

glimpse

of the wealth of mathematical

ideas found in the work of M.C. Escher.

M.C. ISscher mastered the art

of tessellations. He also envisioned a tessellation stamp. The six faces of the cube have a design in six different positions. So as the stamp is rolled on a piece of paper, the design is tessellated. The design reproduced here was not done with a stamp. Instead, a computer was used to rotate and flip the design so that it could be tessellated.

MATH EMATICS & ART

Starting by

modifying one of its lengths,

translate

that modification to the other length's

position. Now

modify one of

its widths, and

translate the modification to the other width also.

Now take the

resulting shape, and tessellate

plane.

XI

x'

x' X

a

83

TE$$ELLATING/ PLANE WITH/ MODIFIEE RECTANOLI

84


TESSELLATIONS OF OLD

From nature's

tessellation of the to

honeycomb,

Roman mosaics, to

the tiles of ancient

Greece, to the marvelous

by

designs

Moslem artists at

The Alhambra, to M.C. Escher's

phenomenal tessellations, to the

simplicity of the tiles— tessellations cross

centuries and

cultures.

Here

we

a

simple yet elegant pattern

appears at the baths

CaracaUa In Rome.

Penrose

have

tessellation

a

design

using pentagons and rhombi created

by artist Albrecht Durer. It dates back to the 15th

century.

uartety of designs. This

Roman, mosaic floors have

of

MATH EMATICS S. ART

85

PROJECTIVI ,*, « *, geometry Projective

mathematics

iis

that

a

figures

as

they

with

therefore

are

ART

with

deals

relations

properties and spatial

GEOMETRY*

flw field off

projected—

of

and

problems of perspective. Many

century often worked these artists, such Battlsta Albert!

In as

a

number of different

Giotto dl Bondone

(1404-1472),

artists of the 15th

disciplines.

(12669-1337), Leon

Plero della Francesca

Leonardo da Vinci and Albrecht Durer,

Some of

(14109-1492),

studied

engineering,

mathematics, science and architecture, while pursuing their art. As

a

result Renalnssance artists connected mathematical

concepts, such

as

These artists

dimensional how

projective geometry, to the creation of their works. developed techniques for painting realistic three

scenes on

things changed

two dimensional canvases.

They analyzed

when viewed from different distances and

positions and developed ideas of perspective. They reasoned that

they perceived a scene outside through a window, it would be possible to project what they saw onto a window as a collection of points of vision if they kept their eye at a single focal point. The

if

would

window their

canvas.

The

would

naturally

affected

the

by

act

as

scene

be

position of the

artist's eye and the position of the artists

canvas.

actually

Some

invented

mechanical devices to aid in

drawing as

Illustrated

perspective,

by

the two

diagrams. Artists'

works

on

per-

An Albrecht Durer perspective instrument.

86


spective influenced the

development of

projective geometry. Just

as

topology

studies the

properties of objects that remain

unchanged after they have

undergone a

transformation,

projective geometry Leonardo da Vinci's

plane figures For

spectograph.

that do not

example, when

ellipse.

Similarly

a a

studies properties of

change

when

they undergo projections.

circle is viewed from the side, it appears square would become

a

different

quadrilateral. Having been influenced by perspective

as an

shaped

in the art of

u

This use

study of The Adoration of the Magic by Leonardo illustrates of linear perspective and a vanishing point

his

MATHEMATICS*. ART

the Renaissance and 17th

of

Girard

projective geometry.

are

He wrote

a

(Desargues' Theorem) study of

point of projection

still essential in the

His work influenced later


mathematicians in this field, Pascal

help artists,

Desargues initiated the

number of theorems

which

to

century architect/ engineer/

mathematician

study

desiring

(1623-1662)

(1788-1867).

To

especially Blaise

I

create

realistic

three

parallell converging

and Jean Victor Poncelet

lines

vanishing point

dimensional paintings Renaissance artists used

87

concepts

from

projective

geometry,

namely— point of

projection, parallel converging lines, and vanishing points.

This work by Jesuit monk Andrea del Pozzo (circa 1685), was painted on the hemi-cylindrical ceiling of St. Ignasio Church in Rome. His mural is an excellent example of perspective which illustrates the concepts of projective geometry of a single vanishing point. In fact, it may be considered to De too exact. A mark is locatea on the floor of the church, indicating where the viewer should stand to get the full effect the artist intended. At this location one actually feels the ceiling is infinite and the reality of St Ignatius Carried into Paradise. Viewing the painting from any other point creates a distorted and uncomfortable effect.

88

TH E MAC l< O F MATH EMATICS

MIXING ART& MATHEMATICS OF

...geometry is the foundation of all painting.

ALBRECHT PURER

—Albrecht Durer Albrecht Durer (1471 -1528) 18

was an

artist of many talents. Of his

children, Durer's father hoped It would be Albrecht who would

follow him In his work

apprenticed

to Michael

artist. After

a

goldsmith. Wohlgemut, who as a

three year

throughout Europe,

apprentice,

As

a

also

result, Durer

happened

was

to be

an

Durer traveled and worked

where he learned painting, engraving,

printing, and making woodcuts, all of which influenced his life's work. He felt the

geometry,

study of mathematics enhanced art,

perspective and ideas of

extensive work

the

on

mathematics

geometric needs. artists

The

on

was

original

drawing

following

on

He also did In

addition,

Geometry.

and

were

this line he in


the mathematics he used. Some of his work in

constructions

Along

particular

proportion of the human body.

A woodcut from Durer's Treatise

he wrote books

In

exact1,

while other work and

approximate and served his artistic

developed

mechanical devices to aid

perspective.

sections illustrate how the work he

mathematical overtone, which helped him and other creations.

pursued

immeasurably

had

a

in his art

MATH EMATICS & ART

j
perspective drawing.

^Durer generated

an epicycloid by following a fixed point of a circle as it the circumference of another circle, but lacking an algebraic foundation he did not analyze the curve. Likewise, he produced spirals from the projection of helical space curves, but did not pursue them

rolled

along

mathematically.

89

90


CUBOCTAHEDRON-TRUNCUM In addition to the many innovative

he

techniques

developed,

Durer is

credited with

describing solids on a plane in unassembled form. Here is Durer's

description of one

such

solid, which, when cut and assembled, forms a

cuboctahedron truncum.

DURER'S CONIC SECTION This illustration

by

Albrecht Durer shows his

interpretation section. His

of a conic

ellipse

is

somewhat egg-shaped, which

implies he either

believed the inclination of the

plane cutting the cone

made the at the

ellipse narrower

top or he made

a

slight error in calculation.

Regardless, interesting

it is

to

see

his

study of an ellipse. From Underqweysurrg der Messuna mit dem Zircklelund Richtscheyt (a treatise on

geometric constructions) by

Albrecht Durer.

MATH EMATICS & ART

THE ORIGINAL PIXEL

Durer can be credited with

launching

In one of his books he illustrates the

Roman letters.

Durer devised his

own

the

concept of topography.

geometric

construction of

mathematical methods for

by constructing each letter from a combination of triangles, trapezoids and/or parallelograms. He

Gothic letters, squares,

systematized the structure of Gothic letters

by using

blocks— somewhat similar to how letters computer's monitor

using the pixel1

as

the

are

these little

formed

building

on a

building block.

These geometric constructions were devised during the Renaissance, although the ideas were not published at that titne.

^Pixel is the abbreviation for picture element It is the visual representation of a bit on the computer's monitor. A bit stands for binary digit which is the smallest unit of information that a computer can hold. Tne bit's value corresponds to the numbers 0 or 1, representing electrical flow as "off' or "on" which translates to a white or black square on the screen.

91

92


COMPUTER ART Over the centuries the tools of artist have

ranged

obscura and the

exploring

a

mathematics

new

form

an

stick to

a

as

the

Today

artists

are

medium of art that is

linked

perspective or

a

to such devices

paint brush, camera

from

window.

Until

the

art

to

recently computer computer. produced by mathematicians, scientists, engineers, and just —

about everyone but artists. deal of art that resembled

Initially, curve

we were

stitching,

flooded with

line work and

a

was

great

optical

Illusions.

Computer generated art Reproduced from Computer Graphics by Melvin Courtesy of Dover Publications,

L. Prueitt

Today, computers play a major role In commercial art. A computer skilled artist with advanced software can transform graphic art ideas for advertisements to

multiple

styles. Introducing various colors, scaling sizes or

flipping objects, duplicating

minutes. In the

graphic

by changing type-

variations

various

of objects,

parts

of

an

rotating

object

In

past, all of these changes would have taken the

artist hours if not

days.

MATH EMATICS & ART

This oil painting

effect was

it via the

converting

achieved

by scanning

a

Engineers, architects and other designers have embrace the a mouse

a

photograph and

computer.

computer

not hesitated to

in their creations. With just

few clicks of

building could easily be modified, a plane could possible perspectives, cross-sections could

rotated to show all

added, parts added was

a

slow and

or

removed

painstaking.

effortlessly.

In the

be be

past this work

93

94


With today's advanced computers and software, Leonardo da Vinci might have rendered this sketch on a computer via an electronic sketchpad.

Artists

over

the centuries have used different mediums to create

their works— watercolor, oils, artists who feel the

acrylic, chalk,

computer

Is

an

etc.

artificial

freedom of spontaneous expression.

There

means

are some

that lacks

They prefer the direct

contact of their hands with the medium of their choice, rather

than

working by

with

stylus

computer With

means

on a

and electronic tablet and

as a

keyboard

and

screen or

Others view the

screen.

challenge.

improved software

screen.

of electricity

and hardware, colors

Rigid computer

lines

can

be

can

easily

be mixed

on

the

softened with any

MATH EMATICS S. ART

shaped a

curve.

Changes

can

be made from

watercolor, and brush shapes be

areas can

erased

or

definitely either

or

create.

Or

painting effect

changed

Parts

easily.

more

or

her creative work. The result

video, paper

perhaps

it to any size.

capturing the we

can

be

elsewhere in the work. The artist is

on

of

to

at whim. Minute

entered

option of enlarging

designed capable

pasted

oil

an

of his

charge

printed

be

magnified and reworked

cut out and in

can

can

be

film. Film leaves the

or

Perhaps

printer will

a

be

texture of the work the artists

should consider this

a new

form of texture

in itself.

Esteemed artists have

displayed

well known international art.

art created with

galleries, simply labeled

Such works have yet to be

What would Leonardo da Vinci think of the art? Given his fondness for

not have snubbed the

inquiry

can

use

innovation1,

of the

use

of

computer.

in

computer

computers

we can assume

in

he would

He said "...no human

be called science unless it pursues its

that he extended this to his art- for example, the

golden rectangle

computer

art.

mathematical exposition and demonstration."

of the

a

but it is still labeled

path through

His works reflect

predominant

in many of his works and the

use

concepts of

projective geometry in his masterpiece, The Last Supper. One form of art should not be considered better than

different. Artists should be free to choose any

another, just

means or

medium.

1His notes and various innovations were used by artists to enhance and facilitate their artwork. Leonardo's mathematical inclinations led him to invent various types of special compasses capable of producing parabolas, ellipses and proportional figures. He is also credited with the invention of the perspectograph, used by artists to help draw objects in perspective.

95

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+

1 °°

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=l-l=i 1V n

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=

0

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1000-1 999

sin(ji/2)

=

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-(£*)

WITH X#Y

THE MACIC OF

NUMBERS DISCOVERING THE MAGIC OF NUMBERS

QUATERNIONS & THE 6AMES NUMBERS PLAY CANTOR & INFINITE CARDINAL NUMBERS NUMBER FANTASIES

WHAT ABOUT PERFECT SQUARES? THE PARABLE OF

%

THE PRIME FASCINATION

CANTOR AND THE UNCOUNTABLE REALS

EUCLID'S PROOF OF UNENDING PRIMES NUMBER MAGIC

PLAYING THE NUMBERS


98

The science of pure mathematics, in its modern

developments, may claim to be the most original creation of the human spirit—A.N. Whitehead

Some people

think that the way certain numbers

results that appear seem to possess

a

operate and the

magical quality. Perhaps the

illusion of hocus pocus Is intensified because there

types

of numbers

numbers which

—

are so

many

to be invented

seem

by

mathematicians' whim. In school was

we

first learned about the whole numbers. Our first task

to learn to count, and then

began

to feel that

perhaps

learned to add, subtract, of the blue

V8 1

3/8

e

—

we

operate with them. Just when

had mastered the numbers

multiply

and divide, then

fractions, decimals, integers,...

0 -7 4

7i

googol

—

we

having

seemingly

out

appeared. They

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THE MAGIC OF NUMBERS

sometimes surfaced when

solutions. Then

suddenly

7 divided

example,

by

2

didn't

particular problems new

was

numbers

neither 3

seem

to have

came

to the

rescue.

4,

so we

learned the

nor

For

number 3 and 1/2 did the trick. What Is

all

the

so

different

individual

reproducing with

amazing Is how many types

some

classifications

of numbers

Numbers

characteristics.

of numbers there are,

almost

along

with

seem

to

plus their

be

list of types of numbers called naturals) numbers (also descriptions— counting themselves.

Here is

a

partial

({1,2,3,...}), whole numbers (include the counting plus the number 0),

({...-3,-2,-1,0,1,2,3...}),

integers

the

rational

(includes

fractional, mixed numbers, improper fractions, proper fractions, decimals

and

decimals),

repeating

Irrational

(Includes

never-ending

non-repeating decimals, radicals without

roots,

§ ),

7t,

e,

real,

imaginary,

the

perfect complex, transcendental,

transfinite, perfect This section seeks to introduce you to

properties and

some

of their

may not be familiar, and to some

youVe

known for

a

some

numbers, their

quirky characteristics with which you

long

acquaint you time.

more

intimately

with

99

100


QUATERNIONS & THECAMES

It

NUMBERS PLAY dimensions. It would

describing

seem

that

hard for the how

having various

number Is just

a

How

two, three, four, etc., dimensional? mathematicians to For

numbers be

comprehend as

a

can

number

—

something

such numbers

are

as

Well, leave it to the

example, mathematicians consider any real number

to themselves that says their amount. Also

line—

a one

numbers

are

composed

of

one,

give another twist to the characteristics of numbers.

imaginary number as one-dimensional because they have a

dimensional

object.

a

real number and

On the other

graphed,

the location for 5+2i on the complex number plane is given by the ordered pair (5,2)

an

imaginary number.

dimensional

complex

on

a

need

part

they

plane (a

figure) called plane. Now

ask how

are

more

are

For

number

the you

why

does

or

complex numbers were

no

they

useful and

shapes

so

to

light many years

have been around. The

exception. Today, for example, they are

used to describe patterns of flows in currents and the

are

invented, their

various uses come

after

one

than these. As with

many mathematical Ideas that

discovered

2-

2-dimensional

numbers used, and -r-r-r-

any

one

hand, the complex

it is located

might

or

they can be graphed on

called two-dimensional numbers because

example when 5+21 is

were

to

layperson

described

particular quantity.

a

is

hydrodynamics, electric

of airfoils. Since these

effective In

complex numbers

solving various types of problems,

it

seemed that the next natural step was to look for three-dimensional numbers.

Although the

to the

discovery

These

were

complex

search for these was not successful, it lead

of four-dimensional numbers, the

invented

numbers

by

they

quaternions.

Sir William Hamilton In 1843. Like the

were

met with

skepticism

and

suspicion.


They are composed of four values x,

y,

values

z-axes

space)

describe direction,

as

do

that

(numbers

three-dimensional

and

a

scalar (

y, and

x,

z

locate

not commutative when it

In which are

they

quantity values). The set of quaternion

comes

makes

multiplied

In

point

a

to

a

complex numbers,

or

multiplication

—

the order

difference In the result. How

these four-dimensional numbers

One of their on

are

—

that does not

a

numbers, unlike the set of real, Imaginary are

characteristics

descriptive

or

everyday

being put to work today? communicating graphic information

is

uses

computers by giving descriptions of rotations

dimensions.

The

following story

like all other numbers, have

a

beginning

themselves

as a

was a

useful.

But,

cliques

already

were

shame that numbers could not think

of

happy family.

Over the centuries when just the

the odd and

quaternions,

personality of their own.

As usual at the numbers convention, the

to form. It

in three

tries to demonstrate that

even

counting numbers

around,

were

numbers would argue about which

was more

they united forces when the integers entered the

picture with their negative numbers. Now sides

were

convention

—

already beginning

to form

over

The counting numbers had always been very whole numbers greater than their natural

garb

—

or

about the

if it fell

since it was neither

negative

Surely the rationals

fractions,

equal

They

into its set. The

quaternion, while

as

big

usual,

to 1.

They

zero, nor

had to

integers

of the

quaternion?

were

study

were

being neutral,

open to

dressed in

only

one

unit

the newcomer,

both hot and cold never

took sides,

positive.

would consider it were

issue

elitist—only

all in increasing order with

between consecutive numbers.

and decide

the

who would accept the newcomer, the

more

more

interested

seriously. in

But the

displaying

their

101

10}


numerators and denominators than to

even

Over the years, the decimals had

used to the fractions.

gotten longer disturbed by the fractions'

were no

knew how much easier

especially

in to say

1/7 jumped

denominators to add

work for

.007

even

"Although

called the fractions passe. But one

subtract us, and

or

has to

we

numbers

of

some

the

included

common

some

we

In fact,

some

fancy foot

prefer

your decimal representations

way out there.

are

can't hold your decimal

find

require

multiplication and division, and while

lowest terms—

They

The decimals

antics.

they were to operate with thanfractions,

calculators.

on

talk to the decimals.

of

to be in

rational

calculator's memories

representation." Thus the rationals, which the

counting numbers,

and

integers, fractions,

decimals, continued to fight among themselves.

Having

seen

the

that went on

bickering

with the "rational"

numbers, quaternion was

understandably

fearful ofagroup of radicals

by the snack

bar that included

f/2, /3,/T5,and/6.

j

Quaternion had heard how irrational could

get.

they

But to its

surprise, they

hear that you have

4-dimensional,"

y^2

a

number

said.

to carry around

instead.

Perhaps

so

you too

of

were

Interested in

parts and

are

in decimal form. All those

I

prefer wearing

can

find

a

"So I

talking.

what

they

"Well, don't feel bad about it, Tm.

ending non-repeating when drag

007 digits

call

never are a

my square root suit

more

abbreviated way

of

expressing yourself." Quaternion wanting

was

a

bit

encouraged and felt

to get quaternion's

hopes up,

j/J

more

relaxed. Not

added, "You'll need to be


cleared with the

complex numbers

set. It

keeps

track

of all of us

—

integers, rationals, irrationals, reals, and imaginary."

the counting,

"But I heard the

complex set has

split personality, and fluctuates

a

between the real and imaginary numbers," said quaternion. the

Suddenly that

complex number 3-5i walked over saying

"You've

got

right, but the

complex number plane gives everyone of us our own

single

point on which to reside. When worse comes

to

worst, lean

always take

refuge there. I know it's my very own

point,

no one

else has that

location,

there I

so

can

be alone, and regroup, relax and meditate.

We each have our own spot 'You

seem

to have

we can

call home."

multiple personality,

a

and scalar," 3-5i said to quaternion. has "I

no

"

Fm

what with your vectors sure

the

complex plane

place for you."

hope I can find

my

own

point and home," quaternion said. With

a

sad note to its voice it continued, "One doesn't know which way to

turn, "It

or

should I say which set to seek out."

certainly

and

saw n.

is

difficult," said a rather deep "It

very hard for

me

Fm irrational as

^~2

was

numbers.

Although

not let

into the reals

me

voice.

to be

Quaternion turned

accepted by

the real

and the others, they would

right away, saying there was

no

way to find

103

104


I heard the complex set has a

split personality.

m*w» qua my exact location

|/5,... what

who

was

use

on

the real number line, unlike the

I to do? I had to do some fast

realized how Important all the circles

j/2, j/j,

the Pythagorean theorem toflnd their locations. So

an

talking, and the realsftnaUy am, especially since

irrational number I

depended on mefor their circumference and area,

and lam

transcendental

toboot." "Well, k, you speak as

The reals finally realized how

important an

irrational

number I

am.

though you're the only transcendental

number," said the number e, who to be

was a

known

braggart.

"I also happen to be


I'm different

Qj

from all of you.

I have

rate*** A quaternion is expressed in the following wau:

more

q=a+xi+y]+zk, where

dimensions.

"a" is

a

scalarfa constant] and

"xl+y]+zk "is a y, and z being

vector, with x, real numbers.

transcendental, the base for the natural logarithms, and found

extensively in nature besides being used in calculus." Quaternion

was

the teasing and said.

"Perhaps

beginning

Perhaps belong

form

set

q=a+xi+yj+zk.

from the

headache and

was

tired

of all

I don't

belong here," quaternion complex numbers shouted. "But

they taunted quaternion.

to my

quaternions. The is

a

bickering. "Perhaps

different from all of you. I

get

you don't," all the

where do you belong?' 'Tm

to

own

I have

more

depth,

set. Yes, thats it! I

more

am a

of 4 dimensional numbers, "

dimensions.

member of the

since my

general

Havtng said this, quaternion began to rise

convention floor, and

gone to another dimension.

suddenly disappeared

as

if it had

105

106


CANTOR & THE INFINITE CARDINAL

We

m

NUMBERS

are

familiar with the

natural numbers, the whole numbers, the

Integers and how and what

they are used to

count. But what about

:

{roffonaf nuimtoers}

Jf:;.;1';!!!*:

Georg Cantor's transfinite numbers? How

.::tk.- ;'--the1ollpwng':s^—^l-lvv%;/:%>

{{wiritihasphere}

many are there and what

type of things do they

'-::si-

describe? Transfinite numbers describe the number of objects In

a

set. For

example, A={apple, orange,

pear} would use

the

cardinal number 3 to

describe how many

objects In set A.

Between

1874-1895 Cantor

studied and

theory. many sets of numbers that a

new

numbers.

are

needed to describe the

were

of Infinite sets. Hence, he created the transfinite

HO, Hi, H2, H3, HA

Hebrew letter

aleph. HO

counting numbers. Any

correspondence

with the

number of elements. The

The

(aleph-null)

symbol H

stands for the

refers to the number of

set that

can be put into a one-to-one counting numbers is said to have HO

diagram above points

out

some

sets with

cardinality of aleph-null HO, aleph-one Hi, aleph-two H2, but has yet come up with examples of sets that use H3 or Hi

no one or

Since there

Infinite, it was apparent to him that

set of cardinal numbers

cardinatlity

the

are

developed set

any higher ones.

THE MACK OF NUMBERS

107

NUMBER The properties and

fantasie:

workings of

numbers at times

almost

seem

magical. Choose any three

digit

number whose ones

and hundreds

digits For

285.

Reverse the order

Subtract the smaller from the

larger number. 582-285=297

The result will have

a

the tens

digit,

the

and

ones

hundreds

always

582

Reverse its digits.

9

as

(582-285) =297

Find their difference.

and

digits

total 9.

Now reverse the

digits

digits.

582

digits.

always

285

Pkk o 3-dlgrt number with different ones and hundreds

different.

are

example,

of the

(

on

the result.

792

729

Reverse the difference's

digits.

Add these two numbers 792+297=1089

792+297

The result will

always

be 1089.

=1089

Find their sum. The result will

1089,

always be

108


WHAT ABOUT PERFECT SQUARES? a

number which

as a

whole number

A perfect square is can

be written

times itself. For

example,

36 is 6x6

and 49 is 7x7.

Did you know? •

The sum of the first n odd numbers is a perfect

square,

n2

—

that is

1 +3+5+7... +(2n-l)

=n2

e.g. l+3+5+7+9=25=52 •

Every perfect square ends 0, 1, 4, 5, 6, or 9.

in a

•

Every perfect square

is

divisible by 3 or is when it is decreased by 1. •


is

divisible by 4 or is when it is decreased by 1. •


is

divisible by 5 or is when it is increased or decreased by 1.


109

THE PARABLE Many, many years

big

party

numbers in all its

ago, there

being

glory.

all the other

the

given for

of the time. One

OFTl

was a

was

there

Two showed up with

even

numbers in tow. And

as

prime numbers

many

could be found had

come.

1 /4 and

of the radicals had showed up like

2/3.

A few

who had just arrived

There

sides

off the

Viand

the sides when

n

of right triangle

n

j/2

and

j/7

with 3. But

with 3.

rolled in everyone asked, "Who invited you?". "What do you n.

on

'Tm

a

number." 'Yes you are,

the number line?" "What about

asked n. "Thanks, to Pythagoras and the

f/2?"

like 1 /2,

just arrived off

a

'Who invited me7', asked

mean,

fractions

of a right triangle

who had

but do you know your location

know

were even some

as

use

of a compass,

I

exactly where I belong on the number line," replied j/2.

felt embarrassed and hurt,

but said, 'Tm

a

little

after the

number

3." "But exactly where?,"

they all chimed in.

factor of every number,

Since 1

was a

give

chance to describe

So

n a

n

began

to tell his

1

felt TV's pain and said,

"Lets

itself."

story.

"As you all know the

Babylonians

110


discovered

probably first circles with

doubling

me.

Some ancient scribe had drawn

different sized radii.

The scribe took the diameter

of each circle.

the radius)

wrap each circle's diameter around it. To his

regardless of the

size

around it 3 and

a

spread quickly

all

Everywhere people

over

knew I

was an

were now a

circle

always wrapped

exciting discovery. The news to Greece to China.

Egypt

about

learning

and distance around

People

me.

Because

devising

by using

me

was no

in their calculations.

offence,

number quite like

me

me.

They

satisfied

with

some

mathematicians are,

It suits

me

just fine.

methods

have

been

so

later

thought

my exact number

on

they

that

name.

Tm

But no, you know how stubborn

they wanted

to be

precise. So for the centuries that followed to the

and

they

never

not able to derive

were

transcendental. You would have

people would have given up on finding n.

but

ordinary number, especially since they had

from any of their regular algebraic equations,

also labeled

of my special

ways to find the

new

anxious to find my exact value. No

were

come across a me

they

surprise, hefound that

the circle, its diameter

the world from

were

connection to circles area

of

little bit. This

(by

And just for kicks, decided to

developed

to

more

and

present,

get

a

more

new means

more

accurate

approximation. The famous mathematician Archimedes found

me

to be between 3


In the Bible I appear twice and my value is

10/71 and 3 1/7. 3.

as

Egyptian

estimated

me as

mathematicians used 3.16

keep

drawing

me

they will never get my exact amount, but they

out to

imagine how much of

more

and

decimal places. You can't

computers

are

used, HI be out

to

ofplaces.

*1

They say I am essential to computing

4, 6,

things, such

various as

more

burden it is carrying around all those

a

decimal places. Once calculus and millions

given

And Ptolemy

3.1416 in 150A.D.

Mathematicians know on

for

me.

volumes,

areas,

V*X

circumferences, and

anything that deals with circles,

*ox

^

cylinders,

*

cones, and spheres. I

also play

a

^

role in

probability. And with

Q '

*»» ZO. W* ~~

my millions decimal

approximation,

Two showed up with' all the Other

modem

even

day

computers

will

rely

on me

them for accuracy and

"Say

no

such

a

all,

number

through

more", shouted 1. 1 continued," Fm

know

as n

each have

we

can ever

not the most

location

put

numbers in tow.

them

their paces and test

speed.

renowned number

we

to

sure we

all agree that

should be counted among

our own

point

on

us.

After

the number line. No

have another number's point, nhas its point. It is

important thing about

a

number to know the exact

of its point."

"Agreed",

shouted 3,

the party

a

bit

one

of the mystical numbers.

of mystery, variety

and

"

I think

intrigue" said

"Welcome", chimed the rest of the numbers. "Lets get under way. Lets start

counting", said n.

n

our

lends

j/2. party

112


THE PRIME

One way to

FASCINATION

classify

describe them A

composite.

or

prime number's only

1 and itself. It is not

factors

are

divisible

by any other

the other hand,

1

numbers is to

either prime

as

number.

On

composite

a

number has other factors beside 1 and

itself

12 is not

(e.g.

1,2,3,4,6, and 12). In

its factors

are

addition,

every

described

by

prime —

number

(12's

formed one

by multiplying

3.

In

be

2»2»3)

is

prime factorization. No

other number besides

JUff WVOKOOWflfl

can

unique product of

a

numbers

called its

prime because

the

12

can

be

two 2s and

1700s

Christian

Goldbach wrote to Leonhard Euler that he believed it could be shown W^P<

that every

even

Integer other than

is the sum of two

(e.g. 8=5+3; 28=13+15). This simple the

i*4» «taMM*fa>

statement remains

unsolved

2

prime numbers

problems

mathematics. Twin

primes1,

primes, Sophie

Germain

clear

of

one

of Mersenne

primes

are

among the other primes

¦

winypfRM

fascinations that mathematicians

explore.

*Twin primes are primes

are

two units

apart, such

as

that

only

3 and 5. As of

November 1993, the largest pair found by Harvey Dubner (l,692,923,232xl04020±l) each with

was

4030

Talk by Theoni Pappas. Publishing/Tetra.

Reprinted from Math © 1991 Wide World

digits.

THE MACIC OF NUMBERS

113

CANTOR i Georg

Cantor's

set

transfinite numbers

accomplishments.

countability numbers was

A

THE UNCOUNTABLI

brilliant

REAL NUMBER:

theory were

His

and

proof of the

of the rational

elegant. Equally fascinating was his indirect proof

showing that the real numbers

are

not countable.

countable if its elements

can

numbers. For the rational

numbers1,

be matched with the he devised

listing all possible fractions, and a method the

counting numbers. He tried the

real numbers, but decided that out to prove

they

were

therefore looked for

arrangement

a

An infinite set is

same

type of method with the

they were not countable.

not countable

counting

method for

to match them up with

by assuming they

So he set were,

and

contradiction. First, he assumed he had

that listed all the real numbers. For

numbers between 0 to 1 it

00

a

example,

an

for the

might be a list like this—

1234867

...

9 8 7 8 1 9 73 0 4 8 3 9... 0 2 8 3 7 16 8 4...

0 O 1 I 2

...

114


Then he looked at the numbers

formed

a

changed have

number by each

digit

placing

a

1.

he

one

contradiction he

digit

to another

diagonal by simply adding

Cantor

of this list. He

the

diagonal,

digit (e.g.

1 to each

and

he may

Then

digit).

by

this number had to be

argued that his

list

was

ostensibly

just formed using the diagonal. This

was

looking

for. This number was

was

supposed

the

to be

list, yet it differed from every term of the list at least at the

that

number

was

was a

along

complete! numbers.

This

the list's

number that

included in the

were

diagonal

to Include all real numbers between 0 and 1, but did not

Include the

in the

this

decimal point In front of It,

between 0 and

supposed

the

taking the digits landing on

on

them

changed

along

was

"complete"

same

diagonal.

Thus, this diagonal

between 0 and 1, and

list.

Therefore, the list

was was

Since the subset was not

can

be

not

proof could be used for any subset of the real countable, the real numbers

not countable!

Numbers that

not

expressed

as

fractions

or

repeating decimals.


115

EUCLID'S PROOF OF UNENDING

PRIME5

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, It

the further out

seems

one

numbers, the primes become because

they

proof

rarer

appear less and less

end somewhere. first

ventures

As

early

as

and

In the

rarer.

frequently,

that the prime numbers

are

Suppose

represents the last prime number.

•

Now form

a

might

think

perhaps they

provided

the

infinite. He used indirect

in the

and

One

that

about 300 B.C. Euclid

reasoning, n

field of whole

following manner.

number from the product of all prime numbers up to n. 2x3x5x7x1 lx...xn.

including the last one,

•

Add 1 to this product and call this number k. k=2x3x5x7xl lx...xn+l

•

kis either prime or composite.

If k is prime, then k is a prime number greater than the assumed largest prime, namely n, because k is 1 more than the product of the existing list ofprimes. If k were not prime, then k must have a prime factor. J^s prime factor cannot be one of the primes from the existing list of primes because we know none of these, namely 2,3, 5,7, 11 n, can divide evenly into k —each of these primes would leave a remainder of 1. Therefore k must have a new prime number as its Since there is no end to this process, the primes are infinite.

factor. As

a

168

matter of mathematics trivia— between 1 and 1000 there

primes, between 1000 and 2000 there

and 3000 there

are

are

are

135, between 2000

127, between 3000 and 4000 there

are

120.

116


NUMBER MACK

Am AS Of IW4—41

YEARS ELAPSED SINCE 1994

0RMI>0K1AN?iV0*?—14 wuiHinmrntmuai ntaam an mm in

i

<§«

14

w*)»WWWm«HIIWIWII>w

3988 Take the year you

were

born. To this add the year of an important

event in your life. To this 1994.

Finally,

sum

add to this

add the age you will be at the end of

sum

the number of years ago that the

important event took place. The answer Is always

3988!


117

PLAYING Every math enthusiast or

at

one

another has discovered

by

tricks

or

been

oddities

delighted involving

numbers. Here

for you to

explore and hopefully enjoy.

or

WITH NUMBER!

time

are

two

SUMMING *£ SQUARING

l+Z+3+4+3+2+l=42 •

••••••••••••

•

••••••••••••

•

•••••••••••a

THE 1'g PYRAMID

x*-x ii2=ui

1112=1Z3X1 IIIIX=IX343ZI

XXXXXX=X2345432X XXXXXX*=X2345*5432X IIIIIIIX=IX345«>7«>543XI IIIIIIIIX=IX345«»787<»S43XI ...^rhen does it

stop?

MATHEMATICAL MACK IN NATURE

WHAT HONEYBEES ARE BUZZING ABOUT MATHEMATICS HEXAGONS & NATURE

CHAOS IS FOR THE BIRDS A CLOSER LOOK AT FRACTALS & NATURE

THE MATHEMATICALLY ANNOTATED GARDEN MATHEMATICS RIDES

THE CREST OF THE WAVE

120


There is

no

branch of mathematics, however abstract,

which may not someday be applied to phenomena

of the real world. When

—

Lobachevsky

try to pick out anything by

we

itself we find it hitched to

everything

else in the universe.

—John Muir

Ever look half,

the

petals,

a

leaf and wonder

notice the

or

of the

at

why

growth pattern

of hair

and bark of redwood trees?

on a

exactly

in

of stars in different flowers'

perfect shapes

spiral growth pattern

It could be divided

of

a

certain shells, of

human head,

or

pinecones,

of the branches

Nature abounds with

examples

of

mathematical concepts. In

our

how

things

look for patterns and similarities that

can

be measured and

used to

are

formed,

we

search to

explain and understand

This is

categorized.

why

mathematics is

us are

oblivious to the

explain natural phenomena.

As nature

puts forth

its wonders, most of

massive calculations and mathematical work needed to

something web is

a

very routine to nature.

simple,

structure is

web

are

but

For

elegant natural creation.

mathematically analyzed,

indeed

example, the

catenary

When this beautiful

the ideas that appear in the

curve

the

logarithmic spiral,

and the transcendental number

is the formation of

e.

Another

tortoise shell. To

plates explain pattern formed, the mathematics of triple junction, hexagonal

example the

spider's

surprising —radii, chords, parallel segments,

triangles, congruent corresponding angles, the

explain

Orb

tiling,

and calculus

are

on a

needed. This

chapter presents

nature's wonders and the mathematical ideas

a

they conceal.

few of


Bees.. .by virtue of

WHAT HONEYBEE: ARE BUZZIN< ABOIT MATHEMATIC

certain

a

geometrical forethought... know that the hexagon is greater than the square and the triangle, and will hold more honey for the same Pappus of Alexandria expenditure of material. —

Bees

have

not

studied

logarithmic spirals. Yet,

as

tessellation In

so

architecture of Insects and animals

mathematically.

Nature

uses

theory,

many things

can

often be

nor

orb

spiders

In nature, the

analyzed

the most efficient and effective forms

—

forms

rrr that require the least this that links

the art of

121

expenditure

of energy and materials. Is it

nature and mathematics?

Nature has mastered

solving maximum/minimum problems,

linear

algebra

122


and

problems,

solutions to

finding optimum

problems Involving

constraints.

By focusing

attention

our

mathematical Ideas The square, the

can

the

on

and the

triangle

self-tessellatlng regular polygons. the smallest when

for

perimeter

a

prisms of square

made up of cells which 30

support

feels

honeycomb 1.5

more

prisms at

of

ounces

in

exactly

sections

A

pounds

the

only three hexagon has

This

area.

same

that bees,

means

In the hive,

use

space than If

less

wax

tessellatlng

comb's walls

1/80 of an inch thick, yet

about

This

weight.

honeycomb

explains why

of about 14.5"x8.8"

of honey, while It

can a

can

only requires about hexagonal

to construct. The bees form the

wax

three rhombic sections, and the walls of the cell meet 120'

angles.

forming

a

The bees work

comb with

downward, and the bees

use

no

fascinating

"tool"

orientation is Influenced

a

by

simultaneously on different

visible

seams.

It is built

of their bodies

parts

instruments. In fact, their heads act

Another

are

triangular bases. The

own

heavy.

so

than five

or

are

their

times

of

wealth

a

Of the three, the

constructing hexagonal prism cells

space with

hold

hexagon

given

and do less work to enclose the

are

honeybee,

be observed.

vertically

measuring

plummets.

as

honeybee

as

has is

a

"compass".

the Earth's magnetic field.

detect minute fluctuations in the Earth's

Bees'

They

magnetic field which

can

are

only discernible to sensitive magnetometers. This explains why the bees of a

swarm

occupying

a new

location

build the hive in different parts of the

directing direction

them. All the bees orient their as

simultaneously begin

new area new

comb in the

same

their old hive.

In the illustration

closely packed,

on

the next page, the cells

the bees

having capped

are

shown to be

the ends with half rhombic

dodecahedrons. In addition, the bees build the cell walls with

slope

to

without any bee

of 13\ which

prevents honey

a

from running out before the


tops

with

capped

are

wax

domes.

Communication

another

is

yet

of interest.

area

Worker bees returning to the hive from

scouting

a

communicate

expedition

direction of the food

the

found

they

source

by

codes in the

transmitting

form of a "dance".

They can

communicate the direction

of the food

and the distance.

relation to the

sun

surprising

gives the direction of the food, while the

dance

of the

duration

indicates

to learn that the

collected

a

the

distance.

a

straight

line.

of the term "beeline". After

full load of nectar

It

honeybees "know" that

distance between two points is

possible origin

The orientation of the dance in

a

is

equally

the shortest

Perhaps

this is the

worker bee has

by randomly moving

from flower to

flower it knows to take the most direct route back to the hive. The

honeybee gets

It is

intriguing

mathematical

honeybee

is no

its mathematical

to

point

analyze

exception.

its

genetic codes.

different aspects of nature from

of view.

and work, tessellation of a

training via

This

Here

plane

we

glimpse

find

into the

life of

a a

optimization of materials

and space,

hexagons, hexagonal

prisms, rhombic dodecahedron, geometric theorem, magnetic

fields, codes, and amazing engineering techniques.

123

124


HEXAGONS The profound study of nature is the most fertile source of mathematical discoveries.

—Joseph Fourier Connections between mathematics and nature abound.

and

shapes from

natural

Objects

various fields of mathematics appear in many

phenomena.

Nature also makes its hexagons in rock. Devil's PostpUe National Monument, Mammouth Lakes, California, Photograph by Margaret Goodrich

What is

so

special

about the

hexagon

that nature

it in many of its forms? The formation and

repeatedly uses growth of natural

objects is influenced by surrounding space and materials. The regular hexagon is one of three regular1 polygons that tessellates

a


Postpile National Monument, Mammouth Lakes, California. Photograph by Margaret Goodrich

Devil's

(hexagon, square and equilateral triangle) the hexagon largest area with the minimum amount of material. Another special feature of the regular hexagon Is that It has 6 lines of symmetry, thereby allowing Its shape to rotate plane.

Of the three

encompasses the

numerous

times without

the most volume the

hexagon.

altering.

The

using the least surface

When

spheres

are

placed

sphere, which encloses

area

side

Is also connected to

side Into

by

illustrated at the bottom of page 126), each surrounded

tangent

to six other

between these

circumscribing as

the

sphere

three

angles

come

angles

of a

at the

spheres is

soap bubble gives

bubbles

spheres. a a

together

When line

a

box (as

sphere

segments tangent points, the shape

are

Is

drawn

regular hexagon. Thinking of the spheres

simplified explanation

of why

at what Is known as the

of 120', the

same

regular hexagon.

The

degree

measure

triple junction

a

cluster of

triple junction

—

of the interior

appears in many

ar-

125

126


such

eas,

as

kernel formation

on

cob of corn, the Interior flesh of the

banana, earth.

Discovering

occurrences

of

of

cracking

hexagons

a

Suppose a perimeter of

a

12 units is

dried

available to form these three regular

new

in nature Is

no

objects. The hexagon's

less exciting than the first time

they a

observed

were

on

area

the back of

the

shape of

or

Today

scientists

fascinated with the

hexagons

are

crystal. equally a

sighting

In outer space.

astronomers have

out to be

6t/3*>10.4. The triangle's area

tortoise, in the beehive's

honeycomb,

would

come

would be

of

4y/3**6.9.

The

square's

area

is 9.

Since 1987

been

focusing much

attention

on

Magellanic Cloud, supernova

not

1987A

was

the

observed.

first time gas

the

Large

where It Is

bubbles

have been

seen

explosions,

but It Is the first time

following

stellar

the bubbles appear clustered in

honeycomb shape. The six lines

of symmetry of regular hexagon.

honeycomb, which

composed

measures

years. Wang suggests that a

cluster,

composed of similarly sized stars which have been

evolving at about same

the

Wang University of Manchester

England

approximately

rate for

several thou-

a

of in

discovered the

30x90

light

of about 20 bubbles whose diameters

light

the

a

Lifan

years, and Is are

about 10


sand years, creates winds of such

magnitude they shape

the bubbles into the hexagonal

configuration.

Lastly,

look

a

nature's

at

snowflake illustrates

symmetry and fractal

hexagonal

geometry. The snowflake the

possesses

hexagon.

growth

of

shape In

the

addition, the

of a snowflake is

Koch snowflake

simulated

by the

curve.

This fractal is gener-

These

are

ated as

by an equilateral triangle snowflake indicated in the drawings f^fj1 into thirds. °

at the

Consequently,

the

first four stages of with

starting

curve,

an

^T9^: divi*:, e<^\ side, Delete the middle third, and

make a point of deleted part

right.

between

the

that

length from the

the connection

equilateral triangles, the regular hexagon and the fractal

snowflake links both Euclidean and non-Euclidean geometries.

Objects in stimulating

nature

have

provided

mathematical

and do

discoveries.

achieving an equilibrium and an exquisite balance The

key

to

understanding

the

workings

to

discovery leads to

1A

way of

is

expressed

a means

the next. What will the

polygon size.

Is

Only

regular

with

it well

language of

by

which

understand, explain, and copy natural phenomena.

outer space lead to?

same

provide

a

in its creations.

he said— "the universe is written in ...the

mathematics". Mathematical tools

try

has

of nature

mathematics and the sciences. Galileo Galilei when

models for

provide

Nature

we

One

discovery of hexagons

in

time will tell.

If Its sides

are

the

same

length

and

angles

the

127

128


CHAOS IS FOR

Ever

TH E BIRDS

become

watching '

¦

as

from

one

by

flight

the air,

direction

to

of seagulls. Aptos, California.

another in

Zoologist question.

perfect harmony?

Frank H. After

Why don't

Heppner wanted the

painstakingly filming

the birds

were

not

guided by

dynamic equilibrium

a

leader.

with the flocks

changing birds at brief intervals. and computers,

he

movement of birds. With

was

the birds answer

to

and

collide? such

a

the movie

studying Heppner concluded

frames he had taken of flocks of birds,

theory

flock of birds in

they swooped through

going

A jlock

a

mesmerized

flew in

that

state of

They leading edge continually

Until he

was

unable to

a

introduced to chaos

explain

concepts from chaos

the

flocking

theory, Heppner


Perhaps has

a

now

school

qfjish

devised

a

also

moves

in

a

state

of dynamic equilibrium.

computer program that simulates the possible

movement of a flock of birds. He established 4 on

avian

simple rules1 based

behavior, and used triangles for birds.

By varying the

intensity of each rule, the flock of triangles would fly computer's

monitor in familiar fashions.

necessarily illustrates possible explanation of

his program does

give

move.

a

It seems the chaos

theory

across

the

Heppner does not claim

bird flock formation, but it how and

is at it

why

flocks of birds

again!

The rules he established were: (1) Birds are attracted to a focal point or (2) Birds are attracted to each other. (3) Birds want to maintain a fixed velocity. (4) Flight paths are altered by random occurrences such as gusts of wind. roost.

129

130


A CLOSER LOOK AT

FRACTALS & Fractals have

NATURE

to

the

as

Though amples

of objects from Euclidean

a

Euclidean

descriptive

know

We

describing such objects pressed

to find

Lavajlowfrom

objects

as

baked

goods, bark

geometric fractals or a

geometry

is

geometry

great for one

is hard

to describe popcorn,

the Kilauea volcano, Hawaii

While

fern leaf

so

provide the best

and bee hives, but

in Euclidean

the

produce objects which elude

Euclidean

crystals

ex-

in nature

triangles...),

In these cases fractals

description. tool.

to

nature.

abundant

are

geometry present

seems

of

geometry

there

(hexagons, circles, cubes, tetrahedrons, squares, often randomness of nature

to be referred

come

on a

tree, clouds, ginger root, and coastlines. are

used to describe such

snowflake, random fractals

can

be

objects

as a

computer

generated to describe lava flow and mountain's terrain.

Euclidean


geometry has

its roots in ancient Greece

Euclid's work the Elements),

(circa 300 B.C. with

while fractals have their origins in

the late 1800's. In fact, the term fractal

was

not coined until 1975

by Benoit Mandelbrot. With their introduction, that can describe the ever changing universe.

Fractals

point

is

are

considered

we

have

a

geometry

dimensions. In Euclidean geometry, a line is 1-dimensional, and a plane is

fractional

zero-dimensional,

a

2-dimensionaL What about ajagged line? In thejield offractals, ajagged line's dimension is considered to lie between 1 and 2. Starting with a line segment and subdividing it into 3 sections as is done with a snowflake curve, its a

dimension will lie between 1 and 2.

2-dimensional

pyramid formed.

over

its

If we start with a rectangle, object and subdivide it into 4 sections and construct a middle section, a fractal of dimension between 2 and 3 is

131

132


FRACTALIZINC THE SURFACE ^

Select

cca i»tu

InttARin

a

America

continent. Let's as an

want to

Its coastline.

In

mathematics,

we

measured with any linear units

measure

the distance of

know that distance

we

they

different units. Yet If we

we

in

the coastline of South America

in kilometers and then in meters, our answers will differ if

consider the coastline to be fractal in nature. units become smaller and

be

choose,

distance—onry expressed

same

measure

can

desire—for example, feet,

inches, meters, miles, kilometers. Regardless of which should represent the

South

example. Suppose

'

we

use

smaller,

we are

As

our

able to

we

measuring

measure more

inlets

and

more

and

bays of this

coastline. In

other words,

theoretically we can

get more

detail

as we

continually decrease the size of our measuring unit which in turn

continually

increases the

length of the coastline.

A coastline, in fractal

length

geometry,

is considered to be infinite In

because every little Inlet and unit of sand Is measured and

the number of these units Is

formation of the snowflake

constantly changing,

curve curve

similar to the

(see page 1270.


"Good morning

gardener,

day!" exclaimed the

she greeted the

as

sunrise and her plants. Little did

she know that strange

lurking Deep

things were

in the leaves and rich soil.

in the roots

of the plants were

fractals and networks, andfrom the cosmos,

irises,

marigolds, anddaisies

Fibonacci numbers

were

staring at

her. She proceeded about her daily ritual

of tending

to her

garden. At each

place, something unusual appeared, but she

was

oblivious, captivated only

by the obvious wonders that nature

Removing the dead fronds to expose

fiddle heads,

she did not

recognize the equiangular spirals greeting her and thefractal-like

formation of leaves on theferns. Suddenly, was

the

as

shifted, she lovely fragrance of

the breeze

struck by the

honeysuckle. Looking

saw

FRACTALS

can

appear

as

symmetrically changing/growing objects or as randomly asymmetrically changing object! either case, fractals are changin according to mathematical rule; patterns used to describe and dictate the growth of an initial

object. Think of

a

geometric fractal

as

endless

She first went to clear out herferns.

new

TH MATHEMATICALL ANNOTATEI CARDEl

an

presented.

the

133

how it

was

taking

over, she

over

the fence

and getting into the peas. She

decided it definitely needed some

judicious pruning. that helices

She did not realize

were at

work, and the

left-handed helices of the

generating pattern —the

pattern

continually

fractal

tree

replicates itself but in a smaller version. Thus, when a portion c geometric fractal is magnified it looks exactly like the original version. In contrast, when a poi of a Euclidean object as a circle magnified it begins to appear le curved. A fern is an ideal exam of fractal replication. If you zen on any portion of the fractal fen appears as the original fern lea A fractal fern can be created or computer.

134


NETWORKS

are

mathematical

diagrams which present a simpler picture of a problem or situation. Networks were used by Euler in the Konigsberg bridge problem ( see 777© Spell of Logic, Recreation, & Games section). He reduced the problem to a simple diagram, which he analyzed and solved. Today networks are tools used in topology. FIBONACCI NUMBERS 1,1, 2,3, 5,8,13, 21,... Fibonacci (Leonardo da Pisa) was one of the leading mathematicians of the Middle Ages. Although he made significant contributions to the fields of arithmetic, algebra and geometry, he is popular today for this sequence of numbers, which happened to be the solution to an obscure

problem appearing

bloodroot

tritium

in his book Liber Abaci. In the 19th century, French mathematician Edouard Lucas edited a recreational mathematics work that included the problem. It was at this time that Fibonacci's name was

cosmos

attached to the sequence. In nature the

sequence appears in:

honeysuckle hod wound around some

of the right-handed helices of

the peas. It required

rose

carejul hand

to avoid damaging her new crop

of

peas. Next she moved to weed beneath

the palm tree she

hadplanted to

give her

garden a somewhat exotic accent. Its branches were

moving

in the

breeze,

and she had no

idea that

involute curves were

brushing against her shoulders. She looked

over

at her corn

"Hal" she

smuggly. hadbeen

hesitant to plant corn,

but

was

encouraged by how well the young

corn was

progressing. Unbeknownst to

her, triple

Junctions of wUd

a

thought.

She


corn

kernels

wovldform within the

'Flowers with

of petals

ears.

How well the entire garden

shaping

up and exploding withnew

growth! Admiring leaves

was

on

the new green

a

Fibonacci numb

(trilium, wildrose,

bloodroot, cosmos, columbine, I blossom, iris)

'Arrangement of leaves, twigs stems is known as phyllotaxis

c

Select

CHERRY

the maple tree,

she knew there

on a

was

something inherently pleasing

a

ste

and

cou

the

num

of leave

in their shape

nature's lines

13!

—

(assumi none ha been broken-i until yoi reach oi

ofsymmetry

had done their work well.

directly in line with the

one you selected. The total number of leaves (not counting the first on

you selected) is usually a Fibonacci number in many plan such as in elm, cherry or pear trees.

And nature's phyllotaxis

was

evident to the trained eye in. leaves

on

only

budding

branches and stems

of

plants.

•

The pine

and

right

cone are

Glancing around, shefocused on the carrot patch. She

they needed

were

doing,

thinning

was

proud of how

and noted

to insure

they

uniformgood

sized carrots. She did not want to rely on

nature to tessellate space with

carrots.

cone

numbers: If th

handed

spirals

on a

counted, the two

p

num

very often consecutive Fibonacci numbers. This also h true for sunflower seedheads a are

seedheads of other flowers. Th same is true of pineapples. Loc at the base of a pineapple cour number of left and right spirals composed of hexagonal shape: scales. They should be consec Fibonacci numbers.

136


SPIRALS & HELICES: Spirals are mathematical forms which appear in many facets of nature, such as the curve of a

fiddlehead fern, vines,

shells, tornadoes, hurricanes, pine cones, the Milky Way, whirlpools. There are flat spirals, three dimensional spirals, right and left handed spirals,

equiangular, logarithmic, hyperbolic, Archimedean spirals, and helices are just some the many types of spirals which

of

She had no idea that the garden

abounded with equiangular

mathematics describes. The equiangular spiral appears in such

spirals. They were in the seed-

growth forms of

heads of the daisies and various

nature

as

the

nautilus shell, a sunflower seedhead, the webs of Orb spiders. Some of the properties of the

flowers. Many things that grow form this spiral because of how

equiangular spiral are—angles

retains its

formed from tangents to the radii are congruent (hence

increases.

spiral's

shape

it

while its size

the term equiangular) it increases at a geometric

It

rate, thereby any radius is

decided she would continue

—

cut by the spiral into sections that form a

geometric progression its shape remains the same as

was

getting

warm so

the cultivation when the —

it grows.

sun

shifted. Meanwhile,

she made onefinal assessment—

admiring the

INVOLUTE CURVE: As a rope is wound or unwound around another curve (here a circle), it describe an involute curve. Involute

combination

is the shape found in the beak of an eagle, the dorsal fin of a shark, and

something escaped her.

the tip of leaf.

a

she

hanging palm

offlowers,

vegetables and other plants she had so

thoughtfully

selected. But once

Her garden

spheres,

was

more

full of

cones, polyhedra

and other geometric

MATHEMATICAL MACIC IN NATURE

shapes, and she did not recognize them. As nature puts forth its wonders in

the garden, most people are oblivious to the massive calculations and

mathematical work that have become so routine in nature. Nature knows well

how to work with restrictions of

material and space, and produce the

Many types of symmetry appear in the garden. For example, in the above photograph one can find point symmetry in broccoli

florettes

and line symmetry in their leaves.

13;

TRIPLE JUNCTION: A triple junction is the point where three

segments meet, and the angles the intersection are each to 120" Many natural occurrences result from restrictions caused by boundaries or availability of spac Triple junction is an equilibrium point toward which certain natun occurrences tend. Among other things, it is found in soap bubble clusters, the formation of kernels the cob of corn, and the crackinc earth or stone. SYMMETRY: Symmetry is that perfect balance one sees and senses in the body of a butterfly the shape of a leaf, in the form c the human body, in the perfects of a circle. From a mathematical point of view, an object is considered to possess line symmetry if one can find a line which divides it into two identica parts so that if it were possible ti fold along that line both parts wc match perfectly over one anothe An object has point symmetry 'rf infinitely many such lines exist fi particular point, for example a ci has point symmetry with resped its center

point.

during each day of spring, the

TESSELLATE: To tessellate a plane simply means being able' cover the plane with flat tiles so there are no gaps and no tiles

gardener will enter her domain with a

overlap, such

most harmonious forms. And so,

gleam the

in her eye. She will seek out

new

growth and blossoms each

day brings,

unaware

of the

mathematical beautiesflowering in her yard.

as

with

regular

hexagons, squares, or other objects. Space is tessellated or filled by three-dimensional objec such as cubes, octahedra.

or

truncated

138


MATHEMATICS RIDES THE CREST

f If you

I

OF THE WAVE

re

a

r , surfer, you know that

sometimes It's difficult to know In

advance when the surf will be up. Sometimes the surf appears perfect from the beach, but when you so

you wait for that perfect

hours. Other times the

wave

perfect

get into the water it's for what sometimes

activity

Oceans

waves.

ocean waves. course

are a

complex system. Many

like

wave

theory

and

factors affect and create

Aptos, California. The wind,

an

earthquake,

the gravitational pull of the

all disturb the water. These

down,

seems

after another and

waves come one

you have many to choose from. Needless to say wave

died

ocean.

moon

the wake of a boat, and of and the

Ocean waves travel

on

sun

causing tides

the surface of the

undulating forms have a somewhat random quality to

them when there

are

multiple disturbances or factors Interacting.

MATHEMATICAL MA6IC IN NATURE

Much

study

was

generated

Observations at

ocean waves.

Czechoslovakia

In

formulated the Initial how water

wave

In

particles

the mathematics of

Interesting when

1802,

In

on

and In controlled

sea

scientists arrive at

experiments helped started

In the 1800's

laboratory

conclusions. It Franz

Gertner

In his observations he wrote

theory.

The water In the

In circles.

a wave move

(highest point) of the wave moves in the direction of the wave that in the trough (the lowest point of the wave) moves in the

crest

and

On the surface, each water

opposite direction.

circular orbit before found to have

was

There

are

depth

of

to its

returning a

diameter

original position.

equal

circles

being generated by the water. But the deeper

circle. In fact, it

was

found that at

(the horizontal distance between

a

particle

to the

height

the particles

moves

In

a

This circle

of the

wave.

throughout

the

the

particle the smaller its depth of 1/9 the wave length crests), the

two consecutive

diameter of the circular orbit is about half that of the circular orbit of a surface

Since

particle.

waves

tied

are

sinusoidal and

into these

cycloid shaped

circling particles also

curves

depend

on

circles, it is not surprising that these mathematical their was

equations

are

used in the

discovered that

other

purely mathematical

curve.

Intensity of the wind, the tides must be considered when are

studied

description

ocean waves are

are

waves.

and

But it

strictly

The

depth

of the water, the

only

some

waves.

or

any

of the variables that

Today

ocean waves

mathematics of probability, statistics,

number of small

complexity.

and

predictions are formulated from the data collected.

large

rotating

curves

sinusoidal

and

A

since

not

describing

by applying the

of ocean

and

waves are

considered

tt«

140

THE MA6IC OF MATHEMATICS

Some other

Interesting mathematical properties of

ocean waves

are:

1) The 2) The

length depends upon the period. wave height does not depend on the period or the length

wave

(there

are some

period

and the

3) The

wave

exceptions

length

are

in which the

wave

the

minute).

will break when the

120'. When the

influence of

angle of the crest exceeds of its energy is then

breaks most

consumed.

4) Another way to determine when the compare its

height with its length. When

than 1/7, the

wave

will break is to

waves

this ratio is

greater

will break.

crest

trough wave

height vertical distance from the crest to the trough

wave

length horizontal distance between

wave

period

two consecutive crests

the time In seconds It takes one wave

crest to travel

a wave

length sinusoidal

curve Is a

Its

regularly repeats

periodic (It

shape)

trigonometric function.

cycloid

by

the

curve

path

is the

of

a

traced

fixed point

circle which rolls

straight line.

curve

smoothly

on a on a

MATHEMATICAL MA6IC IN NATURE

Leonardo da Vinci's notebooks abound with sketches and waves

and water

dynamics.

writings

This sketch from Codex Madrid II

on

on

folio 24

swirling action of waves crashing on shore. Here he wrote, carried away from the shore by the waves of the sea. The sea casts back to the shore all thtngs left free out at sea. The surface of the Illustrates the

"Nothing

water

how

is

keeps

waves

accurate

the imprint of the waves for some time." His descriptions of on shore and rebound onto the Incoming waves were

break

conjectures that conformed

to the the

principles

of wave motion

formulated In later years —namely that each point a wave strikes becomes the point of new disturbances and all disturbances affect the as

shape

of the

wave.

141

\r\

\f

0

h

1' (SI

,u

MATHEMATICAL MAGIC FROM THE PAST BABYLONIANS & SQUARE ROOTS THE CHINESE METHOD OF PILING SQUARES

THE LADDER THAT INCHES ON THE Vl EARLY RANDOM NUMBER GENERATORS EGYPTIAN MULTIPLICATION THE FIRST SCIENTIFIC LABORATORY

PLATO DOUBLES THE SQUARE THE ROMANS & THE AREA OF A CIRCLE

HOW THE GNOMON TRISECTS AN ANGLE UNSOLVED MATHEMATICAL MYSTERIES FERMATS LAST THEOREM GALILEO & PROPORTION

MATHEMATICS & CONTAINERS GEOMETRIES

—

OLD & NEW

WHAT'S IN A NAME? EULERSMAGIC FORMULA— F+V-E=2

144


In most sciences one

generation tears

down what another

has built and what one has established another undoes. In

mathematics alone each generation adds a new story to the old structure. Hermann Hankel —

days of conflicts between ancient and modern

In these

studies, there must surely be something to be said for

study

which did not

with

Pythagoras

with Einstein, but is the oldest and

youngest of all. -G.H. Hardy

Discovering

begin

mathematical magic is not confined to present

things. History and

ideas from the past hold

a

not

for

curiosity

mathematics have

presents surfaced

a

and

progressed

the

desire

to where it is

of

use

irrational numbers, proofs, conic sections. If it

as

human

day

wealth of the magic

of mathematics. One often wonders how the ancients made such ideas

a

and will not end

were

to

learn,

would

today?

This

chapter

few of the multitude of mathematical ideas that have

over

the centuries.

11

iVa ,Pascal [por®l&al&llflOtty Zeno

Laplace

ipaiiraidkDSS Whitehead

& Chu Shin-Chieh

Einstein

•

FsOaiftiWijft^

tOfe©(? Afe®gO Fibonacci §@<§p©(fi)<5© D0 fl, % 30 § 0©„ al Khowarizmi

algebra

flcBg^-s-flQgâlhyipoftQmms©^ s©Q

GOoooits?

•

Pythagoras aS

?

lb?

3

sa

Lovelace

[pirciDgiraiDQiDQSnQg ©©©GÔoixaif©*;

fiffllfimlktoir Pascal & Leibnitz

Apollonius @®si3
•

Newton ©aOOMte© Leibnitz & Seki Kowa

£s araaOsTsfe Noether

•

Diophantus ©apaffl)(B(Da

Eratosthenes ©aurftlh nacsna!i
.

Bhaskara

Dudeney •Descartes ©©tr()©§0@iii)

Napier QffigamSfttais Agnesi

.

Cantor ftramstfQtnBQ© rjammofosffs

Babbage naQ[pQQftcstrs Loyd jpiaasn©s

.

gooimcsftffa©©

Riemann &

Collage of mathematical ideas

Bolyai

&

Lobachesky

and their creators.


History

shows that mathematical

creativity

particular culture, time period, civilization, of

amazing ideas and contributions that

is not

or

to any

privy

gender.

The wealth

developed

over

the

explore.

Such

an

11

12

TTYTYYTTTV7 rrr" WWg < a b r A E q- Z H 0 I

IA

IB

centuries is

incredible and

truly

were

to

exciting

NUMBERS SYSTEMS

0

12

4

3

6

5

8

7

10

9

j^ T

i

ii

in

mi

ii hi

m hi

m mi mi mi

mi urn

A

Ai

An

^

*

5

B

&

k

¦t T\

n

f

t.

•£

K

2

X

i

n

i

•5

K>

3'

I

II

III

mi urn

t

1EC I -I

-II

i n in rv v n

i *5^

0

•

in

••

h

•••

1 10

....

t

n

o

1

TT

vi vnvmixx

•••

»»

j-

_

-3i

<

i

->

A

V

XI

XII AU

/M •

»«»»

••

_

111 1000 1001 1010 1011 1100

11 100 101 110

HINDU-ARABIC-BABYLONIAN-GREEK-EGYPTIAN HIEROGLYPHIC-CHINESE SCRIPT-HEBREW CHINESE ROD NUMERALS-ROMAN-EGYTPIAN HIERATIC-MAYAN-BINARY NUMERALS

exploration would take the reader on a trip that countries all ideas

over

as was

the

a

almost

discovered

were

countries,

the world. Such

simultaneously

with the

case

to all

form

place

value and the number

the world

Maya

with

developed system.

are

—

a

indigenous

first

by

the

Babylonians

modified base 20

and evolved

a

We find that their

standardized

by

peoples.

zero was

some

different

in

of

hyperbolic

numeration

systems of

development

geometries. We know that numbers and some

traverses time and

voyage would reveal that

We find that the

developed

in many

use

of

parts of

with their base 60, later the

system and by the Hindu when they

positional system

notation for the base

was

10

later improved upon and

the Arabs. The Chinese also

developed

a

place

14

146


value

system and used

evolved into

sophisticated

a

in

especially

base ten numeration

calculations.

performing

FERWAT

with their rod numerals, which

zero

Only

a

few of the many

.'HYPATIAC

POiNCARE*

PYTHAGORAS

nBeTHE#

$HASKA»A

OMAR KHAYYAM

EULEff plato

they

system used

-.

i-I&ins Thiemann "xvlobacmevsky

pascal

EUCLID

lagrange leibniz .HILBEftT EUXODUS OEDEKIND

cXyXeY

dAQOBI

PAPPUS

SOCRATES;..BOOLE..

THAliis CA'NT6ft

HAMILTON

.

archimedes

MlOiS^vCAUCHY

& DESCARTES S: KQVALEVSKAYA

^ SEKI

^PTOLEMY

FOURIER

;tCRONECKER

•

KdWA

SAOjKHERJ#KU«MA

BOkYAI

t"'CAVAUERI .^BERNOUU!

.'¦}&.APOLLONIUS

GA£HLE0^ GAUSS :.:|f|;-WEIERSTRASS HERMITEki ;.LE&ENOflEJ*V: De MOIVRE ÂttACEfe mVE$TE&i;: ERATOSTHENES

AGNESt

QdDEL

8AMANUJAN FIBONACCI

.

NAPIER 2ENO ABEL RUSSELL

HetftOH%<. tOfeUCE/^" KEPLEfc GERMAIN^ DIOPHANTli'S ARISTOTLE SOMERVILLE

EINSTEIN

,

AL-KHOWARIZMI

îMlBBtRKHOm -M- CHATELET.^.WiÊÊAD:il&;;: An array

of some famous mathematicians from, the past

thousands of mathematicians and ideas Those I discuss in

chapter. stature

over

sections your

as

appearing. I urge you

of the

magic

to

study and

use

or

these

to enhance

of the past. This is the spirit

of mathematicians and their ideas, and of the

of numbers

systems

is

presented.

mathematical discoveries.

mathematician to understand the

their

mentioned in this

illustrations of the array of famous mathematicians, of

collage

new

are

way reflect their superiority

boards for further

spring understanding

in which

the

those not

no

creativity.

Seek out the

We

are

One does not have to be

essence

magic

of these ideas

in the

diagram

in the midst of many

past

or

a

appreciate

and the present.


BABYLONIANS & SQUARE ROOT!

We often think of ancient

mathematics

as

remote!

just that

Yet

—

WHAT WERE THEY DOING WITI

ancient and

looking back,

it

ACCURATE SQUARE ROOl

APPROXIMATIONS

is ^

surprising to discover you may be using a similar idea, value, of

years

ago.

mathematics

text from to 200

What

comes

or

concept that people used thousands have

we

mainly

from

a

archeological excavations.

B.C.I.

They

learned few

clay

about

The tablets date around 3000

reveal that the

in

Babylonians

unknown

equations

•

systems ofequations

one

approximations2

Babylonian

tablets of cuneiform

following mathematical concepts— •

147

in two unknown, tables

of

dealt with the

148

THE MAC IC O F MATH EMATICS

•

volumes and

•

calculation of the

•

the approximation ofn, 3 circle—

•

areas

area

of a triangle and a trapezoid was

used in determining the area of the

3r%

the volume ofprisms and cylinders base times it

by multiplying area of the

height

•

the Pythagorean theorem

•

aspects of number theory, for example:

l+2+4+...29=29+(29-l) The

Babylonian tablet3

on

the previous page illustrates

an

astonishingly accurate approximation for j/2. It is equally amazing to learn that the Babylonian sexagesimal positional number system

lent itself to such accuracies. The Sumerlan number

system.4

It

was

Initially lacking

to a sexagesimal positional positional number system of

system

the first

a zero

context to indicate

example,

Babylonians updated

and

a

sexagesimal point,

could represent

11(60)+ 12

=

672

number its time.

it relied

on

the intended value of the number.

this number

11+12/60.

the

the For


The

Babylonians later developed

for a zero

What

place holder. the

were

Babylonians

approximations for we see

that the

j/2 ? is

figure symbol

which

the horizontal

with

such

accurate

this cuneiform tablet

square with

of the square has the

Along

doing

Examining

a

i

fc"

diagonals

closely

drawn. One side

^^^^which is how they wrote 30.

diagonal they wrote

^

Y«Yf
By assuming the sexagesimal

represents 1, 24, 51, 10.

point is between 1 and 24, the number converts to 1

+(51/602) +(10/603)

+(24/60) =

l+(2/5)+(51/3600)+(l/216000)

which compares to the estimate the

We

know the

Pythagorean

Babylonians

distance 0;8. Horn

far did the ^wer ":rt is the accepted notation for

indicating the

sexagesimal point*)

repetitive

had

a

good understanding

they

of the

calculated for the vertical

slipped Qi& down from thus tap of the watt

appeared on

The upper end has

a

the beam

TateproMem

slipped down a

used

To arrive at their

by the Greeks.5

theorem. The value

another Babylonian tablet f'Apatu {beam} of length 0,30 (stands against a unity.

1.4142129+

1/2=1.414213562....

Babylonians probably

approximation method often used

-

the

hypotenuse

represents the heamb lengths

this leg of the right triangle is

0;30-0;6 24

0;3O

¦iMfMi' ¦ttw

III'

1§&

t:'>:;ci::i: 'i;t;i

i

the distance thefoot of the beam would be found using the Pythagorean theorem

149

150


diagonal

Is

approximation for the length of the

accurate

an

square's diagonal. Namely, the diagonal numbers 42; 25, 35 convert to

42+(25/60)+(35/3600) «42.42638889 while

42.42640687. In addition to sides the

were

rational numbers theorem

Pythagorean

working right triangles whose (e.g. {3,4,5}, {5,12,13}), they also used

on

1Translatlons

of O.

F. Thureau—

Dangtn

as

whose sides

right triangles

all rational numbers. This explains their such irrational numbers

30/2 «

with

use

were

not

of approximations for

j/2.

Neugebauer, Mathematics Keflschrifiexts, 1935-1937. Textes Mathemattques Babylon/ens 1938, and O. Neugebauer and A. Sachs Mathematical Cuneiform Texts To expedite numerical translation of the tablets, O. Neugebauer developed the notation of commas to separate place values and to represent the ; sexagesimal point. For example, ^W W^ ^^^ translates to 11, 1, ; 10, 3 which .

"

means

11(60)

+1

+(10/60)

+

"

(3/3600).

tablets have revealed tables for approximating various mathematical values. For example: tables of values for reciprocals which were used when division had a remainder, and tables of values for square roots and cube roots have been excavated.

3This

Is from tablet YBC7289 of the

Babylonian

*The Babylonians used

two

and value.

The position of these

one

^

for ten.

symbols

collection at Yale.

to write their

^y

T for numbers. determined their

symbols

°The ancient Greeks used the following method to approximate square assume their first guess for 2 was a (for example, let a=l). Their next estimate was determined by taking 2/a (which equals 2/1=2). Then their next estimate was the average of these two estimates, (l+2)/2, which equals 1.5 The estimate 1.5 would then be averaged with a new approximation, 2/1.5. This process would be continued. roots:

.

"From Science

Awaking by

Inc., New York, 1963.

B.L.van der

Waerden., Jon Wiley & Sons,


151

THE LADDER THAT

INCHES UP

The ancient Greeks discovered how to construct the

irrational

numbers

thagorean inscribing

lengths using

theorem.

and

They

ON THE

of

the

Py-

used

circumscribing regular polygons

infinities and limits to close in

developed

a

x

type

on

the

area

and the concepts of

of

a

circle.

of ladder arithmetic using ratios to

111

IIIIllII

¦in in

of the latter, the rest of the numbers in the first column are generated in the following way

in

¦iii

Hi ¦Hi

ill 70

iiiililili liiiiiii! iiiiiiiiii

i

The ratio 1: the

same

j/2.

Here is how the numbers from the first column generate the the numbers in the second col-

1+1=2

U2=l

2+3=1

1+5=1

5+7=112

5+M=1l

12+17=21)

M+19=41

|||j|p|||;l|||i

liilllili iiiiiiiii iiii»iiiii

29+41=?®

29+70=

y^2 is squeezed in on by the ratios of the numbers on

rung of the ladder. The ratios continually come closer

and closer to 1

//2.

The limit of ratios is the value of 1

NOTE: The two numbers at each rung of the ladder aotve the equation--

y£_2jc2_fci> ^hhj x value* the numbers on the left aide of the ladder. axe

also

approximate

the value of irrational numbers. Here's how it worked for

Starting with the 1 and 1 from the top

They

1//2

//2.

=.707106781...

1/1=1 2/3=.666... 5/7=71428571429... 12/17=. 70588235294. 29/41=.70731707317. 70/99=. 7070...

VI

152


THE CHINESE

METHOD OF PILING Finding experts

ia i>c<

>t"£UAI\t>

who

are

able to

translate ancient Chinese writing Is difficult.

Finding experts

able to

translate manuscripts that deal with mathematical Ideas Is more

difficult. This

mathematical themes

technique algebraic

explains why

are scarce.

examples

even

of

Hsuan-thu, the piling of squares,

was a

that Chinese mathematicians used in order to arrive at

conclusions

particular

Chinese

using geometric and arithmetic

illustration is from the Chinese

The date of Chou Pel is

disputed,

with

means.

This

manuscript, Chou Pel.

possible dates ranging from

MATHEMATICAL MAC IC FROM THE PAST

1200 B.C. to 100 A.D.

If 1200 B.C. is accurate, it would be

.

the earliest known demonstrations of the and the

theorem has

in many civilizations

angle.

means

of

Pythagorean theorem,

predating Pythagoras

appeared architecture. It was one

one

The

Pythagoreans.

Pythagorean thoughout history. In

of assuring the formation of a

In mathematics is has been and is an

right

indespensible

tool

crossing many mathematical disciplines. The

theorem states that

Pythagorean

right triangle the sum of the squares of its two legs equals its the square of its hypotenuse (a2+ br=cP).

for

hypotenuse

leg=a

any

(The

converse

is also

true.)

left diagram below, the interior square's area is indicated as 5x5 or 52 right triangles, each of area (l/2)(3x4) and a square qf area lxl, totaling 25 square units. In the right diagram below, the square is divided into two smaller overlapping squares, one 3x3 and the other 4x4. The part they overlap has the same area as the vacant part qf the 5x5 square they do not occupy, which illustrates that the area qf larger square (5*) equals the sum qf the two smaller squares' area, namely 3* and 4*.

In the =

25 square units. It has been subdivided into 4

'

c/

^

o

_

C

Aw*

A

v

V

/\

b

>

\

"«

>

'

)

This diagram explains how to find the area of the interior shaded square by summing the areas of 4 triangles and the unit square in the middle. In general it shows—

c2=4(l/2)ab + (a-b)2 =2ab + (a2-2ab+b)2 =a2+b2

R

/

>

/v

The

sumofthe two shaded rectangles' areas equals the area of the small shaded square (this is the square created by two overlapping squares). Letting 5, 4, and 3 take b, it shows

on

a2

the variable +

b2=c2.

c, a

and

15J

154


ONEOFTHE EARLIEST RANDOM NUMBER GENERATORS

Although to as

a

during credit for

being

in the National

have

played

one

this die

was

not referred

random number generator ancient Greece, it still

gets

of the earliest remaining die. Today it appears

Archeological

many roles

foretell the future, to

over

Museum in Athens, Greece. Dice

the centuries.

implement

moves

They have been used

of various games such

to as

backgammon and

monopoly, as

or,

in craps,

they are the main elements

of the game.

Mathematicians have

long been intrigued with dice from the

viewpoint of

probability. fact, dice

can

be considered

and Pierre de Fermat to focus their attention

gambling,

Pascal was asked

if the game

were to

Fermat about the

new

probability.

While

by a friend how the pot should be split

stopped before it was over. Pascal wrote problem. In 1654 the two men worked out their their correspondences, and thus launched

branch of mathematics.

random number generators the

on

be

theory of probability in this

In

responsible for getting Blaise Pascal

theory of probability.

are

Today

dice and other shaped

used to teach various aspects of

MATHEMATICAL MACIC FROM THE PAST

155

EGYPTIAN The

of

method

Egyptian

and

civilizations. In

spread

ancient Greek schools it

the Middle

specific

Ages

names

Here is

halving.

its

fig

V

/]

\%\\f\f\ 111 fl/VVt Hi flfl^t

as

need

for

plication tables, on

In

and referred to

by

taught

duplatio for doubling

the Rhind papyrus of how

14 i#j

^**

OA

*iAA

give 24, which

1120

was

by

multiply on

how

doubled to give 48, which in turn is

doubled giving 96. Slashes next to the 4 and the

their

the answer, 144.

glving

drawn

Then

amounts

corresponding

are

8, indicating

twelve.

ls

sum

are

their

added,

The

Egyptian multiplication eliminated

of

memorizing

multi-

ftftftft

since it relied

mainly

ftftftft

OA

<]

800 done

similarly.

To divide

160

80, you find what you would '

80

by to get 1120. Depending large a number you are dividing, the

\W(\ /H

33

/h

11

Ml

divisor

m$

£

47

D/flD=47+33

a sum

1120. If the out

even,

^ 1120

multiplied by 10,

1000, etc. The results

equals come

'

or

3^ O

/l

*J

is

either doubled, 100,

'ÎLy /

¦¦"

be doubled until

+

an

is then

addition.

Division

H+H/I3

and mediatto for

12x12. One first starts with 12. Then

method the

Egyptian calculation.

it is doubled to

——————

<|S)

as

were

Egyptian scribe would have multiplied

M

f°\

r

taught

example, from

an

unno a rtli

was

techniques

such

H

lift

f

for

plication survived centuries and

over

MULTIPLICATION

multi-

the

can

then

is found that

problem

did not

Egyptians used

fractions. As in the example 47-~33.

156

TH E MAG IC O F MATH EMATI C$

THE FIRST SCIENTIFIC

Ancient

Greek

and scholars

LABORATORY

to be

philosophers.

It

set up

is

there the

by Pythagoras

or

the

often considered

predominately

has revealed that the first recorded was

are

mathematicians

Now

theorists and

new

research1

for

experimentation laboratory Pythagoreans (6th century B.C.).

surprising that

not are

no

records from

Pythagoreans

support this claim,

Pythagoreans

to

since the

were sworn

There Is

to secrecy.

supportive evidence from later scholars. This Illustration Is from the

writings of the

Roman scholar Boethlus

(5th century A.D.). It shows

Pythagoras

experimenting with sound,

specifically the relationship between

an

object's

proportions (in this and

the

produce.

case

tones

Theon of

ancient authors,

other Greeks inventions

as

they Smyrna (2nd century A.D.), along with other

wrote of similar

well

of the

bells)

as

the

Greek

types of experimentations by

Pythagoreans.

mathematician

If one considers the

Archimedes

(287-212

B.C.)— laws of lever and pulleys, methods for comparing volumes of

objects by submergence

catapult

—

one

realizes

in water, the Archimedean screw, the

the

existence

and

importance of

experimentation. 1These new findings were presented by Andrew D. Dimarogonas of Washington University in St. Louis in the Journal of Sound & Vibration, May, 1990.


157

PLATO DOUBLES ArEftMETPHTOZ MHAEIE EIZITQ Let

no one

ignorant of

geometry enter here.

These words doors at the In Athens.

were over

the

Academy of Plato

Although Plato

(428-348 B.C.) Is not famous for his mathematical

contributions, he is renowned center where

for

providing

he

guided, encouraged and

a

inspired mathematical

thought. This elegant method for doubling the area of a square appears in Plato's

dialogue Meno.

The

diagram

illustrates how to double

a

square and how not to double a

square.

The upper shaded square's area double that of the white square. The lower shaded square has

is

area

Jour times that of the white

square. Its side is twice

Note that although the size of the white square's side, its area is quadrupled.

THE SQUARE

158


HOW THE ROMANS FIGUREDTHE AREA

To find the

OF A CIRCLE

area

of any

given circle,

the Romans used the square whose

area

diagonal

of

was

a

1/4

than the circle's diameter.

longer

Here's how It worked, and Its accuracy.

Suppose this circle's diameter Is d. Then this makes the square's

d+.25d= 1.25d.

diagonal

Using the Pythagorean theorem, the square's side Is

1.25d/y^2~.

Squaring the side of the square gives the square's

area as

1.5625d2/2. Since the circle's radius Is .5d, and

the

area

7tr2,

formula for any circle Is

the actual

area

of the circle Is

(.5d)27t=.25d2Jt. Using

3.1416 for the

ROMAN METHOD

approximation of it.

we

get what

&

.25d2n

approximation

—

ACTUAL AREA METHOD

vs

.25d2jt =.785d2

1.5625d2/2=.78125cj2

Equating .78125d2

We get

and

solving

this relation for n

the Romans had torn,

.78125d2 ¦78125d2

.25d2n

=

=

tc

.25d2 3.125

3+1/8

=

71

=7t

namely—

MATHEMATICAL MACK FROM THE PAST

159

HOW THE GNOMON Trisecting

an

three famous construction

angle was one Impossible

TRISECTS AN ANCLE

of the

problems of antiquity which

v

launched many mathematical discoveries. cannot be trisected

trisected

gnomon used

using can

using only a

a

compass and

Although as

be used to make and determine an

angle

straight edge,

tool the Greeks referred to

by the ancient Greeks to trisect

an

angle

it

right angles. in the

The gnomon has two marks, R and S, so that /7fl/=/7S/.

can

be

the gnomon. The It

was

following way: 1: The gnomon is used to make a line parallel to a side of the angle, as shown in

Step

Step

2.

3: The gnomon is lined up as shown with one mark on the side of the angle, one

Step

the parallel line, and the shaft running

on

through the angle's vertex.

Step 4: The clotted Stepl

Step

2

lines are drawn in to form three triangles. APCBs APAB

by Hypotenuse -Leg. APCBSAPCD by SAS. Thus, APCB3APCDSAPAB, which makes

z1«z2sz3., andzP trisected.

Step 4

160


UNSOLVED MATHEMATICAL

Mathematics

MYSTERIES

abundance of

problems.

mathematics

and

certainly offers

an

In fact,

problems

are

shows that mathematical Ideas have been

Inseparable. History

catalysts for mathematics problems,

and that math

problems have

stimulated many mathematical Ideas and discoveries.

The three

impossible construction problems of antiquity 1; the Konigsberg Bridge Problem2and the Parallel Postulate Problem3 are historic examples of problems that have stimulated mathematical

posing

and

and the

pondering

scrutinizing

been resolved and in the process

Ideas and discoveries. The

thought,

of mathematical

of solutions and

problems

proofs

are

and

questions

driving

forces for

mathematicians. Here

are a

few famous "unsolved" mathematics

problems:

THE UNSOLVED PRIME NUMBER PROBLEMS •

Is there

one

number is •

Is there

formula

or a

test to determine whether

whose

difference

are

to the

sum

is two. For

even

perfect

A

perfect number is

one

which

an

a

example

number because 6=1+2+3. Other

examples are B.C., Euclid showed that if 2n'1 was

prime number that 2n"^(2n-l)

18th century,

pair

of its proper divisors (a proper divisor is

28, 496, and 8128. Circa 300 a

a

3 and

example,

divisor other than the number itself). The number 6 is of an

given

5 & 7, 11 & 13, 41 & 43.

The odd perfect number mystery.

equal

a

infinite number of prime pairs? A prime pair is

5, since 5-3=2. Some others

is

not

prime?

an

of consecutive primes

•

or

Leonhard Euler

is

a

perfect number. Then,

proved

that

any

even

in the

perfect

number must comply with Euclid's form. For example,

But odd

8128=26(27-1).

perfect

numbers remain

a

mystery.

As

yet

no one

has


found

an

odd perfect number,

has anyone

nor

proved

that all

perfect numbers are even. GOLPBACH'S CONJECTURE Is every

even

number greater than 2 the

sum

of two primes?

In 1742 German mathematician Christian Goldbach wrote to Leonhard Euler even

number except 2

(1707-1783)

was

the

sum

the

of

6=3+3; 8=3+5; 10=5+5; 12=7+5,

4=2+2;

conjecture

is believed to be true, no

Thus far the

following

can

conjecture that

two ...

Examples:

Although

Goldbach's

has proven it

be written

as

the

apparently proved sum

primes—a far cry from two primes.

of not

more

Ivan M.

primes.

In

1973,

Chen

Jing-run

sufficiently large even number is the that is either prime

or

so as

yet.

has two

sum

that any

than 300,000

are

showed

the that

of a prime and

prime factors.

even

Vinogradov(1891-

1983) proved that all sufficiently large odd integers three

every

primes.

ideas have been shown: In 1931, Soviet

mathematician L. Schnirelmann

number

one

(1690-1764)

a

sum

of

every

number

161

162

THE MAG IC O F MATH EMATI C$

FERMATSLASTTHEOREM In the 1600's Pierre de Fermat one

(1601-1665) wrote

In the

margin of

of his books—

To divide

cube into two cubes,

a

fourth power,

a

or

in

power whatever above the second, into two powers

denomination,

is

wonderful proofof this, restated:

If

is

n

a

but the margin is too

eluded

challenge.

as a

even

the most

The next section latest

z

any

same

assuredly found narrow

an

to hold it.

natural number greater than 2, there

positive whole numbers x, y, went out

and I have

Impossible,

general

of the

are no

such that xn+yn=zn. Fermat's note

For centuries the

proof or disproof

has

prominent mathematicians.

gives additional background and discusses the

information

Last

Fermat's

on

discoveries that have been Fermat's Last Theorem

are

Theorem.

outgrowth

an

probably

more

Some

of efforts

of the

to prove

important than the

theorem. *

The

study of mathematical

interesting

as

delving

*

Ideas that have not been resolved is

Into what

we

sample of unsolved mathematical simple enough

background,

to

explain

their solutions

to

are

*

do know. This is but

mysteries. Although

someone

with

no

a

a

as

small

few

are

mathematical

elusive.

iThe three impossible construction problems of antiquity, which were to be solved using only a straightedge and compass, are: trisecting an angle (dividing an angle Into three congruent angles), duplicating a cube (constructing a cube with twice the volume of a given cube), squaring a circle (constructing a square with the same area as a given circle). A few of the discoveries that these three problems helped evolve were the conchoid of Nlcomedes, the spiral of Archimedes, and the quadratrlx of Hlpplas. 2The challenge of the Kontgsberg bridge problem was to find a path over seven bridges of KOnlgsberg without crossing any bridge more than

the

once.

Euler

developed

the notion of networks when

solving this problem.

3The Parallel Postulate dealt with determining whether Euclid's fifth postulate was indeed a postulate and not a theorem. Attempts to prove this postulate led to the discoveries of non-Euclidean geometries.

MATHEMATICAL MAC IC FROM THE PAST

165

FERMAT'S LAST There

are no

THEOREM

positive whole

numbers that solve

xn+yn=zn when n is a natural number greater than 2 century mathematician

When 17th

^

Pierre de Fermat scribbled this note

•

in

a

margin of

metica

a

translation of Arith-

by Diophantus,

•*r'T*r'v*-T--

little did he

.

•¦'¦•¦"¦'L'.TT'T-

know the impact his comment would

have

mathematics for the next 350 years. Had

he

really solved

playing

a

joke?

No

mathematics. construction

tr«?f.'.'. .&fa(U)
we

will

¦

txarfitc is.

know for

ever

do know Is that it

of the famous unsolved

one

problems

*-i.-*r*-.-.a.-._.i.

it? Was he just

one

sure, but what

became

the

history of Like the three famous problems of antiquity, in

the

Konigsberg bridge problem, and Euclid's

5th

Postulate,

mathematical

last

Fermat's

theorem

has

stimulated

thought and discoveries for centuries.

The mathematical

community

Is very excited and enthusiastic

about Modular Elliptic Curves and Fermat's Last Theorem, the 200

page work of Andrew J. Wiles, professor of mathematics at Princeton

University. Presenting his work at talks at Cambridge

(June 1993), he climaxed his last talk with the announcement that he

had proven

the

Shimura-Taniyama-Weil conjecture, which

mathematicians felt held

a

key

In mathematical circles the

work has

•

development of

the

on

.fjouma. ¦.

put

to

proving Fermat's last theorem.

feelings

are

very positive that Wiles'

Fermats Last Theorem to rest.

164


Over the centuries there have been thousands of Fermat's

problem,

but

none

thus far has held up to

not that Fermat's last theorem Is

said "I have found

proof)

Is the

beauty

number theory,

a

truly

so

extraordinary,

wonderful

It Is

but

Fermat

as

proof —the solution ( Its

of this theorem. It has

cryptology and codes,

of

"proofs"

scrutiny.

discoveries In

sparked

to name

some areas.

Wiles had been

with

Intrigued

Fermat's theorem

as a

teenager.

But did not delve Into Its

until he

saw a

possible

Wiles considers his

proof

means.

proof

a

collaboration of the many mathematicians and efforts that went before

him.

plateaus 18th

historical

Among

toward this end

we

have

century mathematician

onhard Euler

who

Le-

the

proved

theorem for n=3. German mathematician

Kummer

E.

Ernst

proved the theorem for all but three numbers less than

Andrew Wiles

Present

day computer proofs

have shown that

no

100.

solutions exist

for the first four million natural numbers. In the 1950's Yuktaka

Tanlyama made his conjecture involving elliptic structures In

a

hyperbolic plane.

that if Taniyama's

conjecture

curves

In the 1980's Gerhard

were

true for

a

certain

and their

Frey posed

type of elliptic

(called semlstable), then Fermat's theorem could be

curves

proven. When Kenneth A. Ribet

proved Frey's proposition, Wiles

decided to undertake Fermat's theorem. From that time worked

Barry

intensively

for

seven

Mazur of Harvard

described

a

years. In

came

a

on

he

paper

by

to Wiles' attention. The paper

numerical technique that

old which was very

May

of 1993,

was over a

hundred years

important in the finalizatlon of his proof.

MATH EMATKAL MAC IC FROM TH E PAST

165

GALILEO*. There

are

PROPORTION

mathematical

many

concepts that have

no

restrictions "

In the realm of

mathematics, but

have limits when is

one

applied

such idea that is For

problems.

a

to the real world. The idea of proportion

very useful tool In

how many boxes

pounds, for

168

proportion

pounds? is

one

(3 boxes/42 lbs But not all

if

example,

identical boxes of marbles

three

weigh

are

Setting

solving many types of

42

needed up

a

way to solve this—

=

?

boxes/168 lbs).

problems

of

proportion

may be realistic to solve. Can any tree be scaled down to any size and be

functional? Can exist? The person, size.

a

person of any size

composition of an object, be it

a

tree

or

the bones of

a

play vital roles In deciding the upper and lower limits of its possible, since the structure

A hundred foot person is not

and materials that make up the human such mammoth size.

heights,

dictated

Even

body were not Intended for

giant redwoods have limits to their

by their roots system and the properties of wood. problem of scaling an object up or

One of the first records of the

down

was

tackled

by

Galileo in 1638 In his work

Dialogues

Concerning Two New Sciences. Here he states "... if one wants to maintain in a great giant the same proportion of limb as that found in an ordinary man he must either find harder and stronger material for making the bones, or he must admit a diminution of strength in comparison with men of medium stature; for if his height be increased inordinately he will fall and he crushed under his

weight. Whereas, if the size of a body be diminished, the strength of that body is not diminished in proportion; indeed the smaller the body the greater its relative strength."*

own

'Dialogues Concerning Two New Sciences. translators. Macmlllan, 1914.

Henry

Crew & Alfonso De Salv

166


MATH EMATI C$& CONTAINERS

It Is often

discover were

There

are

vessels for

startling how

amazing

objects

mathematlzed

to

and

Ideas

the

past.

In

of varied vases, containers, and

examples storage that have been designed numerous

and

Into many forms. This

design and

drawing of a chalice

by Paolo

Uccello Is from the collection on

display In theUfflzl

Gallery of Florence, Italy.

Although It was done

during the

first part of the 1400s, Its

precision and exactness Is reminiscent of a

computer

analysis of a goblet, and Illustrates linear

perspective, constant ratios

of proportional

measurements, and

use

of

geometric solids.


GEOMETRIES

things that I

was

amazed. ...out of nothing I

have created new

a

strange

universe. —Jdnos

When most of

geometry

Bolyai

think of

us

course

from

geometry,

letter to his father, I823.

a

we

think of

"memorize", the excitement of

thing

been

about

Little did

elements.

taking place

more,

much

we

geometry

realize that

more

postulate

was

school

a

area, one

The

visualize its

silent evolution of ideas had

present. Those of us was

geometry than Euclidean geometry.

We

study to

volume.

can

of mathematics learned that there

learned how geometry and

system by Rene

is that

from ancient times to the

who continued the

high

"deadly" theorems we had to doing our first proof, points, lines,

triangles, quadrilaterals, circles, solids, beautiful

our

all those

—

algebra We

Descartes.

challenged by

were

saw

linked via the coordinate

how Euclid's 5th (Parallel)

mathematicians for centuries.

mathematicians felt the Parallel

postulate

was not

an

Many

independent

idea, but could be proven from the existing materials of geometry. As

has

history

in

independent attempts

as

to

Parallel

Euclidean geometry

really separate

mathematicians

mathematics is too

the

postulate definitely —

knowledge.

intriguing

one

get

was

but the unsuccessful

led to discoveries of non-Euclidean

impossible rest, for

shown,

geometries.

It is

field of mathematics from the ideas

they rely

on

their entire

A time line of the evolution of geometries

to pass up.

Space limits this

time line to

an

evolving of the fields of geometries rather than specific ideas within a

particular geometry. Hopefully

your own research.

—

OLD & NEW

/ have discovered such wonderful

167

it will

serve as a

springboard

for

168


600 BC—Thales Introduces deductive geometry. It was developed over the years by such mathematicians and philosophers as Pythagoras, the Pythagoreans, Plato, Aristotle. 300 BC—Euclid compiles, organizes and discovered and proven, Into thirteen

systematizes geometric Ideas, which had been books, called The Elements.

140 BC—Posedonlus restates Euclid's 5th

3rd century AD—Proclus

postulate.

(410-495 AD) Is one of the first recorded critics of Euclid's 5th

postulate. Countless attempts

are

made

over

1637—Rene Descartes formulates

the centuries to prove Euclid's 5th

postulate.

analytic geometry.

Gerolamo Saccheri (1667-1733) Is the first to try an Indirect proof of Euclid's Parallel postulate. Unfortunately, he does not accept the results of his work. Prior to his death he publishes a book, Euclldes ab omnI naevo vlndicatus (Euclid Freed of Every Flaw), which came to the attention of Eugenlo Beltrami one and a half centuries later. Had Saccheri not rejected his findings he would have sped up the discovery of a non-Euclidean geometry by about a century. 1639—GIrard Desargues (1591-1661) publishes a work on conies in which he discusses his new work on his discoveries of projective geometry. 1736—Leonhard Euler (1707-1783). His study and solution to the problem launches the field of topology.

KSnigsberg bridge

1795—Gaspard Monge (1746-1818) describes structures by plane projections. 1822 Jean Victor Ponceiet (1788-1867) revives formulates the duality principle.

1843—Arthur Cayley (1821-1895)

projective geometry with his treatise and

begins the study of n-dimenslonai spaces

in

analytic

geometry. Cantor (1845-1918). His set theory provides a basis for topology, which is presented in 1895 by Henri Poincare (1854-1912) in his Analysis Situs. Develops the Cantor set, an early fractal.

Georg

1871—Christian Felix Klein (1849-1925) does extensive work In projective geometry and topology, and proves consistency of Euclidean, elliptic & hyperbolic geometries. 19th

century—Nicoiai Lobachevsky (1793-1856), Jonas Bolyal (1802-1860), and Carl Gauss (1777-1855) independently discover hyperbolic geometry.

1854 —G.F.Bemhard Riemann

(1826-1866) presents elliptical geometry.

Mobius & Johann

1858 —August Mobius strip.

Listing independently discover one-sided surfaces, e.g.

1888—Guiseppe Peano (1858-1932) creates 1904—Helge

von

Koch

Peano

space-filling curve (fractal).

(1870-1924) creates Koch snowflake curve (fractal).

1919—Felix Hausdorff defines fractional dimensions in fractal geometry. A.S. Besicovitch generalizes Hausdorff's work.

1971—Vladimir Arnold links

algebraic (n-dimensional analytic geometry) geometry and

topology. 1951-75 —Benolt

development.

Mandelbrot, coins the word fractal and works nearly singlehandedly on its


600 BC

1637

EUCLIDEAN GEOMETRY

Analytic Geometry NON-EUCLIDEAN GEOMETRIES

1639

Projective

1736

Geometry Topology

r

1829

1854

I 1860's

Hyperbolic Geometry Elliptic Geometry Fractal

Geometry

present

169

170


WHAT'S IN A NAME? HYPERBOLIC GEOMETRY IS NOT NAMED AFTER THE HYPERBOLA

Ever wonder how certain

get their

names?

example, consider hyperbolic and elliptic geometries.

In both

mathematical fields For

their creators had

to do with the

that

were nothing eventually adopted. Hyperbolic geometry was discovered independently by both Nikolai Lobachevsky (1793-1856) and Janos cases

Euclidean

Bolyal (1802-1860). fifth

geometry's

postulate

states that one and

parallel line a

can

only

pass

P not

given point

names

one

through given

on a

line L. Mathematicians unsuccessful efforts to show this

postulate

was

and thus

provable,

a

theorem, led to the discovery of non-Euclidean geometries. It

hyperbolic geometry

discovered that there exists than

one

P and

line

parallel

comes

Hyperbolic

from the Greek word

case

number of

geometry —

it

and

had

to the

originally

pangeometry.

the

non-Euclidean

parallel

geometry

name

by

as

to L.

imaginary

which it is called

given by the well known

—was

elliptic geometry

geometry

passing through

But the

P and

creator of the Klein bottle. In addition, Klein

term

P.

through

referred to his

hyperbolic geometry

coined

SSttt^0"

lines passing

geometer Felix Klein, also

excessive.

applies

parallel

Lobachevsky today

more

passing through to L.

hyperbole, meaning In this

In

was

In which

no

for

Georg Rlemann's

lines exist

parallel

to L

The Term elliptic is from the Greek term

elleiptis, meaning to lack or fall short.

MATHEMATICAL MA61 £ FROM THE PAST

171

EULER'S MAGIC The

r

special thing about mathemat-

leal Ideas Is that been

should hold for all first k

have

they

once

established

true,

as

cases.

For

they example.to

sum

counting numbers, l+2+3+...+k, all

the

method

impossible

call

Induction.

^v

^—j—~^^

mathematically by It is physically

>i^^

V

tetrahedron

cube

to test his formula for every

possible counting numbers starting

set of consecutive

with 1, but the

beauty

that

not

do

they

/f\

/I

one

needs to do is plug into the formula k(k+l)/2 This formula has been proven a

O l\AAU LA

of mathematical

proofs

is

require brute force. Swiss

mathematician Leonhard Euler is credited with many mathematical discoveries, field of

especially in

His solution to the

topology. bridge problem is

said to have launched the

of

networks.

topological

properties of

object

objects

is distorted.

distorted into

a

faces, vertices

or

For

K6nigsberg study studies

Topology

that do not

example,

change a

the

cube

the

as

can

tetrahedron

be

and

by stretching squashing the cube, or vice versa. The size of the cube obviously changed, as did its number of

edges.

As

result,

a

wonder, what type of properties and remain

unchanged?

a

type of

number

invariant

offaces

a

interior

on

rhombicostdodecahedro,

polyhedra, of

its

point

on

the

point of the tetrahedron.

theorem that Euler

property of potyhedra,

then subtract the number

F+V-E=2. Try it out

an

fascinating

of the

might

left to observe

One observation is that any

interior of the cube remains

Outside of topology,

are

one

is that

to the number

proved about

if you

edges, the result

is

the Platonic solids shown. If you

energetic try it on the rhombicosidodecahedron.

add the

of its vertices and always are

2.

feeling

MATHEMATICS PLAY ITS MUSIC MATHEMATICS & MUSIC MUSICAL SCALES & MATHEMATICS MATHEMATICS & SOUND

174


Music is the pleasure the human soul

experiences from counting without being aware it is counting. -Gottfried Wilhelm Leibniz

Sounds,

whether In the form of noises

the vibrations of objects. Once

piece

they

of wood, cause

a

wire,

or a

surrounding

objects,

or

music,

such

column of air In

a

are

as a a

flute

all caused

by

rubber band,

begin

a

to vibrate,

molecules of air to vibrate. These vibrations travel outward from

the

source

In

three

dimensions. When these

vibrations reach

eardrums,

the

our

eardrums'

vibrations send

signals

to

brains which create

our

the sensation of hearing. Each musical instrument has

a

method of

creating

vibrations which in turn cause

vibrations

throughout the structure and

material of the instrument. For

when

example,

guitar's

string

vibrations

is

a

plucked

cause

its

the

other strings and entire instrument to vibrate.

Since ancient times, mathematics has been used to

explain music.

Today computer modeling and digitization, the quantization of music and sounds coupled with the study of acoustics and acoustical architecture

are

producing new sounds

sensations.


175

MATHEMATICS & MUSIC

May not Music be described as

the Mathematics of sense,

and Mathematics as the Music of reason?

—

J.J.Sylvester

Music and mathematics have been linked

During the medieval period

through

arithmetic, geometry, astronomy and music modem

Score

the centuries.

the educational curriculum

together.

grouped Today's

computers are perpetuating that tie.

writing

its influence

is the first obvious

on

music.

time, 3:4 time, and

where mathematics reveals

In the musical

forth), beats per

so

1^ ETfl/i

?

area

script,

we

measure,

*£

p

find tempo (4:4

whole notes, half

>*ffin f f

kf

f

i .>fc

^m S3e

±s±

m

nf

¦y?

t—t

IS m notes, quarter notes,

Writing

eighth

notes, sixteenth notes, and

so on.

music to fit x-number of notes per measure resembles the

process of

finding

a

common

denominator— the different

length

176


notes must be made to fit

a

particular measure

at

The composer creates music that fits

so

together In the rigid structure of completed work Is analyzed, every

the written

a

certain

tempo.

beautifully and effortlessly

measure

number of beats using the various desired

When

score.

has the

lengths

a

prescribed

of notes.

In addition to the obvious

connection of

mathematics to the musical score, music Is

linked to

exponential

ratios,

curves,

periodic

functions and

computer

science.

With ratios, the

Pythagoreans (585-400 B.C.)

were

the

first to associate music and mathematics.

They

discovered the connection between

musical

harmony

sound caused

string. They

and whole numbers

by a plucked string depended

as a

produced.

For

note C, then

4/3

ratio of whole numbers.

string of whole numbers ratios, example, starting with

that the

length of the

were

given off by

In whole number ratios

In fact every harmonious combination of

expressed

upon the

also found that harmonious sounds

equally taut strings whose lengths were

the

by recognizing

plucked strings could

By Increasing the length an a

—

be of

entire scale could be

string that produces the

16/15 of C's length gives B, 6/5 of C's length gives A.

of C's gives G.

3/2 of C's gives F, 8/5 of C's gives E, 16/9 of C's

gives D, and 2/1 of C's length gives low C.


Have you every wondered

grand piano

Is

shaped

Actually there

Is?

are

whose

Instruments structures

are

exponential by

an

and

concepts. and

Is

equation

one

curves

An

concepts. curve

where k>0. An And its

many

shapes

Exponential functions such

a

linked to various

mathematical

are

why

the way It

described

of the form

example

graph has

Is

the

y=kx, y=2*.

shape

illustrated.

y Musical Instruments that either

suing

formed

from

reflect

the

exponential

Instruments

columns

In

or

of air,

of

shape curve

are

an

their

structure.

> 1

The

study

X

2

of the nature of musical

sounds reached Its climax with the of

work

the

19th

century

mathematician John Fourier. He

proved

that all musical sounds

instrumental and vocal

described

—

could be

mathematical

by

expressions, which

—

were

the

sums

of simple periodic sine functions.

Every sound has three qualities

pitch,

loudness and

musical sounds.

quality

—

—

which

distinguishes

It from other

177

178


Fourier's

discovery makes It possible for these

sound to be to the

frequency

quality to Without In

graphically represented

the

an

of the

curve,

using computers

loudness to the

would

In musical

amplitude

not

have

mathematics of music,

composition and the been

and

design

were

headway design of

Mathematical

possible.

discoveries, namely periodic functions, modem

properties of a

and distinct. Pitch Is related

shape of the periodic function1.

understanding of the

instruments

three

essential In the

of musical Instruments and in the

design

of voice

activated computers.

Many instrument manufacturers compare the periodic sound graphs of their products to ideal graphs for

these Instruments. The is

also

closely

tied

fidelity to

of electronic musical

periodic

graphs.

reproduction

Musicians

mathematicians will continue to the

and

play equally important roles production and reproduction of music.

The

diagram shows a string vibrating in sections and whole. The longest vibration determines the pitch and the smaller vibrations produce harmonics. as a

lA periodic function is one that repeats its shape at regular

intervals.

In


179

MUSIC SCALES & The

of

speed

examples

These

of

our

commercial.

our

numbers which

are

equations and formulas which define

have vital roles In

objects

all

are

of constants of

MATHEMATICS

(p and

n, e,

c,

number

Avogadro's universe.

light,

world

—

Among

be

octave should be

included

chemical

or

concept of

an

they geometric, physical,

these famous constants, the .

r,

various

&JLLL4A

as a

constant of a

special nature. The octave

plays a

vitally Important part

world of

In the

music. It

establishes the unit a

or

distance of

scale. Just

as

the ratio of a

circle's circumference to its diameter

always produces the constant n, the ratio of the

number of vibrations of a is the ratio

length

of

1/2.

an

plucked string

These notes have the

octave.J

to

a

same

string half Its length sound, and form the

The shortened string vibrates twice the

amount per second than the or

subdivisions of

an

original string. The number of notes octave Is arbitrary, and is influenced by a

number of factors. Consider the various factors that

play when creating

a

particular

scale.

Sounds

or

come

notes

are

Into

what

180


comprise

a

mentioned,

scale.

Each

two notes

has

are an

double that of the other.3 different sounds with

this

or

notes in

there

were

an

a

octave.

But to

one

be

ludicrous,

that many notes.

octave,

a

whole note

jg two

As

of one Is

hear about 300

produce

a

scale

since

traditional

For

example, If

piano of eight octaves would

JJ

o

frequency

ear can

produce

300 notes In

apart

If the

The trained

many notes would

Instruments cannot

frequency.2

particular

a

octave

JJJJ

half notes

jg

4

quarter notes

JOTOT jgj

8

eighth

notes

—

16 sixteenth notes

JJJJJJJJJJJJJJJJ oJ*J A033J^ d J J ±- JJ Musical equations. The illustration above shows the relaionships between a whole note, a half note, a quarter note, an eighth note, and a sixteenth note. A dotted note is one way of conveying fractions notes, because a dotted note equals one and one-half times its value.

keys. Can you imagine a pianist running back playing a keyboard this long? Thus, the number of

have 2400 white

and forth

possible notes Is restricted by the physiology of our ears and by the

capabilities of

our

discernible sounds of a scale Is

How and which of the 300

Instruments. were

analogous

to

selected for

selecting

base should be used and which

represent the numbers?

—

a

a

scale?

Selecting

numeration

symbols

the notes

system.

—

What

should be made to

With the scale, the

length

of an octave


string needed

to be selected and the number of subdivisions

comprising the scale) had systems

we

to be determined.

(notes

As with numeration

find these evolved

differently In various civilizations. letters of their alphabet to represent the

The ancient Greeks used

of

notes

seven

These notes

their

scale. Into

grouped

were

tetrachords (four notes), which

modes.

groups called

Into

put

were

The modes

the

were

forerunners of modern Western

major and

minor scales.

The Chinese used

(five note) scale. sic

and

was

within defined

a

In

is

pentatonic India,

mu-

improvised

boundaries

specific

This octave is

by ragas.

divided into 66 Intervals called

srutls, The oj

score

a

of

14th century missal page

a c

rego

are

17

22 notes.

only

two

practice

22 srutls, from

basic

seven-note

formed. The Persian scale divided the octave Into either We

constant, different

see

although the

that

musical

musical Instruments from

to

In

although

are

which

scales or

there

octave

systems evolved.

one

culture cannot

In

was

a

given

addition, the

necessarily be

used

statues,

and

perform the music of another.

artifacts

Archeologlcal frescoes

Sumerian

There

clay

instruments,

vases,

vocal and instrumental musicians have been

depicting

unearthed.

of

are

many

early examples

of written music

tablets excavated in Iraq appear to show

note scale (circa 1800

B.C.); fragments written

and pieces of papyrus from ancient

(circa 100 A.D.);

a

Greece,

on

stone

an

—

eight

carvings

Greek text books

Greek manuscript in which notes

were

written

181

182


A fresco

using

depicting

their

musicians from the tomb

alphabet

(circa

Moslem-Arabic chant from 8th

300

qfDJeserkara

A.D.);

in Thebes.

manuscripts

of

a

century Spain.

During the 6th century B.C., Pythagoras and the Pythagoreans the

were

first

Pythagoreans

to

associate

believed that

things. Imagine their delight note, the

periodicities

instruments. own

music

numbers, when

came

In addition,

to be known

believed In the music of the each of the the known radio

mathematics.

some

they discovered

they

believed that

as

The

way, governed all the octave of a

of notes, and the ratio of notes

music, that the celestial bodies

This Idea

and in

planets

on

string

had their

produced

musical sounds.

the "music of the

spheres". Kepler

spheres, and

planets. Today,

signals carried by solar winds.

In fact wrote music for

astronomers have received

These sounds,

including

whistles, pops, whines, hisses when synthesized at Increased


were,

how

would birds

sing?

musical tune

have

a

written written

1The seven

Yet most verbal renditions of

change slightly

a

story

or

of

with each oral communication.

composition performed, scales

language

of music,

language

of mathematics.

even as

a

To

essential. Scales

are

the

equations and symbols

are

the

are

term octave comes from the Latin word for 8. A diatonic scale, has distinct notes from C to B and high C, the octave, is the eighth

note.

^Frequency

is

the

number

of vibrations

per

second.

There

is

a

correspondence between sounds and numbers (its frequencies), although our ears are not designed to discern all the possible distinct sounds. Â plucked string produces a certain note, e.g. C has 264 vibrations per second. When the string is depressed at half its length, its octave is produced, which vibrates at 528 vibrations per second. one-to-one

183

184


MATHEMATICS & SOUND

Mathematical

twisting and sound

waves

ideas

turning

have

been

music

and

for centuries. A walk

around the Interior of the dome In St. Peter's Cathedral In Rome will convince you that the curve of

the dome's walls carries one's

St Peter's

amphitheater

at

have studied and

on

a

listener

on

the

Attending a Greek tragedy at the ancient Epldaurus illustrates that its designers must experimented with the mathematics of acoustics

designing and locating

spectator pin

to

Cathedral, Vatican.

opposite side.

before

whispers

this

seated in the farthest

the center of the

phenomenal outdoor theater.

row can

easily

hear an actor

A

drop a

stage. Specific mathematical shapes

are


used to

185

design sound reflectors which are hung from the ceilings of iall

symp

focus

For two parabolas situated

as

shown, sound originating

off the parabolic ceiling, travels parallel to the other focus point

at

a

focal point bounces

opposite ceiling where it bounces

diagram is a schematic design of the ancient Greek amphitheater ISpidaurus, Greece. The photograph below is of the amphitheater today. Both the design and ideal location enhance its acoustics.

The above at

to the

186


(south of the Rotunda

in the United States

reflects the conversations of Individuals the

parabolas.

Two

from the two focal

people

can

carry

standing at

on a

points, undisturbed

Capitol building)

by

focal

points of

normal conversation the noise level of the

hall.

As

we see,

finding ideal points for emitting and receiving sounds is Acoustics and sound are linked directly to

not coincidental.

mathematical shown to be

objects

and ideas.

simple periodic

Sound

waves

themselves

were

sine functions in the 19th

mathematician John Fourier.

He

century by proved pitch, loudness and

quality of sound were related respectively amplitude and shape of sinusoidal functions.

to

the

frequency,

recently, QSound has been invented by Danny Lowe and John Lees, recording engineers and mathematicians. QSound And most

produces sound

in multi-dimensions. Unlike stereo which has

different sounds coming from different at you from all directions.

3-dimensions. that

It

requires

You

no new

speakers, QSound

literally

hear the

recording

equipment other than

CD that has been

comes

a

imaged with QSound 1.

plays

a

cassette

With such

a

recording just position yourself and your speakers

or

in

stereo

Illustrated, and let the mathematics of sound do the rest.

as


1As of February 1991, Madonna's The Immaculate Collection and Sting's The Soul Cages were the only commercially available QSound albums. Future releases by Janet Jackson, Paula Abdul, Bon Jovi, Europe, Richie Zambora, Stevie Nicks, Julian Lennon, Freddie Jackson, and Wilson Phillips are scheduled to use the patented process.

187

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THE COMPUTER

REVOLUTION THE COMPUTER, S1ST CENTURY'S TOOL FOR CREATIVITY

A i'OZJK AT THE Vt-^T OBSOLETE CALCULATORS /-.

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COMPUTERS ARE IN OUR TREES MATHEMATICS BECOMES A PRIVATE EYE WHAT'S MY SECRET? PICKING OUT PRIMES

CRYPTOGRAPHY, ANARCHY, CYBERPUNKS, & REMAILERS

COMPUTERS, IRRIGATION & WATER CONSERVATION COMPUTERS FIGHT FOREST FIRES j-. loo J'. A.r-j'rJi F'jruj'.F. CYBERSPACE/VIRTUAL REALITY

HYPERTEXT LITTLE FERMAT COMPUTERS &A-LIFE

OPTICAL COMPUTERS FUZZY LOGIC & COMPUTERS

190


To

err

is human, but to

requires

Whether

are

we

ready

not

or

or

a

whether

the tool of the 21st

computer

is

impacting

all facets of our lives both

has

sped up the gears of change.

can

be the pen, pencil,

teller it

and

architect the

professor

a

drafting table,

computer or

bank

For the artist

keyboard.

laboratory,

a

—a

for the

engineer design tools, for the

research tool, for the librarian an

card

and

catalog incredibly fast a

and

planned, the computer becomes the scapegoat, and

as

told— "it lost your file," "the

system is down," "there

seems

we

to be

Nevertheless, since they have touched almost every

glitch."

a

It

calculator/computer. However, when things don't go

accurate

are

the

For the accountant

reference desk, for the mathematician

exactly

not, the

negatively.

today's writers

scientist

for the

or

The computer is and

positively

typewriter.

palette, for the

like it

we

century.

For

be the calculator and

can

paintbrush

or

really foul things up computer. —Anonymous

aspect of our daily lives, we are now dependent on computers. BASIL LORAL FAC! LOT

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THE COMPUTER REVOLUTION

191

& i©@K &1T ™^ P&S1T OBSOLETE •The ten digits our

ofour hands

CALCULATORS

were

^

earliest counting device.

•The Chinese devised box to

use

This box

a

II

with their rod numerals.

was

used to write systems

of equations. •The abacus

was

used in calculation

by many cultures including the Chinese, the Greeks, the Romans, and the Japanese. •The Incas used the knotted quipu

as

their accounting device.

•Napiefs rods Napier in the

were

invented

by John

1600s to aid

computation. •The slide rule 1620

was

invented around

by Edmound Gunter.

•Thefirst adding machine was invented

by Blaise Pascal in

And in 1673,

Gothfired

Leibniz invented

multiply •In the

one

1642.

Wilheim

von

that could also

and divide.

early

1800s Charles

Babbage's designs and work on the difference and analytical engine furnished thefoundationfor the modern computer.

1 iiiii

segmented

ill

III in

T JL ii ¥ 1 T

II 1

—

zs

mi


191

A D D 1 T 1 O N.

Matter,

This illustration Is

be eaficft toap in tljis arte ,»3 to acts but ttoo fummes at ones togrttjee: J>ott> be it, pou marc a&oc mc:e,as J toil tci rouanont* ttyctcfoit toxemic pou topllc BDDe ttoo fortunes,pou fl)flll fpitte fct Dottmc

^Y *•-*-

a

re-

productton/rom the early English arithmetic book The Orounde of Artes, In by Robert Recorde. addition to the lines representing Os, 10s, 100s, etc, the places between these lines also were used to represent 5s, 50s, 500s following the Roman numerals. The V on the line was initially used to mark the 1000s but later it was line, used to indicate a comma when writing such numbers as 23,650. When

of tbem^tfoicet¬tolncbc.anDttjcti it Dzato a Ipnc croffe tbc otber Ipjtte.tf no aftcttt>«t>c I'ettcDotincHKorbcr famine, fo one

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five

counters

accumulated on a line those beads would be removed and tDc oic to tijc otber m tlje famr place. 02 els one counter would be a :icto carried to the above the place: to&t'cl) \faap,bpraufc if xa molt plpiica space. Hence, possible origin of the term "carry". In addition to writing this book, Robert Recorde (circa 1510-1558) introduced the symbol "=" for equality, wrote an important algebra text book called The Whetstone qfWitte and the geometry book. Pathway to

•poumapafcortl):mborfjcro&itlKcin

Knowledge. 9

3

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9 3

9

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4

T

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These tables are

from

a

book that

*

I z

9

5

r/\o/

was

4 ¦

Treviso,

Italy.

o/|o/ o/ 9 /9I/5 /4 < 1

6

9

5

4

5

V i\

multiplying 934 and 314.

4 6

R9|\5 i

oX|o\ [\7l\9

*

*

0omms-

*

9

S

It

shows four methods for

T7\T7 X X

printed

in 1478 in

/

193


NAPIER'S CHESS BOARD The

uses

only

the

key

to

with

communicating electronic

CALCULATOR

two),

Os and 1 s to write

held

numbers,

its

(base

binary system

which

computers

since Os

—

-'

and Is could indicate the "ofi"

position of

and "on" The

electricity.

famous

Scottish

mathematician John

Napier

(1550-1617) utilized the advent of is

electricity.

Napier

for

known

best

74

two before the

concepts of base

99

46

rods (or

Napier's based

on

used

were

,

known

bones)

•

His

logarithms.

calculating rods,

• •

• ••

m

revolutionizing computation by his invention of

m

• ••

as

128 64 32

16

8

4

2

1

were

logarithms

and

1"

¦

by merchants to

j

perform multiplication, division, and could be used to

find square roots and cube roots. Less well known is his

chess board method of

calculating. Although he did use

binary

not

notation to write

numbers, the board does illustrate

how

he

expressed

the numbers in base two. For

example,

to add 74

+

99+ 46,

sum

%•

* •m •

128 64 32

16

8

4

2

1

sum's final form 128 64 32

16

8

4

2

1

each number is written out in a row

of the chess board

by

194


placing sum

markers in the appropriate squares of the

of the markers values (indicated

the number

64+8+2=74. After each number is

on

the bottom

one one

added

are

row.

row so

the bottom

expressed

by gathering

Two markers

on

the chess board,

the markers'

sharing

the

vertically down

same

square

marker to their immediate left. So two "2" markers, "4" marker.

sharing in the

the

same

adjacent

square will have

from

Working square

that the

line) total

74 has markers at 64, 8 and 2 since

they represent.

the numbers

along

are

right

equal

produce

to left, any two markers

removed and

marker

replaced by one

square at the left. At the end of this process, more

than

one

marker. The

the remaining markers represents the

sum

sum

no

of the values of

of the numbers.

m

M

:¦-:

*

¦

.'.

i'¦}¦•£] *••'%;('

*> »

•

14 -

'
v-*.

.¦

tBl ••

',:

:

•".:

'

.'*V'

128 64 32

•

:

¦•

v'

16

.:•••.

:•¦

• • • •• 8

4

32

¦'

•y-t

ffl* ffl v&l

¦¦¦¦¦¦¦¦:

'"

•A

^R Q • {'/*•

';••

$•;

64

16

¦/''h:

'•;•¦:.'

128

2

8 4

2 1

1

multiply using Napier's chess board calculator, the number representations along the horizontal and vertical column are used.

To


:¦"

128

'.

-

128

•

.

.........

64

m

:".

64

•

32

'

:¦

32 ..........

16

€£&£* '*•¦•

4

8

8

i

t$

4

#

2

•

1

'

we

want to

with markers column

intersecting with

a

simply

.¦

diagonally

1

2

1

128+32+4+1=165

multiply

right.

Then

square where

15 xl 1. One number is

a

a row

by sliding

the

to the vertical

row.

markers occupy the

marker is

2

•

4

8

new

marker is

with

a

marker. After this is done, the done

4

•

¦*..

expressed

along the bottom row and the other along the vertical

the

at

8

':¦¦¦}

v

15x11=165 Suppose

¦

'-"

..

1

2

16

'.:-\

placed

same

marker meets

multiplication

markers

from

the

square,

they

are

product

of 15 xl 1.

a

at

the

column

process is

bottom

As with addition, any

place

removed and

in the square above them. The

column will represent the

placed

row

two one

resulting vertical

195

196


A [L©©K AIT THE PRIES (HOT COMPUTERS ARE IN lfis

rtiiRTRCCc

OU R TRE E>

unwortr)y of excellent men

fQ lQse hour$ jjke slQves jn fhe '

labor of calculation which could safely be relegated to anyone else if machines used. —Gottfried Wilhelm

"I'm

dinosaur." "I refuse to have

a

can't understand how one."

"They're

so

a

person

Impersonal."

many times have you heard statements ?

But it must be

can

our

lives easier in

Invading before The

our

privacy.

networks

or

In

Now

computers.

growth

some

or

being

spend

are

here to

made

fact, it

be

even

In

extensively

urban forests.

can

computers. Washington

days

has

a

be used to

to either

computers

out

put

are now

Many cities, in efforts

to preserve and maintain the health of their trees

D.C.

others, and

patterned using mathematical

thereby develop strategies log

some

making

are

trees!

our

drawn using fractals. Computers

to

in

How

is difficult to remember the

create controlled burns. In addition,

used

of how

stay. They

"I

of these

some

complicated

more

ways,

can

computer Illiterate."

perhaps

computers are

of trees

model forest fires, and fires

to do with them."

much time In front of

so

accepted, regardless

people may feel, that computers

Leibnitz

anything

"I'm just or

von

were

are

resorting

to

database of about 109,000

street trees, while Paris, France also has its 100,000 trees

on

computer. What kind of information is entered? Each city usually decides what data is essential for its

particular situation.

database Includes—the tree's location (each tree is

numbered curb and

along

consecutively

the street, and its distance from the

adjacent

trees

are

The Paris

building,

noted), its vital statistics (species,

sex,


age, trunk size, statistics

height), type

(includes

mulching),

of

type,

health

tree's

environmental

soil

pruning received, environmental

planter dimensions, drainage,

pollutants

are

an

diseases

and effects. The initial

information is very time consuming, Since trees

and

(status

as

is the

noted),

logging

of such

updating

of flies.

invaluable asset to any community, both

aesthetically and environmentally, the

time invested

and

long

range outcome make such databases worthwhile. In San Francisco, California, the city

gardeners

using laptop computers.

Washington D.C.,

Paris

maintain the

followed

in which the Initial

a

city's inventory

method

inventorying

was

used in done

by

engineers using hand-held computers. Without

junior forestry computers, such sophisticated

databases would not have been

197

198


possible. Storage and filing space would have been too formidable, let alone the necessary updating,

As the

computer evolved, computer functions slowly Invaded

aspects of

lives.

our

wondered how and

searching and sorting of data.

they

Inca scribes

Inventory

of the

without the qulpu,

Even

as

early

merchants and navigators,

had functioned without

must have

population so today we

ISBN

a

set of

Napier rods,

speculated how they

and are

products at

a

had

kept

of the Inca empire

similar point.

D-T3317M-A7-X 51495

9"780933"1 74870"

Bar codes for

labeling & tracking such things trees, groceries and packages.

As In

we see

with the urban forest,

storing, processing,

devices used

by

are

by

computers play an Incredible role

retrieving data. Today,

are

data

Inventory,

and

fewer

civilization will be bulb.

are

tracking.

Invaluable In the sciences for analyzing, comparing

computing Information, thereby saving tremendous

of time with

Imput

restaurants to process orders, bar codes

stores and businesses for sales.

Computers and

used

and

as

as

errors.

great

This

as was

evolving tool's the

Impact

amounts

Influence

of the electric

on

light


199

MATHEMATICS BECOMES Mathematical wavelets «

l...

.

i.

,

are

a

new

fighting technique

being

used in the United States.

Mathematicians

have

wavelets

to process

developed

fashion. The method divides then isolates the

reconstruct the

Consequently,

images an

in

image

image

a

quicker

into basic

more

accurate

components. It

important and relevant information needed

to

image and discards the non-essential information.

wavelets reduce the amount of data necessary to

store, transmit and retrieve the image. saves

~-..WA__

This data compression

time and money. In addition, the wavelets do not distort the as

other methods do. This allows the FBI to

process, identify and match the

image

-w»

A PRIVATE EYE

j.i

and tool

crime

analysis

.

of a person's

more

efficiently

fingerprints.

200


WHAT'S MY SECRET? Governments, revolutionaries and financiers have

needed to

always

send and receive top secrets. Many methods of

messages have been devised the Spartan

over

scytale1, Polybius'

coding

the centuries.

decoding

Among these

cipher2,

square

letter substitution, Thomas Jefferson's

and

are

Julius Caesar's

cipher wheel, and the

\F \0\ \U\M\0\P\L\J\G\D\S^ K P xlv\ B M) QZXJ Yl ^

VP\I

\Q\Z\

A s F E\ Ul Gl HI J WE M U s T D E T E RM 1

\L\T

Wl H N E WH 1 C H WA Y

G F D s A H J K M N B V C X L W E Y T R UO PI A R F F II ft H .1

P Y V R M n 7 y .i Y R W 1

/j/m/k/l/0/

/A/B/N/l

portion of Thomas Jefferson's cipher wheel. It was made with 36 wooden wheels of the same dimension. Each wheel had the letters of the alphabet printed In various arrangements. A message would be lined up along the metal cross-bar. Then the sender would write the letters of any other horizontal line and send it off. The recipient would line up these nonsense letters along their horizontal cross-bar, and look around the cylinder until a line of letters was found that made sense.

A

German

military code Enigma.

literature where

are

also many

examples

cryptography is important to the plot,

Allan Poe's The Gold

Doyle's

There

Bug,

Jules Verne's La

as

in

Jangada,

Edgar Conan

twenty years the encryption formula, DES,

has been

used

by banks, government agencies and private companies to

secure

information.

which

in

The Adventure of the Dancing Men.

For the past

theoretically

This method

uses

56 bits of

computer data

would take 200 years to decode using

computer. Intelligence

and law enforcement

H

agencies

a

super

fear that the


present methods not

are

adequate and

intercept communications from

criminals, terrorist

foreign

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governments.

Consequently, new

53

;806*

would not be able to

and

Is

system

The translation is: A good glass in the bishop's hostel in the devil's seat twenty-one degrees and thirteen minutes northeast and by north main branch seventh limb east side shoot from the left eye of the death's-head a bee one from the tree through the shot

being

formula using 80

fifty feet out

bits that would

approach

take

than

billion years to crack. This

new

special microchip which would be inserted

communications.

Agency.

one

in

satellites, fax machines, modems etc. to scramble

mathematical

coded

more

uses a

telephones, secret

The substitution cipher in Edgar Allan Poe's The Gold Bug.

a

proposed with a new

theoretically

In

formulas

The

would

chip

developed by

the

use

classified

National

Security

addition, mathematical keys that unscramble the

data would

determined

be

kept by the

FBI

and

other agencies

by Attorney General of the United States. The manufacturing process of the chip makes it nearly impossible to take

the

apart and decode. There

are

people,

the Electronic Frontier Foundation

such

technology

will not and should not

as

Mitchell

Kapor of

(a public policy group

Washington D.C.), who say "A system based American

on

in

classified secret

gain the confidence of the

public"3

* Plutarch describes how messages were exchanged by winding a narrow band of parchment or leather in a spiral manner around a cylindrical shaft. A message would then be written, and the band removed and sent to the recipient who had a cylindrical shaft of the exact same size to wind the band around and decipher the message.

2(Circa200

201

B.C.-118 B.C.) the Greek Polybius devised a substitution around a 5x5 square, which converted each letter of the two digit number using only the digits for 1 through 5.

cipher designed alphabet into a

3Secnet Phone Scheme Chronicle, May, 1993.

Under Fire,

by

Don

Clark,

San

Francisco

lOl


Ol en

PICKING OUT

o

PRIMES

One of the earliest methods _

2

_

^ #% #% ^%

—/ OOOwV

by

the

Greek

and

identifying prime

given number. Ever

some

mathematicians have been

In

was

math-

devising

for

new means

since

then

seeking

out

Fn=22 Fermat

numbers

were

prime. This

But the

(for n=0,1,2,3 and 4). 1880

the Fermat

274177 times 67280421310721.

(F22)

shown to

was

Mersenne wrote

would

be

an

yield primes.

^

following century

J*?

times

^{

factored

into

^ft

no

other Fermat numbers

%J\ ^|

(F5) was

In 1644, the French monk Marin ***

expression— 2P-1, with

But not all the numbers

p

a

prime number— that

Jj^J

this expression ^^ 33 known Mersenne primes. Working with ^^

prime. There are currently expressions (such as those for numbers,

m

00 00

of 1993 the 22nd Femat number

as

composite.J

are

Carmichael

00

into 641

number.Fg,

Currently

have been found to be prime, and

^1

the first five

is true for

Leonhard Euler factored the 6th Fermat number Then in

XJ

£?

numbers.

(for n=0, 1, 2,3,...)

+1

ro qq ro

(275-194 ^

1640 Pierre de Fermat claimed that all the numbers of the form

6700417.

23 37

devised

numbers

numerical sieve that would eliminate multl-

a

of numbers up to

pies

11

prime

ematiclan Eratosthenes

B.C.). He created

5

finding

for

and

Mersenne

produced by

numbers, Fermat numbers,

number

others),

theory,

and

computer

^^ Ol

programming techniques, mathematicians using either supercomputers ^* or

strings of rewed-up personal computers seek

prime numbers

—

and look for

quirky

WHY ALL THE INTEREST IN PRIMES? —

setting

new

new

records

—

J*J

patterns.

QQ

-

mathematical curiosity

QQ

or

testing the efficiency and hardware of CO CO

computers— using primes numbers

641 for encoding confidential

larger and larger

out

characteristics

material

to form

multidigit

numbers XT

Today's computers, along

with

ingenious programming and problem solving have made

possible

to find

literally

enormous

Projects inSctentific Computation by Richard Santa Clara, CA 1994.

2859433-1.2

2Found by

CO

©* Ol ^J

E. Crandall.

Springer-Verlag, Crandall.

Q*

prime numbers. WHATS THE ^

CURRENTLY LARGEST EXPLICIT PRIME? 1

It

David Slowinski in 1993. Verified at his request

by

Richard


lOJ

CRYPTOGRAPHY, Ever wonder,

much

you receive

so

catalogues

you

offers for

name

and

the

&> REMAILER$

—

requested, How

"super" buys?

your

mail

junk

never

ANARCHY, CYBERPUNKS,

how it is

irritatedly,

were

and address accessed? Welcome to the electronic age

loss

electronic

of

data

Most

privacy. about

them,

individuals have considerable

which

can

be

accessed

by

a

computer/mode m user a

to create

profile

based

on

you have

of you

things bought,

\

your travel

tickets, your medical records, your traffic

tickets,

Although the loss of privacy was created by today's technological methods for storing and accessing information, perhaps there Is a way to restore that privacy by outstanding loans,

etc.

new

using these have

come

same

to the

the individual,

methods. rescue.

Cyberpunks,

They

advocate

as

they call themselves,

restoring the privacy of

by using sophisticated cryptographic methods to and hinder Its ready availability.

encode one's Information,

Naturally

there

Some feel that to

right

are

pros and

government

eavesdrop,

and

obstacles. Remailer is

using

advanced

cons

to this

cyberpunks

an

cryptographic anarchy.

and law enforcement new

example of

cryptography

method developed

allows

one

information via modem without the sender.

right

use

to

send

to trace

leaving any "footprints" Many feel that if governments can use methods

devices to encode important information, to

similar methods to Insure

a

methods would present

one new

which

agencies have

an

and

individual also has

personal privacy.

a

204


COMPUTERS, IRRIGATION &

WATER

It Is eerie to observe lush crops

CONSERVATION

growing

from

cracked earth. use

drip irrigation, water, fertilizer delivered

Irrigation,

directly

raisin

cut his water

farm while

of

now

with the

computerized subsurface

and

to the roots of

occasionally pesticides can plants. With subsurface drip

be

grape farmer Lee Simpson of Fresno, California has

consumption

cutting back the

In half and doubled the use

of

project

cotton field of Harris Farms, also In Fresno,

of cotton per

acre as

yield

pesticides to one-sixth.

California Department of Water Resource test

pounds

sometimes

dry, But

opposed

to 930

on a

of his A

160-acre

produced

pounds per

previous years, and used 7 inches less water per

acre

acre.

1600

from

Claude


cutting edge of this

Phene has been at the

(drip irrigation Phene had been

by

has

actually

advocating

conserving water and

produced

make

sense

100 tons per

to flood

a room

claim that California

supply, with

one

to water

26 tons per acre,

potted plant.

a

uses

His statistics

85% of the state's water

computerized drip

acres

would be

irrigation1

enough

The method also reduces the

weeds do not grow

as

readily between

use

water to

were

used

supply

on

Los

of herbicides, since

the furrows because

plants

watered at the roots and not flooded. In addition, the amounts

of fertilizers and pesticides used would be cut His

some

his tomato

on

crop— e.g. cotton— the 6 inches of water saved

each of the 1.4 million

Angeles.

improving yields

Phene points out that It does not

agriculture

and If subsurface

just

formerly produced

acre.

years).

not until 1987 that

was

Phene tested his methods

notice.

crops in California. Acres that now

are

been around for about 20

this method for years, but it

farmers took

type of farming

"new"

by

computerized techniques allowed his operation

interaction

applications

between and

different

their affects

amount of fertilizer for

nutrients, on

crops

factors from soil texture to

to discover the

timing

yields,

optimizing plant quality,

than half.

more

of the

and

nutrient

optimum numerous

rooting characteristics.

irrigated by underground pipes placed 18 inches below the The very sensitive lysimeters detect morning dew, measure amounts of water that plants absorb and emit, measure surface weather conditions. The computer gathers and analyzes data, and delivers minute amounts of water throughout the day and occasionally fertilizer, herbicides, and pesticides. Without investing in a computer system, small farms could monitor their fields via modems at a centralized computer location. Crops

surface.

are

205

206


COMPUTERS FICHT FOREST FIRES

Today's computer modeling is a very powerful tool, used by sclentlsts and professionals across a spectrum of fields.

by

economists to

and

predict

theory

to

the

predicts spread of

describe

heart

sociologists along with seems

almost

a

cycles, by contagious disease

arrythmias.

It

statistics to observe

unending.

It can be used

doctors to monitor

economic

a

has

or

with chaos

been

used

by

social trend. The list


Not too

long ago firefighters'

poles,

pike

chemicals.

chain

ropes,

Today

computers

tools

some

were

firefighters

complete with personal computers. people

and

fireproof blankets,

saws,

are

tent/laboratory

or even a

also

equipped

axes,

water,

with

laptop

in the middle of the forest

In addition to

the computers

supplies,

protective garments,

are

keeping track of

also used for forest fire

analysis. While

for

working

the

United

Patricia Andrews

conditions (wind

burning

trees or

the computer.

Service's

Inter-

in Montana, mathematician

Laboratory developed the program

The

Behave in 1984.

typography speed and direction, dryness, etc.),

the climate the

types

grasses and much

are

fed into

how the fire

"might"

of the fire, the

location

Forest

States

mountain Fire Sciences

of the area,

The program then

more

information

predicts

of

behave, thereby helping fire managers decide how to best fight the blaze.

it cannot

Naturally,

Behave

can

comprehensive

be

predict

continually

as

new

all

possible outcomes,

modified

predicaments

occur,

Yellowstone "crown fires" in which the fires

treetop. Effects forest

high

is

a

management

tech

spread

now

become such from

being

tools

Spanish,

are now

being used

in

but

more as

the

treetop

used to

with decisions for controlled burns.

firefighting

been translated into

by Italy.

companion program

to

to

help

These

China, have

and information has been

requested

}07

208


A tL@®K AY TMd $mU \R[ CYBERSPACE

It is the

perennial youthfulness of mathematics itself

VIRTUAL REALITY

which marks it off with

a

disconcerting immortality from the other sciences.

Temple Bell

—Eric In the 1500s

people

obscura. Actual room were

devices

were

moving

projected on

were

needed to

treated to the marvels of the

scenes

from

things taking place

the wall of a darkened

bring

these

scenes

optical

about how

physical

our

Physicists

minds

were

and

distortions which tricked Some of their

our

No electronic

room.

Then,

in

surge of interest in

psychologists

studied and wrote

tricked

by

structure of the eye and the

process information from the eye

a

a

the second half of the 19th century, there was illusions.

room.

into the

camera

outside

were

what

we

The

perceived.

analysis of how

our

minds

studied in efforts to explain

minds into believing

they existed.

findings were—

(1) The placement of certain angles and segments eyes inward and outward and

thereby make

an

can

lead

object

our

appear

to be shorter or longer.

I

(2) Horizontal objects have

a

tendency

appear shorter because the retinas eyes

are

curved.

to

of our


(3) A light region's

image

radiates into the dark

on

the retina

region's image thus

making the dark region appear smaller. Identical

4)

-Egg

c ,-Jwr

different

In

objects

locations in

a

I

perspective drawing

appear to be different in size.

y

(5) If

an

interpreted

/ way, then

our

can

in more than

be one

mind oscillates the image back

and forth between .V

image

A

interpretations.

(6) Diagonal

/

segments

on

parallel lines make them appear non-parallel.

/

(7) An empty space and

\

/

an

identicallyfilled space appear

/ s

/

different in size. (8) A diagonal broken by

/

f

a

vertical bar does not appear to be

aligned.

Now, In the 20th century, computer scientists, mathematicians and Inventors

optical worlds.

are

illusions to

carrying optics, computer technology and new

Equipped with

longer simply

an

heights with the creation of artificial paraphernalia, the observer is no

various

observer— but

actually

enters worlds created

by the computer. In these artificial worlds, one

—

an

gold —

one can

be

a

participant.

For

example,

may choose to be:

"Olympic

"

runner

experiencing

the thrill of running for the

metal

an

air traffic controller

directing planes flying

around In the

3-dimensions —

a

weather forecaster

flying

around the world

experiencing first

}09

210


hand weather

conditions

that

have

been

programmed and

processed into the computer. -

-

an

an

atom about to be bonded with other atoms into

architect viewing his

walking through

Virtual

or

and

suddenly

find

a

small

oneself

room

design by actually by room.

artificial reality

expressions coined to describe this In the confines of

molecule

her most recent

the computerized image

reality, cyberspace,

a

new

room one can

form of don

are a

few of the

optical illusions.

computer gear, 1 and

walking English countryside, project's progress thousands of miles away, or learning how a bee collects pollen by becoming the bee. The inspecting

a

in

the


r Computer generated living room awaiting a visitor to virtual reality Photograph courtesy of Autodesk, Inc. Sausalito, California.

applications

of such

still In Its Initial

technology

are

mind

stages with much

boggling. Cyberspace Is yet to be developed and

refined2. Hopefully Its evolution and popularity will not means

of mind control, but rather of mind expansion.

scientists

and

mathematicians

are

breaking

new

serve as a

Computer

ground

In

computer graphics, especially by using fractal geometry

to create

these

The Next

special effects. Perhaps the holodeck of Star Trek—

Generation Is not

so

far fetched.

The computer gear comes in many forms which gear, data gloves, data body suit.

can

include

special

eye

o

A few of the universities and businesses involved in these artificial worlds are University of North Carolina; Autodesk, Inc. of Sausalito, CA; HITL at the University of Washington; VPL Research Co. of Redwood City, CA; Carnegie Mellon University. Connecticut State Museum of Natural History has an installation created by Myron Krueger called Video Place. At U.C. Berkeley's ASUC, one can play Dactyl Nightmare, a virtual reality game.

211

212


HYPERTEXT Speculations

on

the fourth

dimension surfaced In the 19th

when a

made Into

through

a

shadow of

left handed shadow

lead

would

to

they

are

not

are

Idea to another

analogous

directly

interactive

For

example,

brings before you

with Interactive

information

on

learn

day

present

and how It that

were

It

to

jump from

be

can

thought

one

of

as

The

computer transports them

computer immediately topic, which may include a

video.

Suppose you

of World War II. The

newsreels from that

popular during

as

hypertext. Though

ability

historical Information, maps

proceeded,

terms

computing, after you decide

that

even a

about V-E

now

view, the

sound tract, graphics and

something

one's hand

realize that the term

related to the 4th-dlmenslon,

to go and the

or

hand could be

dimension to another.

one

what you want to do, read

able to

and

computing.

moving from

to

participants decide where there.

one

related to the computer and Its —

right

generation of such

a

hyperspace, hyperbelng, hypercard2 the last two terms

a

simply by passing

Little did

the third dimension.

hypercube1

century August Mdblus noticed that

showing

period,

that time. You choose

want to

computer

Is

the Invasion

and

even

songs

topics by clicking

your choice3.

Hypertext draws active

on

participation.

in the traditional

tool to

explore

evolves. With

clicking a

on

Interactive

manner.

other

even more

beginning to

Instead, the computer functions

possible

hypertext

highlighted

computing by allowing

A hypertext is not read from

outcomes, and

thereby

the reader spins his/her

Ideas that

are

own

interconnected in

a new

tale

a

end as a

story

by

story called

web, which has been spun into the original program. While

reading the story on the computer, you which you want the story line to

storyline,

but rather choose

can

move.

choose the direction in You do not create the

options along the way, and

see

what


they unfold. location,

A click

a new

key word

on a

idea

or

or even a new

how the author unfolded the story

image may take you to

plot line.

along

the

t

paths

are

able to

see

you chose. The ^

reactions to interactive literature

curiosity that in its initial

You

a new

is far more

stages,

are

interesting

and thus far

mixed. Some feel it is in

once a

a

theory than practice. story

novel It is

has been "altered"

by

the reader, it cannot be undone. In addition, it may also be difficult for the novice to tell when the story is finished, since there may be

key words

the reader to

or

phrases available throughout the web

continually go off to something new or perhaps even around in a circle by mistake. The web stories may require time and/or practice for inexperienced hypertext readers allowing

213

214


to orient themselves to this

programming

and

new

writing techniques

refined if it is to work

professor George P. potential to be

for this

new

explore

Landow of Brown

its

As

potentials.

makes the reader

a

anarchy,

or

is it

a new

kind of creator?"* There way of

is

art form.

Most of the focus thus far has been

a new

the direction of the

question

reading form that

are

many who feel

writing, and

hypertext

introducing

English

University says,"It really has

the next wave of storytelling...The next

is, is this total chaos and

helps shape

Improved

media need to be

and if the reader is to have

optimally

unencumbered freedom to

the

form of reading.

on

consider it

an

the reader who

But consider what

story. story writing entails for the author. The writer does not develop a single plot line, but instead a family of possible plot lines now

such

and their relative outcomes, called the web.

the web

computing,

sound and video.

literature, released

or

thing

modern

It is too

be enhanced

to follow the outcome

is certain,

As with interactive

by computer graphics,

early to pass judgement on interactive the hypertext stories are just being

especially since posted on electronic

fascinating One

can

bulletin boards, but it will be .

Will it catch

on as a new

craze?

hypertext would not exist if it were not for the

computer.

1

Hypercube

2

HyperCard

is the term coined for

a

4th-dimenslonal cube.

has two meanings. It can be the trade name for a computer program, which allows you to access and create stacks of cards with information In various forms (Including graphics, video, sound) simply by clicking on a particular topic. HyperCard can also be a card (which Is 2dimenslonal) that has been cut and folded Into a 3-dimenslonal model, similar to the book illustrated on the computer screen. It Is hard for one to pass up trying to recreate a hypercard from a 3x5 file card. 3 It

easy way to pursue learning. What are some drawbacks? depend on the material that Is accumulated in that particular piece of software. Sources other than those on that program If you are using the program for more than just general are not at hand. knowledge, you have to be careful not to rely on this as your sole

You

seems an

have

to

research material. Hopefully this form of available information will not hinder creativity or the ambition to do additional research, but rather stimulate It. 4 A New

CA,

April

Way 1993.

To Tell Stories,

San Francisco Chronicle, San Francisco,


215

LITTLE FERMAT gets the feeling that the

One often

layperson

now

feels that computers

have evolved to their ultimate state. but

scientists,

computer

mathematicians and scientists know this Isn't

Charles

Babbage, M.M.(Monty) and

Chudnovsky

Saed

the

to

handle errors

Using ideas

computers.

spirit of

Denneau, George V. and David V.

Younls

number

computer designed without

G.

In the

so.

created

Little

Fermat,

gigantic computation problems with

associated

from

number

conventional

theory—specifically

modular arithmetic and Fermat numbers—the computer carry out

computations virtually

error

free.

called Younls.

Little

programmed

in

numbers

divisors in modular arithmetic, It

as

a

language

certain kinds of calculations and avoid the

Thus far Little

Fermat is

creators feel It is

a

a

Fermat is

By utilizing use

can

can

Fermat

speed up

of real numbers.

one-of-a-kind computer, but Its

digital signal and Imaging Ideally the solving of various problems from processes hydrodynamics, chemistry and aerodynamics that require differential suited for

and

it will

equations.

Furthermore, they hope

enhance the

performance of the supercomputers.

serve

as a

model to

216


COMPUTERS & A-LIFE

With a

at such

technology changing rate and with

rapid

new

ideas

'

and

applications

springing-up

overnight, computers Impacting or

Many scientific laboratories

not.

utilizing Alije,

every aspect of our lives whether we

its

as

modeling

seem

the

are now

computer,

a

way to simulate lifelike

forms, their behavior, reproduction and evolution.

£/

example generated bats

Utilizing logic, living thing,

of Its

uses

the

were

lifelike

How

in Batman Returns.

One

computer

was

it done?

mathematics and computers, the habits of such

These

steps.

of it

and simulation abilities.

It is often referred to, Is

recent

to be

are aware

as a

bat,

steps

analyzed

are

developed

are

Into basic

into

a

logical

computer

a

simulation which captures the movements and habits of the life form. The

applications

are

far

One

reaching.

example

Is

the research in disease done at Scripp Research Institute. Here Gerald a

Joyce

uses

laboratory procedure to how

^v

trial

^^î

on i

\9o ^ YW"? CE?

and

designed by predicting

these

develop

they

not

their

own

solve certain

economic

explore

enzymes

only

information,

a

cure.

programs based

problems. Programs

this method include

that

according

genetic code of an AIDS virus. By

steering

a

vehicle,

outcomes,predicting planetary

Computer pioneer, John

contended

could

in the

can

how well

motion.

in which enzymes evolve

error,

Computers

^*

vA

they slice

computer simulations and develops

von

Neuman,

machines process

but

j,

^^L^ J^

"VVV

^r9^

%}


reproduce themselves. Some

proponents of A-life feel that

the essence of life is

a

set of

rules directing the interaction

of cells, atoms,

molecules, etc. A close connection between A-life and fractal

mathematics is also

developing.

For example,

computer simulations have been

developed of The robot used at Stanford University Hospital (Stanford, California] to carry records back and forth. Photograph courtesy of Stanford University Medical Center,

plant cells that grow and divide

Visual Arts Services.

according to a set of instructions. One such grew into cell formations almost

identical to those of a fern. ants

Studying how different insects such as

navigate, simulations of miniature robots have been created.

These robots when

they

are

come

instructed to

to

an

obstacle

move

they

and

change

explore Mars.

3.6

pounds

and will

It comes to

finding designed by MITs hopefully be used to

their way around obstacles. In fact, Atttila,

weighs

only

cannot traverse. Such robots

outperform the traditional larger designs when Mobot Lab,

direction

217

218


OPTICAL COMPUTERS

I

Our

present computers

electricity. working new

But

on

type

some

the

run

on

scientists

are

of

development

a

of computer that is made

up of optical fibers, lasers and switches that process data and

perform calculations via electronic

computers

light1

rather than

in which data and

electricity. Unlike are

programs

stored either

memory

drives,

on

chips, hard or

disks the

optical computer circulates data

light

as

pulses through the optical fibers.

"For the

first time,

have

we

a

computer in which the program and data are

always on the move in the form of light,

eliminating for static

Harry

Optoelectronic Computing Systems Colorado at Boulder, Colorado. serve

data

the

as are

computers

encoded in

thus

far

light pulses

of Colorado. The

does

components of

says

Center at the

University of

Over three miles of

optic fiber

and circulate the

head the

optical fibers.

development

at the

computer that they have developed

"demonstrates a

storage,"

F. Jordan of the

main memory, where instructions and

Jordan and Vincent P. Heuring

University

need

the

the

general-purpose

principle machine

that can

of

the

done

in

all

be

optics."2 1 The

only

light Is not used occurs when the optical switches are light pulses momentarily converted to electrical current.

time

activated and

2Harry Jordan,

quoted

from Science News

,

January 23,

1993.

HO


term

"fuzzy" logic was coined in the 1960s by Berkeley professor paradoxes can probably be blamed for the birth

Lofti Zadeh, but of this

glitches to

new

Over the centuries

logic.

in traditional

explain such paradoxes

Bertrand Russell's class

The Mitsubishi HSR-IV is

True

logic.

as

or

logic could not be used J pile of sand paradox or

Eublides'

membership

designed

to

have created

paradoxes

false

paradox.2

avoid accidents

These

paradoxes

by utilizing

I

a ni

fuzzy inference system. go hand-in-hand with be answered have its

for

room

origins

precisely

simple

gradations

Traditional

must

logic

either true

"0" of the

yes

of

a

or no.

Traditional logic does not

situation.

and scientists. Just

describe

things

Although fuzzy logic had by Eastern

and or

in

into the

as

mathematical

world,

so

objects

traditional

computer programming rely false

(electricity being

binary system).

come

our

logic

do not cannot

to the real world and real world situations.

perfectly applied

being

many real life situations which cannot

so

in the United States, it has been embraced

philosophy be

by

a

To humanize

picture. Fuzzy logic

"on"

or

on

statements

"off'

—

"1"

or

computers, fuzzy logic tries to imitate how the

human brain would work. In other words, it attempts to make


219

FUZZY LOCK &

COMPUTERS Suppose

I take

a

log

the fire in my wood It

begins

to bum

and add it to

burning

stove.

immediately.

what precise moment will it

At

longer be considered a log? One person might say right caught fire. Someone else might feel when half of it is burned away, and yet another person might contend it's a log until its last ember. no

after it

Fuzzy logic

may be inside—

washing machine, if

•your

determines

machine

your

best

the

washing cycle from water

your input of level, load size, fabric and ,

dirt and stain levels. After a wash cycle. The machine will repeat the cycle if the water is not clean enough. •

your

vacuum

automatically based

through

on

its

cleaner, if

adjust

it

can

suction

information gathered infrared

sensors.

No matter how

logical

one

tries to be

about answering this question, there is

definite

no

right

or

answer.

wrong

answer.

the

way to

quantify matter of degree. appointment

one

a

pm

for

same

set

appointment

an

3:30 pm

mean

host of

are a

depending

True

cannot

situations status of

or

or

deal

and

is

a new

ever

on

logic deals

Fuzzy logic

with

with the

the universe. These

you

with

life and

false, yes

everything

that fuzzy

case

so

for there is

life, things nature to subjective universe.

possible

whom

upon

in

many

things

5:00

A

asked. And such is the

logic

no

It is all

person, 5:30 pm to another.

In fact, there

times

There is

the

for late afternoon may to

no

answer.

means

But

everyone.

There is

a

the or

no

such

changing

earth and in

are

the

things

with.

way of

analyzing

looking

at

the world. The


artificial

intelligence real intelligence. Japanese and Korean companies are leading the way with innovations that make use

of

fuzzy logic.

Among

other

things,

air conditioners, cameras,

computers,

they

parts, televisions and washing machines, which with

fuzzy logic technology.

Show,

the

computer

in

are

In the October 1993

enhanced Motor

Tokyo

Mitshubishi's prototype HSR-IV

to imitate the information

fuzzy logic

producing

are

dishwashers, automobile

uses

processing of the driver's

brain. The computer studies the driver's normal driving habits, and then selects occur.

For

response for various situations that

a

example,

if built-in radar detects

fuzzy logic system decides if the driver on

is aware of it

past driving patterns). If the driver does

predicted,

it

can

then

automatically

an

might

obstacle, the or

not

not (based

respond

as

control the brakes to avoid

a

collision.

Up until

now

many Western scientists have been reticent to

fuzzy logic, feeling

that it threatened the

possibilities

of

of

a

particular product.

Foundation

rejected proposals

perhaps

future

especially Otis

expands the

computer programming by considering

scope of variables for the solution of

design

integrity

Other scientists feel it enhances and

thought.

projects

In the

a

a

broader

specific problem

or

the

past the National Science

that dealt with

fuzzy logic, but

will be considered in another

since more United States

use

of scientific

companies, such

as

light, Ford,

Elevator, Pacific Gas & Electric, Motorola, and General

Electric

are

expressing interests

in its

uses.

Mth century B.C. Greek philosopher Eublides considered how many grains of sand make a pile. ^Russell's paradox Bertrand Russell's ( 1872-1970} paradox deals with —

the idea of membership of a set. A set is either a member of itself or not Refer to a set which does not contain itself as a a member of itself. member as regular. For example, the set of people does not include itself as a member, since it is not a person. Refer to a set which does contain itself as a member as irregular. An example is the set of sets with more than say five elements. Is the set of all regular sets regular or irregular? If it is regular, it cannot contain itself. But it is the set of all regular sets, thus it must contain all regular sets, namely itself. If it contains itself, it is irregular. If it is irregular, it contains itself as a member, but it is

supposed

to contain

only regular

sets.

551

phosphate

T=thymine base A=adeline base an

example of

one rung from the DNA ladder

The two strands of the double helix wind in

opposite directions. This is why the sugars and the phosphates are in opposite locations on the two strands.

®

® © ^hydrogen, carbon, nitrogen,

® oxygen,

II phosphc

The DNA molecule with its genes provide the blueprint Jor the life of a cell.

MATHEMATICS & THE MYSTERIES

OF LIFE

MATHEMATIZING THE HUMAN BODY

MATHEMATICAL MODELS & CHEMISTRY MATHEMATICS & GENETIC ENGINEERING BODY MUSIC

SECRETS OF THE RENAISSANCE MAN

KNOTS IN THE MYSTERIES OF LIFE

224


only be possible to imagine life or beauty as being "strictly mathematical" if we ourselves were such infinitely capable mathematicians as to be able to formulate their characteristics in mathematics so extremely complex that we have never yet invented them.

It would

—Theordore Andrea Cook (1867-1928)

The

scientific world Is

to unravel the

mysteries

of life. This

chapter presents

Ideas to a

try

few of the

In which mathematics Is used In the search for answers to

areas

such

constantly using mathematical

questions

seems

a

Inanimate answers

as

how

mystery

In

objects

and

to these

our

bodies work and how life

Itself that

objects

questions.

of

a our

subject

began.

It

which deals with

Imagination could

hold the

MATHEMATICS S.THE MYSTERIES OF LIFE

225

MATHEMATIZINC THE HUMAN BODY blood pressure: 120/80 cholesterol: 180

LDL/HDL: 179/47

trigllcerites: glucose:

189

80

body temperature: 98.7T In

present day medicine,

numbers and ratios that are

functioning.

which

our

electrical

analyze

patients, our

bombarded with

are

health and how

bodies

our

Doctors have tried to pin point ranges of numbers

normal. Numbers and mathematics

are

fact. In

we, as

seem

pervasive.

bodies the networks of our cardiovascular

Impulses

our

bodies

use

In which cells communicate, the

molecular

structure

of

genes

to

In

the

system, trigger movements, the ways

design all

—

effort to

of

our

bones, the actual

possess

mathematical

quantify bodily functions of

elements.

Consequently,

humans,

science and medicine have resorted to numbers and

other

in

an

concepts of mathematics. For example, Instruments have

been devised to translate the sinusoidal curves, from

Readings ultrasound

—

body's

thereby making

an

curve's

outputs.

electromyogram,

shape, amplitude, phase shift.

All of

Information to the trained technician. Numbers, ratios,

this

gives

and

graphs

are

aspects of mathematics adapted

Let's consider other mathematical

linked to the

to

our

concepts and how

bodies.

they

are

body.

If you think

hieroglyphics unravel

to

Impulses

it feasible to compare

electrocardiogram,

display the

electrical

that is

deciphering codes, ciphers,

exciting and

molecular

codes

and

challenging, Imagine being

the

body

uses

Mayan able to

for communication.

226


Science has

now

discovered that white blood cells

brain. The mind and

body

Deciphering these intercellular

biochemicals.

are

linked to the

vocabulary

of

codes will have

an

communicate via

a

increased

medicine, just astounding impact understanding of genetic codes is unveiling so many ramifications as

on

discovery phenomenon. the human body.

mathematical

present many

of

areas

change

as

in the

of the double helix in DNA was another

health field. The

in

our

growth

it grows.

pattern of the hair

But the helix is not the The

equiangular spiral

possibly

—

Look for the

because its

only spiral

is

present

shape does

equiangular spiral

in the

your head, the bones of your

on

cochlea of the inner ear, the umbilical cord,

or

perhaps

in

not

growth the

body, even

your

fingerprints. The

physics and physical aspects

mathematical ideas. The to balance and

the three

a

curves

center of

illustrate

as

the

amazing

the human

as

are

important

very

to lift its

own

to

body's conformity as

it may

weight

the

various

proportions and

golden mean. in

the chaos

theory is being explored study of a heart's beats and arrhythmias. the heart of some people to beat irradically indicates example,

The

functions of the brain and brain are

also linked to chaos

Exploring the body mathematics.

on

Geometric

geodesic domes,

invading

in fitness and the

and other burdens.

that it appears to conform to ideas of chaos.

disorders

lends itself

sound, the chaos theory also has its place

For

body.

causes

also lead to other

symmetrical, which

in the area of heart

what

body

Leonardo da Vinci and Albrecht Durer, tried to

measurements such

As

of the

gravity. Along with enabling balance,

of the spine

body's physical ability Artists, such

body

is

are

viruses.

a

In

the

and treatment of brain

theory.

molecular

shapes,

present

waves

In addition, the

such

level, as

we

various

find

inks

polyhedra

to

and

in the

shapes and forms of various

AIDS

virus

(HTLV-1) icosahedral


Theodore A. Cook published this analysis qfSandro BotticelUs The Birth of Venus. In his book, The Curves of life, he states "the line containing the figurefrom the top of the head to the soles of the feet is divided at the navel into exact proportions given by the golden section ($1 we have seven consecutive terms of § in the complete composition." ...

symmetry and present

in

mathematical formed

by

a

geodesic

DNA

dome structure

configurations

findings

from knot

DNA strands. The

from various

geometries

...

have

theory

findings

led to

are

found.

scientists

Knots to

study loops and

from knot

theory and

have proven invaluable in the

use

knots ideas

study

of

genetic engineering. Scientific research and mathematics to

are an

essential combination

discovering the mysteries of the human body and analyzing

functions.

its

227

228


MATHEMATICAL MODELS &

Mathematical

CHEMISTRY

in many natural

deoxyribose, tetrahedrons in

crystals

the

Studying

pentagons

shape of

in the

in

in

polyhedron shapes and other molecular Thus it does not

formations.

are

finds

molecules, double helices

DNA,

as a

one

present

are

occurring

substances.

chemical

molecular structure of objects, silicates

objects

come

surprise that when chemists

creating new substances they

rely

on

mathematical models.

Carbon atoms

are

well suited

building blocks because way

they

can

chains and formed

be

looped

into

structures. For

linked into

as

of the into

rings

or

molecular

3-D

example,

1983 Leo

in

Paquette,

University

chemist,

resembling

a

of 20

and

was

Ohio State

atoms

surrounded

atoms. It resembled

called

Euclidean models

a

dodecahedrane.

are

not the

) of Colorado formed

only

shape

20

ball

But

used.

ones

University

compound they named

(tetrahydroxymethylethylene)

takes the

Mobius in

a

by

a soccer

In June of 1983, chemists at the

tris

molecule

a

dodecahedron which consists

carbon

hydrogen

an

formed

of the Mabius

which

strip (the

single-sided single-edged discovery of August 1848). Tris is composed of chains of carbon and oxygen

atoms and ends in alcohol groups, which lend themselves to

clipped easily together

after the chain is

given

a

half-twist.

being


229

MATHEMATICS*. has

Park

Jurassic

general public

very

made

aware

wonders and possible horrors of

life unfolds within

living cell.

a

It Is not

human, fish, foul, plant, insect of

home

are

helical

DNA

a

mathematical how

^

engineering. The drama of

genetic is

GENETIC ENGINEERING

the

of the

genes

spiral doing —

or

important if that living cell

bacteria, every

molecule. inside

a

But

what

cell is

living is

this

a

3-D

cell? What is its function? And

the transmitters the characteristics of life forms

related to DNA? Here enter the mathematical concepts of

—

patterns, sequences,

relations, one-to-one

correspondence, all

playing a

role In

unraveling the codes and

mysteries of the living cell. Within or

a

plant

animal cell,

we

find

a

nucleus1

in which resides chromosomes. DNA is

into strands called chromosomes, and

molecules.

RNA molecules and

genes

proteins

are

are

found

on

split DNA

also in the nucleus

along with other minuscule materials and water. Different species have

a

different number of

humans have 46,

a

types of chromosomes.

crab 254 2. It is amazing to consider that every

composed

of the

same

For

certain flower species has 4, while

six elements

phosphorus, nitrogen, and sulfur.

—

carbon,

living

example, a

hermit

cell is

hydrogen, oxygen,

In the cell these atoms combine to

THE MAdC OF MATHEMATICS

}30

form molecules of water,

phosphate,

and sugar. In addition,

they

form macromolecules (enormous molecules made up of thousands of atoms

strung together) complex

even more

with its genes

lipids, starch,

cellulose and the

a

cell.

Even

building elements are the same, dramatically different genetic blueprints.

every cell's essential

though

"symbols" (elements to form the living things are formed by combinations of sequences of these "symbols". codes are deciphered by the same type of

although genes

cell's

as

blueprint for the life of

the

provides

different organisms have And

such

nucleic acids and proteins. The DNA molecule

use

the

same

code), the different structures of different

involving

In every cell the

mechanism used to

replicate cells and create proteins and other

cellular structures. The formation of genetic codes lies in Nucleic acids

and

phosphate, referred to

by

understanding nucleic acids.

formed from nucleotides

are

a

the

base (there

are

—

bases A, C, G and T appear. Each base 1

Vz

-

fe^\

U-J

/î. /f~^\

1

\z—v

^^—//

1

.^C

l/=V

IU—-J\

7V^

1 1

:

/23k /

i

'

\

other

shape

together

of the

spiral

and

takes of

a

specific

place

over

one

base —A with T and C

strings of nucleic

at their bases and form

pentagonal shape

turn.

requires

10

rungs

On the other hand,

to

RNA

complete

a

the bases A, C, G, and U and is

only

diagram

a

single rather

than double strand

"^

DNA-

illustrates how the

code. When gene

^ DNA

string of bases

copying (or DNA replication)

the DNA double helix is unwound at breakneck

8000rpm,

strands.

are

the four

has

^dMSb^sZ double forms

only

a

of deoxyribose which makes the

and much shorter

form

sugar,

pair with only

the famous double helix. It is the

usually

helix.

can

complementary

DNA is made from two

a

In DNA,

with G— and in DNA two

acids join

a

five kinds of bases which

A, C, G, T, U)3.

symbols

made from

and

While this is

the bases

speeds

into two

splits along separating happening unattached free nucleotides

the cell with the proper base

"symbol" attaches

in

their bases to the

MATHEMATICS fcTHE MYSTERIES OF LIFE

Enormous numbers

of nucleotides reside

in

a

ceVL There

living

are

four

types of nucleotides which are made up of a base and part of the helix. Note the sugar phosphate molecules are identical and can therefore fit

together

in any order and to form

DNA's bases

along

DNA double helices

adding

each are

genes

functions. Genes exposure to

a

half and

addition to DNA

In

protelns4 and other

cell

various stimuli

(e.g.

Into action

particular hormone).

by

How does

gene orchestrate

a

The thousands of genes alined

Its activities?

helix.

In this way two Identical

halves.

formulate

triggered

of the DNA

by splitting the original In

to the

also

are

strand

split strand.

formed

free nucleotides

replication,

one

the DNA

along

analogous to the "on" and "off' switches of a computer's binary Instructions genetic programs or algorithms.

molecule

are

—

A stimulated gene

uses a

chemical reaction to

other genes of the DNA molecule. Its

molecule turning genes "on"

transmitting their

own

signal,

a

manipulating

engineering

of

a

transmitting

processes

taking place

is

replicate, alter, splice,

cell.

They

can

DNA, DNA that consists of

a

in

the DNA

start

on

the structure of DNA molecules.

sequences to the DNA molecule,

along

Those turned off stop

Imagine all these

Genetic

have discovered ways to

blueprint

"off'.

while those turned

instructions.

simultaneously.

or

signal

trip the switches of

runs

the

process

These

of

engineers

and attach code

order words

change

the

make what is called recombinant

combination of bases that

are

taken

231

232


from

different organism, e.g. from

totally

human cell and

a

a

bacteria cell. Besides the

shapes

study

of sequences, codes and helices, mathematical

and structures

are

also

Important

in

cells.

Since these 3-D structures, such as

proteins

are

computer modeling

are

playing

genetic engineering. Chaos theory since

simple minuscule changes

enormous

an

is another

in the

a

rigid

theory

increasing role

In

Important field here,

gene stimuli could result In

difference in the outcome, e.g.

about fractals? Cannot the life of

sophisticated fractal-like

ever

living

not

but often appear to be knotted— the mathematics of knot and

the

categorizing

Infinite combination of atoms and molecules that form In

a

mutation.

And what

cell be considered

creation where the

six basic elements of life and its rule the

given objects

genetic

a

super

are

the

program of the

DNA molecule?

With

genetic

discovered at

relatively

new

engineering a

techniques

being developed

frantic pace, biotech companies

science to discover new

cures

are

racing

to

and use

and to manufacture


miracle

Food manufactures

drugs. to

techniques

manipulate genes

in

2JJ

seeking to use genetic various produce5 to enhance are

shelf life, size, taste and

pests. The

resistance to

altering

of genetic codes

be done in

can now

minute fraction

a

of the time it has taken the process of natural selection to take

The questions

place.

remain. Are these

damaging?

Will the uninformed

consumer

be at risk from po-

tential hidden

responsibility are

changes

effective,

desirable,

allergens

that

Genetically engineered the vine and

added

as

tomatoes

must bear is

it has evolved

can

ripen on

bruise and frost resistant

through gene splicing?

genetic engineers

experimenting with life

are

over

The

enormous.

They

millions of years!

1 Protocyotes cells such as bacterial organism's cells do not have nuclei, while eucaryotes cells which are those found in plants and animals each have a nucleus. 2 The

varying numbers are influenced by many factors including the environmental stages that species experiences during it life.

various

3 A is

adekine,

4 Like nucleic

C is

cystosine,

G is

guanine,

T is

thymine,

U is uracil.

are also long chains of smaller units called types of amino acids have been identified). Every protein is identified by a specific sequence and number of amino acids. Enzyme proteins are responsible for directing the chemical reaction of a living cell. Where does RNA enter the picture? When the is a messenger RNA DNA is splitting during its replication process, constructed along the DNA strand matching complement bases on the

amino acids

acids, proteins

(thus far

20

DNA strand. (This process is called transcription.) The messenger RNA carries this genetic code from the DNA to the translators of the genetic code called transfer RNA molecules. Here a string of amino acids are connected in the translated sequence by ribosomal RNA and the protein is formed. Many steps and parts have been eliminated for simplicity, but

the wonder of this process mind boggling.

taking place simultaneously

in

living

cell is

5 These include: tomatoes with flounder genes to reduce frost damage, on the vine and not easily bruise in shipment; potatoes with silkworm genes to increase disease resistance; corn with firefly genes to decrease insect damage. Consumer action groups, such as Pure Foods Campaign based in Washington D.C. have organized to monitor

ripen

genetically engineered

foods.

234


BODY MUSIC We have heard of

created we

give off by the way we

body

move or

by

posture

music? Some scientists! have been

in DNA from another

our

bodies. But what's

exploring patterns

found

different point of view— the music of

totally

genes. Two helical strands

body language

the various messages

joined at their compatible bases2 to

are

form the double helix chain-like formation of DNA. The bases hold

genetic codes and along with the

^MjflBbk ^»0

fc^^^f n^^|

\ j/Kb âwî ^BP^ ^thviB^MH ^^^^/

blueprints

these bases

hopes

of

petitive

are

the

being intensely

studied in

their

apparent, when

musical link becomes the

recurring

put

one as

sequences

recurring melodies of a song. them have been

been

have

the gene. A

throughout

recur

Re-

meanings.

bases

of

sequences

considers

are

patterns of

living

deciphering

observed to

DNA

cell. The

for the

In fact, many of

to music, both In

an

octave and other intervals. As mentioned earlier, music and math have

been linked

Pythagoreans scales.

linked

the

over

linked

numbers

Even Johannes

the

elliptical

although scientific

velocities

orbits

The

centuries.

in

planets

musical

this connection

significance.

musical

Kepler (1571-1630)

of

with

and

today

their

harmony, is

no

Therefore, it

given is

not

unusual for scientists and mathematicians to seek out the music of the

body.

iDr. Susumo Ohno of the Department of Theoretical Biology at the Beckman Research Institute of the City of Hope, Duarte, California.

2Only certain bases pair up with each other. These are A with T and G with C. The way

they appear on the chain, forms a DNA's genetic code.


235

SECRETS OF THE

RENAISSANCE MAN

This famous da Vinci Divina

drawing by Leonardo appeared in the book, De

Proportione,

Leonardo

Illustrated

which for

mathematlcian Luca Paoll in 1509.

Leonardo

wrote

section

on

the proportions of the

human

body

extensive

an

in one of his

notebooks. He determined measurements and

proportions for all

of the

body, Including the

parts

head, eyes, His

proportions

numerous

and

hands and feet.

ears,

were

based

studies, observations

measurements.

notebook, reference

on

he to

also

the

his

In

made

works

of

Vitruvius, the Roman architect

^îSM^M with the

proportions

he was influenced

'"¦

"•

')

(circa 30 B.C.) who also dealt

of the human

body.

Leonardo writes of how

by Vitruvius:

Vitruvius, the architect, says in his works on architecture that the measurements of the human body are distributed by Nature as follows: If you open your legs so much as to decease your height by 1/14 and spread and raise your arms till your middle fingers touch the level of the top of your head you must know that the center of the outspread limbs will lie in the navel and the space between the legs wUl be an equilateral triangle. ...

Leonardo adds, The his

height.

iRichter, Dover

length of a

man's

outspread

arms is

equal

to

*l

Jean Paul, editor. The Notebooks of Leonardo da Vinci, voll. Publications, 1970, New York.

236


KNOTS IN THE

MYSTERIES OF LIFE

.

_

A1 t. the tl the r Prom t. time Alexander Greatt

cut the Gordian knot, knots and

their many

pervaded

facets

many

philosophers

have been

knot, which has

of

our

intrigued by

beginning

no

or

end.

such knots

Today

to knots

and forms have

shapes

Magicians, artists, and

lives.

the trefoil

as

scientists

looking

are

possibly holding keys to the mysteries

some

as

These

of

of life.

theories range from

the

the

to

cosmic

microscopic worlds. At first

glance

think there

theory,

or

rigging

.

as

boat. there

entire field called knot

discoveries

and

Celtic knot design from Gospel of ., St John. Book of Durrow. From constantly Celtic Design by Ed. Sibbett, Jr. this theory Dover Publications, 1979.

on a

mathematics

in an

keeping

secured, such

things

exists

nothing knots,

than

other

shoes

was

about

special

But

would

one

,

.

,

are

.,

.

,.

,

made that link

_

being

directly

to the

physical J

world.

Knot

theory

is

a

very recent field of

linked to the 19th

century

topology.

knotted vortices that existed in ether, which invisible fluid that filled space. knots and arrange his

theory was

current

a

was

origins

can

classify

be

were

believed to be

He felt he could

an

these

periodic table of chemical elements. Although

not true, the mathematical

topic today.

Its

and Lord Kelvin's idea that atoms

study

of knots is

a

very

MATHEMATICS &THE MYSTERIES OF LIFE

What one

distinguishes mathematical knots from the everyday knots they have no ends. They are a closed type of loop,

ties, is that

which cannot be formed into been

trying

to do is

different knots.

circle. What mathematicians have

a

classify

knots and be able to

distinguish

important ideas that have been

Some of the

formulated thus far are: knot cannot exist in

•

a

•

the

than three dimensions

simplest possible knot

crossings. are

more

mirror

It

comes

in

a

left

is the

and

trefoil knot,

which has 3

right handed version which

images of one another •

there exists

only

one

knot with 4

crossings •

there

are

with 5 •

over

types of knots

crossings

identified with

crossings, trefoil knot

two

12,000 knots have thus far

been

the

only

not

13

counting

or

less

mirror

images

$& From left to right: thejirst knot has 4 crossings, the second has 5, the thwd has 6, the fourth and fifth have 7. •

Consider the

diagram,

posites of each other. opposites,

they

together. Try remain

it!

below. These knots

are

mirror op-

You would think that since undo

when

they

are

would each other brought (They simply pass through one another and

unchanged.)

237

238


Now look at the

•

Chefalo knot false knot.

or

happens when

What

the ends

are

pulled?

(The knotfalls apart.) actual models of knots

Originally, them. to be

To test if two knots

equivalent;

The

If it

shape.

if not,

no

study of knots in

various

created in order to

were

equivalent,

one

properties.

transformed

was

conclusion

Geometry Supercomputer Project to

technology

they

even come

1 is

to transform

identified

were

possible.

was

the field of topology tries to

Computers have

study

of the models had

manipulated by hand and an attempt was made

it to the other's as

were

explain these

into the

picture. The

using advanced computer

and produce visual three dimensional

study

representations of mathematical forms and

equations, such

as

torus

knots and fractals. Mathematicians have

testing

knots.

casts and write

They an

theory

now

some

the

exciting

to work in molecular

new

classifying and

look at the two-dimensional shadow it

in these fields have used the

applied

other methods for

equation describing it.3 Many

have been made and knot

developed

techniques

recent inroads

connections discovered

biology

findings

in knot

and

linking

Scientists

physics.

of mathematicians and

theory

to their

study

of DNA

configuration. Researchers have discovered that DNA strands loops which sometimes are knotted. Now scientists can

form

findings

from knot

theory

to decide if

a

DNA strand

viewing has

appeared

determine

sequence of steps in which DNA strands

a

transformed to produce unobserved very

a

before in another knot form.

particular configuration

configurations

of DNA. All of these

helpful in genetic engineering.

they

They can

and to

can

can use

are

also

be

predict

findings

may prove

MATHEMATICS &THE MYSTERIES OF LIFE

in

Similarly,

studying

the interaction of

configuration can

take

knot

physics

theory

place. Physicists exploring

played

The Theory

that would

model

(electromagnetism, gravity,

a

TOE

can

devoted their life's work in search of current TOE,

physicists

some

universe is tied into

particular TOE

as

most

of the

actions

spatial dimensions and

1

strings. The theory further speculates that

Big Bang the 9-dimensions were equal. Then,

expanded only

3 of the

dimensions

spatial

with it. The other 6 dimensions remained entwined and

across

Hence, these

(i.e.

6-dimensional

are

are

measure

laid end to end

10

strings

Scientists

strings

today's

essence

superstring theory describes

encased in compact geometries that

them.

In

answers.

have

Some

time) with its building blocks of matter and energy

the universe

centimeters

be found.

Big Bang resulted from the

10-dimensional ( 9

at the moment of the

expanded

a

contends that the universe, all forms of matter and

being these infinitesimal

as

develop

of nature

forces

believe that the

and interactions of superstrings. The

dimension of

to

trying

called superstrtngs. A version of this

objects

energy, from the instant of the the universe

are

the

unify strong force and the weak force).

the

believe

Physicists certainly

ofEvery thing{TOE)

vital role in the creation of the universe.

a

In their incessant search of TOE, scientists

mathematical

when

helpful

be used to describe different interactions that

can

believe that knots

proves very

that resemble knots. A knot

particles

only

measure

10

1

centimeter).

believed to possess 6 dimensions within

now

topological

trying

to describe them

by using

models. It is believed that these super-

may either be open

closed in

or

a

loop, and make up the

different forms of matter and energy of the universe their vibrations and rotations.

In other words,

by altering

superstrings

are

distinguished among each other by how they vibrate and rotate. The importance of the superstring Einstein's hard

of General

Theory enough to envision.

theory

Relativity. Einstein

is

being compared

Four dimensions

posed

that

to

were

length, width,

239

240


height

and time

In the universe. one

were

needed to describe the location of

Ten dimensions

thinks of dimensions

as

out of the

object

question. But if

descriptive numbers that pinpoint the

location and characteristics of become more

seem

an

an

object

in the universe,

they

comprehensible.

These ideas of TOE have been

evolving

for

over

20 years.

Gravity

had been the wrench in the gears because the computations

(involving gravity) needed to support the various forms of this theory produced mathematical Infinities4. The breakthrough came

In 1974, when John Schartz and Joel Scherks considered

gravity

as a curvaceous

piece of geometry

in the lOth-dlmension

similar to how Einstein described gravity in the geometry of the 4th-dimension.

The mathematics behind the superstring TOE is very powered. The results

have been most

convincing.

high

Schwartz and


Michael Green

had been

were

working

encouragement

or

two of the main Initiators of this on

It for over

decade In

a

caused

physicists

scientists contend

published

had been

Some seriously. avoiding ("wasting" time)

the idea because the mathematics is too difficult.

researching

Super symmetry, six-dimensional topological models, dimensional

universe,

infinitesimal

strings

concepts needed to describe and validate the this

theory

vice versa.

paper

take the idea

to

physicists

spite of little

who found the

support from colleagues,

10-dimenslonal world hard to accept. Their

finally

theory. They

has turned These

are

some

just

physicists

the

are

a

of the

some

theory.

As

ten-

a

result,

into mathematicians and

beginning

discoveries and

applications of an emerging mathematical field— knot

theory.

An International group of mathematicians and scientists who work on the same supercomputer via telecommunications network with pure mathematics to solve challenging problems of geometry. Science News, vol. 133, p. 12, January 2, 1988 issue. A "knot" that can be transformed to is a circle or unknot.

no

twists

or

crossings

i.

e.

into

a

loop, 3

The first such equations were done by John Alexander In 1928. In the 1980's Vaughan Jones made additional discoveries on the equations for knots. Science News, vol. 133, p.329, May 21, 1988 issue. 4

Mathematical Infinities can be caused by such operations number by zero and raising zero to the zero power.

as

dividing

a

241

The Oracle

office building. Redwood City,

CA.

MATHEMATICS & ARCHITECTURE BUCKMINSTER FULLER, GEODESIC DOMES & THE BUCKYBALL

21ST CENTURY ARCHITECTURE —SPACEFILLING SOLIDS

THE ARCH—CURVY MATHEMATICS ARCHITECTURE & HYPERBOLIC PARABOLOIDS THE DESTRUCTION OF THE BOX & FRANK LLOYD WRIGHT

244


"Mechanics is the paradise of mathematical science we come to the fruits of mathematics."

because here

—Leonardo da Vinci

For thousands of years mathematics for

design

architectural

design

eliminate the trial and

comprehensive

It has been

and construction.

and also the

as

has been

means

error

by

an

Invaluable tool

a resource

which

an

techniques of building.

It may seem, this Is but

a

partial

for

architect could As

list of some of the

mathematical concepts which have been used in architecture the centuries:

Temple ofKukuikan.

Chichen Itza, Yucatan.

over

MATHEMATICS £. ARCHITECTURE

•*^Jb-

A pyramid theme is carried out in the

design of this modem office building in Foster

City, California

*"jwv- i^Nr

•pyramids •prisms

•golden rectangles •optical Illusions •cubes

•polyhedra •geodesic domes •triangles •Pythagorean theorem •squares, rectangles, •parallelograms •circles, semicircles

•spheres, hemispheres

•polygons •angles •symmetry

•parabolic •catenary

curves

curves

245

246


•hyperbolic paraboloids •proportion •arcs

•center of gravity

•spirals •helices

•ellipses •tessellations

•perspective The

design of

availability resource

St

a

and

type

of materials, and

upon which the architect

can

by its surroundings, by the by the Imagination

and

draw.

Sophia. Istanbul, Turkey.

Some historical

•

structure is influenced

examples are—

The task of computing size,

shape,

stones for the construction of the

number and

pyramids

arrangement of

of Egypt, Mexico and

MATHEMATICS*. ARCHITECTURE

the Yucatan which relied squares, •

The

been

•

Pythagorean

regularity

of the

on

knowledge

of

right triangles,

theorem, volume and estimation.

design of Machu Picchu would

possible without geometric plans.

The construction of the Parthenon relied

on

the

use

of the

rectangle, optical illusions, precision measurements

knowledge

of proportion to cut column modules to exact

(always making the diameter 1/3 •

The

not have

geometric

theater at

exactness of

Epidaurus

was

of the

layout

golden

and

specifications

height of the module).

and location for the ancient

specifically

calculated to enhance

acoustics and maximize the audience's range of view.

•

The innovative

use

of

circles, semicircles,

hemispheres and arches became the main mathematical

ideas introduced and

perfected by Roman architects.

•

Architects of the

Byzantine period

elegantly incorporated the concepts of squares, circles, cubes

and

hemispheres with

arches St.

as

used in the

Sophia church in

Constantinople. •

Architects of Gothic

The Roman Coloseum, Rome,

Italy.

cathedrals used mathematics to determine the center of gravity

247

248


Each of the Jloorplans of the three levels of this house is designed from two overlapping equilateral triangles. The triangle motif is carried out throughout the interior supports and windows.

to form

meeting

an

at

a

adjustable geometric design

ceilings

point that directed the massive weight of the stone

structure back to the

•

of vaulted

ground rather than horizontally.

The stone structures of the Renaissance showed

a

refinement of

symmetry that relied on light and dark and solids and voids. With the ideas

discovery

were

materials.

adapted

of

new

building materials,

and used to maximize the

new

mathematical

potential of these

Using the wide range of available building materials

—


stone, wood, brick, concrete, iron, steel, such

glass, synthetic

plastic, reinforced concrete, prepounded

as

architects have been able to

modern times

we

design virtually

concrete

shape.

any

have witnessed the formation of the

paraboloid (St. Mary's Cathedral

in San

materials —

In

hyperbolic

Francisco), the geodesic

structures of Buckminster Fuller, the module

of Paolo

designs

Soleri, the parabolic airplane hanger, solid synthetic structures of the

mimicking

the

tents

supporting

the

Olympic Sports Hall

octagonal

home with

an

nomads,

elliptical

This tent-like structure Illustrates the constructions. Fashion Island, Foster

Architecture is reuse

an

an

Tokyo, ceiling.

curve

and

cables

even

an

of new materials and methods of City, California.

use

evolving field. Architects study, refine, enhance,

ideas from the past

analysis,

dome

catenary in

as

well

as

create new

architect is free to imagine any

ones.

design

so

In the final

long

mathematics and materials exist to support the structure.

as

the

}49

250


BUCKMINSTER FULLER, GEODESIC DOMES & BUCKYBALLS

Richard Buckminster Fuller an

inventor,

gineer, architect of ideas, time.

Among

a man

whose visions

an

a

designer,

author. He

were

his ideas and inventions

was

an

en-

was an

often ahead of their

we

find the

following

creations for which he is best known.

—

the

dymaxion car designed with

a rear

engine and front wheel

drive —

the

Dymaxion House and Wichita House, precursors of

prefabricated

housing

minimized

use

for

mass

production,

whose construction

of materials, and maximized space.

designed to be completely transportable living

units.

They were


—

in the field of cartography he created the World Energy

Map of

1940, the Life magazine World Strategy Map of 1943, and the

DymaxionAtr Ocean Map of 1954 —

—

the

geodesic dome (triangulated

space

enclosing dome)

tensegrity

The

geodesic

the

one

dome

was

greatest commercial

Fuller's

with which his

patent application he described the geodesic dome invention relates to to the

a

to shelter

a

conventional wall and

square

good index

is the structural

the figure is

weight

the weather. In

foot of floor from

roof designs

A

often

2500

kg

square meter. I have discovered how to do thejob at around 4 square meter

by constructing

a

material.nl Buckminster Fuller's he

saw

sphere,

a

dome.

subdivided

Let's reconstruct

its

The

sphere.

domes.

geodesic

faces

circumscribed this with

geodesic

of plastic

important insight was that

polyhedra2,

the

scenario of the

possible

Beginning

with the icosahedron,

equilateral triangles. and

Now

he

its vertices onto

the surface of this

shape

more

and thus also the

has the least surface

area

dome encloses

geodesic

a

sphere projected equilateral triangles do not remain congruent.

structure's

sphere,

into

a

Suppose he truncated the

most

skin

a

connection between the traditional

evolution of his

the

frame with

per

kg per

and architecture. This connection materialized into the

geodesic he

and

follows: "My

as

framework for enclosing space.

performance of any building frame

required

success,

has become synonymous. In his

name

for

more

solid. Now the

closely approaches

properties a

new

of the

the

sphere.

shape A

sphere

given volume. Consequently the space with less

expenditure

materials than conventional architectural forms. In addition, new

type

tension

of

stability

pulls

inward

results.

of

of a

As in the soap bubble, surface

against the outward compression of the

equilibrium is reached between tension and The compression. design of traditional architecture requires the consideration of weight and support. Gravity plays a prominent role. In the geodesic structure, gravity's role is nearly irrelevant. enclosed air—

an

251

252

TH E MAC K O F MATH EMATICS

There

was

spheres,

so

catch, however.

one

their sizes

combined the ideas of

integrity ), the

action of the

geodesic

tensegrity

To

invention and

patent of light-weight

the

of these

scope

proposed hemispherical

megastructures,

dome with

diameter that would enclose part of New York described its installation

as

follows, "A fleet

Sikorsky helicopters coiddfly kilometer

high,

all the

a

In 1960 he

city.

of sixteen into

segments

3 kilometer

of the large

position for

3 kilometer wide dome in three months at

$200 million...an

These

supporting columns,

nearly eliminated size limits of these

illustrate

consider Fuller's

(tensional

enormous.

buttresses, but with tensile forces (the pulling

load). Fuller's

construction

structures.

could be

not

were

But when Fuller

tensegrity3

domes with

themselves up not with

or

domes

geodesic

somewhat limited.

size of these domes

structures hold

beams, arches

were

The

area

of fifty blocks which includes

all

a

1.6

cost

of

of upper

Manhattan

skyscraper city.

A dome

this kind would

prevent

falling

of

and rain from

snow

on

the protected

area

and control the effects

sunlight air..."4 Fuller's

and

quality of

geodesic

not without

dome

of the

was

numerous

setbacks in both financial

aspects

and

acceptance.

Although

his

Dymaxion

House

the American Institute of Architecture

was

rejected

During

honorary

the

subsequent fifty

architectural

Equally important 300,000 geodesic domes

the

a

driving

and acclaim.

recognition brought by the

that

to

in 1928, his

years of his career, he received

degrees, fellowships, awards

was

the

'patent'

philosophical vision of providing economical housing was force.

of

gift

were

based

on

Fuller's

over

patents


between 1954 until his death in 1983. Fuller's not confined to the

are

geodesic

megastructure

have also been found at the molecular level. The called buckminsterjullerene

synthesized

in the 1990s.

carbon

60

form

for

as

uses

scanner, or

the

modifying

C6o

buckyball

possible

microscopy also

The

is

located

atoms

icosahedron.5

,

in

new

a

is

chemical a

is now

battery

architectural

also was

consisting of a

truncated

catalyst,

in solid

the

tip of

storage cells. Scientists

structure

to how Fuller modified

the

produce

to

of

that

being synthesized

(somewhat analogous

polyhedron)

polyhedron

vertices

lubricant,

a

buckyball

carbon molecule

the

at

structures

size. Geodesic structures

of the

are

buckyball

new

theoretical molecules that would

be

stable, lighter and

more

stronger

than

a

buckyball.6

The conservation of

materials, lightness, stability and of

strength dome

are

characteristics

researchers

the

the

geodesic

same

that

interest

these

in

new

molecular forms.

truncated icosahedron 1 His patent was approved In June of 1954 and he received royalties on all geodesic domes that were built during the 17 years it was in effect. From Buckminster Fuller by Martin Pawfey, Taplinger Publishing Co., New 2

York,

1990.

Polyhedra

are

geometric solids whose faces

3

In November of 1962 Fuller received integrity structures. 4 From Buckminster Fuller

by

Martin

a

are

polygons.

patent covering

all tensile-

Pawley, Taplinger Publishing

Co.,

New York, 1990. °

It is also called

a socceme

because of its resemblance to

a soccer

ball.

6 The 168-carbon molecule called the a buckygym jungle-gym-like researchers at the IBM Thomas J. Watson structure was created Research Center in New York. —

—

by

253

254


21ST CENTURY ARCHITECTURE SPACE FILLING SOLIDS

Over the

ages the triangle, the

and

square

played major

the

rectangle

Wood and stone

design.

have

roles in architectural were

among the first natural materials builders

used

shelters.

Since the

right angles from what

shapes

make

triangle

gave the most

was

as

and

their

and

stability

then known, these

utilized

were

structures

to

the

the

in

such

pyramids

Yucatan.

of

New

Egypt jj shapes and designs evolved as --ij_ knowledge, understanding, and u_: d materials improved. For example, A

diagramfrom Leonardo da Vinci's

over

notebook, showing his work on trying resolve the forces acting on an arch.

to

the

the centuries discoveries of

dynamics

of curves and arches

allowed stone and wood to be

used

introduce

to

these

features in such structures Roman

and Gothic

aqueducts

cathedrals.

With

introduction of

steel, iron,

the

concrete, and bricks,

designs

were

plastics

and

materials,

along

structures, to consider

of

as

glass,

new

bold

possible. Later, synthetic

with

tensegritory

allowed architects a

whole

new

family

Our mathematical

shapes. knowledge, coupled

with

corn-

Gothic plans by the master architect of the Dome of Milan, Caesar Caesartano.


This model

of Paulo SolerCs visionary project

is

displayed

at Arcosanti in

Arizona.

puter modeling, and

physical forces Yet the dimensional

shapes

objects

such

geometry,

as

In-depth understanding of

an

which act

structure have

on a

of mathematics.

rectangular Others

Many

are

from Euclidean

exotic forms

are more

All these

architect to both fill space and to create

21st

increased.

square solids, pyramids,

or

solids, tents, and geodesies.

involve

greatly

and forms of architecture remain the three

spheres, cylinders.

What

the various

a

objects

using curved used

are

What

objects

can

prefabrlcatlon, adaptability

fill space? and

by

the

living space.

type of structures and living spaces will be designed

century?

cones,

in the

If design features

expandibility,

then ideas of

plane and space tessellating play prominent roles. Any shape that tessellates and other

a

plane, such

polygons

as

can

the

be

triangle,

adapted

the square, the for

hexagon,

spatial living

units.

255


256

Additional units

can

shared walls later arises.

the need

Design possibilities

be very or

be added to

as

with

exciting

can

courtyards light and

alcoves left open for

exterior

On the other

access.

hand the architect may want to solids

consider space,

the

being the cube parallelepiped. designs

which

most

and

pack

traditional

rectangular

Some

module

may include

rhombic

dodecahedra

or

truncated

oc-

Plane filling Penrose ties & space truncated octahedra

filling

tahedra.

The choices and

challenges

in

options

now

determining

space In such

a

way to

available to architects add to

which solids work best

optimize designs, aesthetics and

comfortable livable areas. Designs and projects as

today's together to fill create

by such architects

Paulo Soleri, David Greene, Pier Luigi Nervi, Arata Isozaki, I. M.

Pel, and others lend themselves to the out

geometric shapes.

Now,

as

use

in the

of new materials and far

past, the feasibility of structure

dictated the

laws

a

is

by of

mathematics and —

as

physics

which act both tools

and

measuring rods.

left illustration shows how these pentagons, tessellating plane, can be adapted to pentagonal prisms that become module spacefilling units. At the right is a structure formed by modifying the shape of a cube.

The a

MATHEMATICS S. ARCHITECTURE

257

THE ARCH —

gods play; they play with numbers, of

Behind the wall, the

which the universe is made. -Le

CURVY

MATHEMATICS v

Corbusier(1887-1965)

centuries elegant architectural triumph. Over the curves mathematical the arch has taken on the shape of many the catenary) to (such as the circle, the ellipse, the parabola,

The arch Is

an

parabolic arch arch the elliptical arch the pointed or equilateral arch segmental the arch transverse the the squtnch arch the stilted arch,

the become— the semicircular arch, the ogee arch

the relieving arch horseshoe arch the trefoil arch triumphal arch the corbelled arch, the diaphragm arch the half arch the

triangular

or

false arch.

semicircular arch

stilted arch

horseshoe arch

A pointed arch

A^ ogee arch

elliptical arch

258


IIIIMMI The Roman aqueduct at Segovia, Spain, constructed with 148 arches 90

feet high. In essence, the arch Is

architectural method for spanning

an

space. The nature of the arch allows the stress to flow more concentration

throughout, thereby avoiding (wedged shaped stones), also

voussolrs

the arch's form

a

gravity

At the center Is the

curve.

locking causes

mechanism

The thrust is countered Until the invention and

relied

on

stepped stones, were

as

use

seen

use

spread

in

as

out

The

pull

of

(the thrust force).

found In Greek architecture;

Egyptian pyramids.

extensively

Add to the arch their

concrete and brick, and

With the

All the stones

of the arch, architectural structures

the first to

semicircular arch.

keystone.

by the force of the walls or of buttresses.

columns and beams,

architects

called arch stones, form

triggered by gravity.

the sides of the arch to

evenly

the center. The

on

an

use

or

The Roman

and

discovery

develop the and

architectural revolution took

of the arch, vaults and domes, the Romans

use

of

place.

were

able

MATHEMATICS & ARCHITECTURE

to eliminate

horizontal beams and interior columns. The arch allowed them to relocate the

weight

of the structure to fewer yet massive

more

supports.

Consequently interior space

opened up. the arch,

Prior to

a

structure was

by

necessity spanned with columns Grace

inside and out, and

Cathedral San Francisco, California.

the distances calculated

between the columns had to be

carefully

have the

under excessive stress.

spanning beam

The Roman arch

was

centuries, architects

collapse

based

began

enclosed

became

more

higher,

space. The

of the structure bore the arch bore the load

shape

of the circle.

allowed

shape

weight.

pointed arch. more

to

light

arch had the load flow

through

thus

not to

Over the

first to the As

a

result

enter, and

of the arch dictated which

parts

While the semicircular Roman

the span to the walls, the

over

building's buttresses,

the

to deviate from the circle,

then to the

elliptical (or oval) arch, structures

on

so as

pointed Gothic

the arch to the outside of the

making it possible for higher ceilings.

The arch is not passe. As with all architectural ideas, its concept and

use are

still

evolving.

With the invention and

of building materials, architects of mathematical

curves

and

can

shapes

combine and

use

of new

use a

in their creations.

types

multitude

259

260


ARCHITECTURES* THE HYPERBOLIC architectural

Some

DA l> A Of> I I r>

structures

have been

In less

designed recognizable shapes. A striking example

is the

hyperbolic paraboloid

used in the

Cathedral in San Francisco. The Cathedral A.

design of St. Mary's

was

designed by Paul

Ryan and John Lee and engineering consultants Pier Luigi Nervi

of Rome and Pietro Bellaschi of M.I.T.

At the

unveiling when Michelangelo

what have

of

thought

Cathedral,

Nervi

could not have

This

design

geometric

asked

would

the "He

replied: thought

comes

theories

of it.

from not

then

proven." The top of the structure is

St

Mary's Cathedral, San Francisco,

California.

2,135 cubic foot

hyperbolic

paraboloid cupola

with walls

rising and

concrete carries

1,680

a

supported by

which extend 94 feet into the

weight

of nine million

pounds.

concrete coffers

ground.

The walls

involving

The dimensions of the square foundation A

200 feet above the floor

pylons

prepounded

a

are

4 massive

Each

pylon

made from

128 different sizes.

measure

255'

by 255'.

hyperbolic paraboloid combines aparaboloid (a parabola revolved ofsymmetry) and a three dimensional hyperbola.

about its axis

hyperbolic paraboloid equation:

jr2

-

_x2

=

*

c2

a,b>0, c*0

MATHEMATICS & ARCHITECTURE

261

THE DESTRUCTION

OF THE BOX The work of FVank has

a

tures

definite are so

Lloyd Wright

rSiiuomwScHri

his struc-

THE LIBERATION OF SPACE

style, yet

diverse that the

style

does not lie in the similarities of his

buildings,

but rather in the

philosophy that the structure projects. In his words, "Architecture is the scientific art of making structure express ideas." His architecture has

come

to be called

organic architecture

—

encompassing

landscape, materials, methods, purpose and imagination In

a

special way.

Civic Center was one of his final Marin County, California

Wrights Marin County

Wright's designs space become

designs.

of structures in which internal and external

one

designed buildings

had so

a

profound impact

that the outside

came

on

architecture. He

inside. He called this

262


the destruction be

Dwellings, or

by

as

a

Frank

box.

they private

commercial,

viewed

of the were

Lloyd Wright

conglomeration

boxes

or

cubes.

Space

of in

Euclidean geometry Is

defined

the

as

points. Although often

used

geometry space,

box architecture

has limits.

Wright wanted his works to

flowing of points He

separation architecture.

sought

a a

we

a

cube Is

Euclidean

represent

know that space

boundaries

no

feeling

from the inside to the outside.

lines, he discovered

designs.

give

the

in

to

of all

set

or

of space— the

Thinking along

these

way to eliminate the traditional box from his

change

from the

feeling

of confinement and

from the outside world which characterized box

Wright

realized that the

potential

of certain

materials had not been utilized. These materials

—

building

steel and

glass

—

with

along innovative design changes furnished the

means

to do away with the

box, thereby

allowing the merging of interior and exterior

space.

Wright's designs did away with

corners

of

the box by removing the them eyes

supports from the

corners

along the walls by using cantilevers. Thus,

were

not bound

or

led to

corners.

and

relocating

the inhabitant's

Space was allowed

to flow.


By replacing post and beam supports, walls were

no

construction with

longer viewed

independent and unattached. modified

Wright

by either

as

Any

one

shortening, extending

did not stop with

freeing

sky.

designs

eliminated the

of these walls could be

or

redlvlding.

the horizontal

vertical. He did away with cornice, and His

canuleverage

enclosing walls, but rather as

opened

stacking

and

plan,

but also the

up the

top to the

duplication of boxes.

Instead, he used columns, and made them part of the ceiling,

thereby creating

a

continuity of form.

outside of a structure could

allowing space

to

move

organic architecture. which you feel and space alive

1

move

Now space inside and

in all directions. The freedom of

in and out of a structure is the essence of

"So see

organic architecture

all this happen

as a

is architecture in

third dimension

...

1 by way of the third dimension."

Frank Lloyd Wright, An American Architecture, ed. York: Bramhall House, 1955.

Edgar

Kaufman. New

263

have always played games and done puzzles. Books on the various games give numerous examples of ancient games that are still played today. This illustration is from an Egyptian papyrus dating back to 1200 B.C.. It is a humorous rendition of a goat and a lion playing the game of senet Senet was one of the most popular games of its time, and was played by all sectors of society. Unfortunately, there is no remaining documentation on exactly how it was played, but a possible version of senet has been devised using the findings and work of

People

origins of

archedlogists.

THE SPELL OF

LOOK, RECREATION & GAMES \B

MATHEMATICAL MYSTERY TALES PUTTING YOUR LOGIC TO WORK THE GAMES MATHEMATICIANS PLAY

SOME MATHEMATICAL RECREATIONS

MAGIC SQUARES & MORE RECREATIONS THE KONIGSBERG BRIDGE PROBLEM UPDATE

CHECKERBOARD MANIA A FEW OLDIES

266


Logic is the art of going wrong with confidence." —Morris Kline

Logic

and mathematics

people and

definitely go

hand in hand. But most

do not consider mathematics fun and games. Yet, games

recreations

are

an

of mathematics.

integral part

development of many mathematical ideas person's pursuit

of an

to be driven

by

puzzles for hours

on

seem

enjoy mathematics one

may

have

intriguing

an

or

are

They belong

are

fascinated

spent hours

even

the result of

problem.

invisible force to work

end.

and

notion

The

on

a

Some

people problems and

to those who

by

it.

Before

inherently realizing it,

days exploring

different

ramifications of what

ostensibly

started out to be

a

simple pastime. As

history attests problems,

challenges, games and pastimes have sometimes lead to

remarkable

discoveries, and Circa 212 B.C. Syracuse fell to the Romans. At the time, Archimedes was working on a mathematics problem in his home. When a Roman soldier entered and ordered him to stop working, Archimedes did not pay attention. Angered, the soldier killed Archimedes with his sword.

even

to the

creation of new

fields of

mathematics. In

fact, the famous Greek mathematician Archimedes

absorbed in reveal

some

a

mathematics

of the games,

mathematicians love to

problem.

puzzles

play.

was

and

killed because he The

following pages

was

will

mental calisthenics that


267

MATHEMATICAL

MYSTERY TALES Mathematical mysteries have been around

for

centuries.

Lewis Carroll's works into

this

Some

of

would fall

category. Today,

mysteries have entered the

these

with such books

The

Puzzling Adventures of Dr.

popular

scene

Ecco

Dennis Sasha and Visitors From The Red Planet

by

as

by

Dr.

Cryton. Logic problems,

as

illustrated below,

can

also be embellished with

stories.

*

*

*

The teacher started to read-

It

was one

of those week s at the circus,

go wrong. First,

one

when

everything seemed to

of the horses for the acrobat dancers went

lame. Next the head clown threw

a

fit because thefat lady's child

got into his makeup. Then Madre and his wife had an argument over

the trapeze artist. The final blow,

came

found dead under the big top. Next to his

when Madre was

body

was

his cane, which

he occasionally used. An overturned glass of water was on his

desk, and

a

The teacher

minute pile of sawdust

question

caps, and

his

body.

stopped reading, and asked, "Well, what do you think

happened to Madre ? I

was near

can answer see

immediately.

How did he die? You

with

a

yes' or 'no'.

can

So

ask me any

put your on thinking

where your logic leads you". Hands began to go up These

shyest student.

logic stories always seemed to spark even the

The questions started and the drama of the circus

death began to unfold.


A student asked,'What did Madre do at the circus?" "Remember,

only yes or no questions," 'Was Madre

a

the teacher cautioned.

manager?" Carol asked.

"No". 'Was there any

sign of

violence?" "No," the teacher replied. 'Was there

anything else near the glass?" 'Yes,

Important to know what It was?"

there was." "Is It

'Yes"

'Was It

a

sandwich?" Bill blurted out. "No."

'Was It

a

pill?" Tom asked.

'Yes."

"He died because he had not taken his medicine?" Tom continued.

"No, not exactfy," the teacher's replied. 'Was there

a

pencil sharpener on his desk?" Terrl asked.

"No"

'We have to find out how he died," Terrl told her fellow students.

The students

began discussing the situation amongst themselves,

continuing the questions.

before

'We know he was not well, since he

was

know he was not killed

act," Tom said,

death

by

a violent

taking medicine. And we

analyzing the

scene.

"Right.

He must have died from his Illness,"

point he directed

a

Gary said. At which

question to the teacher, "Did he die from a

heart attack?" 'Yes."

'Well, that was

an

easy one to solve this time."

bit smug. 'We're not done,

Gary said, feeling a

Gary," Barbara pointed out.

'We

haven't found out what caused his heart attack."

"Sure we

have, he didn't take his medicine. Right Mr. Mason?"


Gary asked. 'Wrong! Gary," Mr.

Mason

replied.

The students went back Into conference. "Have

the Initial facts of the case?" Terri asked her

"No," Bob, who

we

considered all

colleagues.

rarely talked In class, replied.

'We haven't

considered why the sawdust information was mentioned, what his

job was at the

circus, and what about the

cane,

and Madre's wife

and the trapeze artist."

"You're

right,

excitedly. marital

Bob. We need to address all these," Terri said

"The cane, the sawdust, the heart attack, and the

problems." Tom muttered thinking out loud. and the sawdust go

The

cane

was

shortened with

teacher

a

saw,

together.

thus

For

"IVe

some reason

leaving the sawdust."

—

got It!

the

"Yes," the

replied. Tom continued," Perhaps Madre got upset

because

someone

wrecked his cane."

"NO"

cane


"Perhaps he didn't know his "That's was

right,"

very

Mr. Mason

upset,

cane was

shortened?" Bob asked.

replied. 'Then, when he stood with It he

but why?" Bob wondered. "Because It didn't fit him. He

was

too

tall for It!" Carol

shouted. The excitement that

the solution

was

close at hand

spread among the students. "So far,

so

good,"

the teacher said,

encouraging them to continue. "But

why would

thinking he

had

grown taller

upset him so?" Gary asked the other students. "This time I've hand In the air and "Madre

was a

finding

midget!" The

really got It," Tom yelled, waving his

It difficult to contain his enthusiasm.

student's excitement was

contagious.

The teacher shouted, "YES! What else?"

This time Terri's hand shot up in the air. The teacher nodded to

her, and she began to summarize the

midget. one was

His wife and the

day when he wasn't

scenario— "Madre

was a

trapeze artist sawed off some of his

using

too short for him. Madre

It. When he went to

use

thought he was growing taller,

aggravated his precarious heart problem.

He couldn't

cane,

his cane, It

which

get to his


medicine in time. How does that sound Mr. Mason?"

"That's it! You've all done you with

one

a

great job

on

next week." The teacher

this one!

Maybe

I'll stump

replied as a smile crossed

his

face.

Here

are a

few

logic stumpers to try out with a friend or group of

friends.

?

(1) It

was

came

*

*

Tuesday and Tom and Jerry

timefor Tom to go home,

were

at the

job.

same

When it

Jerry wouldn't let himgo home.

WHY?

(2) Eric walked into

a

bar, and asked the bartenderfor a glass of

water. The bartender looked at Ericfor gun and pointed it at Eric. Eric

was

a

moment,

startled for

a

thenpulled out a

moment, and then

said thank you and walked out of the bar without having drunk the

glass of water. WHAT IS THE SCENARIO? (3) When Mary came home she walked into the kitchen. She

suddenly let out a scream when she discovered her dead husband on

thejloor. Along side was waterfrom a bowl that had been on the

table and

was now

kitchen table

was

tipped over on thejloor.

ajar.

The window

WHAT HAD HAPPENED?

over

the

271

272


PUTTING YOUR LOCK TO WORK

RECTANGULAR VOLUME ILLUSION

Our eyes

to

jump with this

Determine

puzzle. 2x1x1

begin

blocks

structure.

how

make

up

many this

THE SPELL OF LOGIC RECREATION i. GAMES

273

THE IMPOSSIBLE How Is It

string

possible

paper trap without buttons from the

string,

POSSIBLE

to remove the

with two buttons from the

or

tearing

removing

the

string, cutting the the

paper?

STACKING THE DIE

Assume

around and

see

Find the

you

this all sum

could stack

walk

of dice

exposed

faces.

of the pips

on

the hidden faces of the dice.

274


\

r

ARE ALL WHEELS ROUND?

A rouleau Is

To

fascinating object.

a

construct

with From

rouleau,

a

equilateral

an

each

start

triangle. the

set

vertex,

compass to the radius of the side

of the

Make

triangle. the

through

two

arc

an

opposite

vertices. If you make the rouleu out of cardboard, you

how well It rolls. If It

smooth

as

center

of the

were a

test

can

fact,

In

It's

as

circle. The

rouleau

the

Is

Intersection of medians of the

triangle. stick a pencil through equilateral and

use

the

pencil

The rouleau inscribed In

placed

can

can

this center

as

an

axle.

also be

square, and when a

circumscribing

can

rotate within the

in

square, it square.

a

You

As

a

result, it has been

used In the Wankel

engine,

in

which the rouleau-shaped piston rotates inside

enclosure,

as

automobile engines.

a

square

In Mazda

shaped

THE SPELL OF LOGIC. RECREATIONS.GAMES

275

THE BLOCK LETTER PUZZLE

Cut out these pieces,

need to be rotated, of the

or

rearrange them In your head. Some may

flipped or translated. They form a capital letter

English alphabet.

Good luck!

276


HOW A PUZZLE

CAN BEATURNINO Simeon Poisson (1781-1840

POINT

difficulty trying that suited him.

him to pursue medicine

or

and/or desire.

Apparently

story has It that

on a

one

find

to

His

a

) had career

family urged

law, but he seemed to lack the talent a

puzzle put

Poisson

on

track.

The

trip someone gave him a puzzle similar to the below, which he easily solved. Realizing he had an aptitude

and interest In

things mathematical,

he discovered that he wanted

to work with mathematics. He became well known for his work

celestial

mechanics, electricity,

and

magnetism,

and

on

for his

discoveries of applications to

addition,

he studied

integrals and Fourier series. In probability theory, and discovered the

Poisson distribution.

_^->_

The milkman had two 10 quart pails of milk. Two customers want two quarts each in their containers. One has a five quart container and the other has

a

four quart

solve the problem? This version

Loyd.

container. How does the milkman

of the problem was created by

Sam

THE SPELL OF LOGIC, RECREATION S. GAMES

277

1$ IT POSSIBLE? Using only a straightedge distances

can

on

be Indicated,

bisect the given

which

can

you

and prove

angle,

why It Is bisected?

CAN YOU MAKE A If no number Is In any row

or

of

square

tiles, It

column of a domino

18

Is called

Latin

a

square. The square

below Is not

a

LATIN SQUARE?

repeated

••• •

•

••• •

•

•

For

rows

or

example,

repeat

In

columns.

the top

•

and the column

and

•

•••

••

row

4s.

rearrange this, new one

square.

on

the

has two blanks

two

•

• • •

•

•

•

•••

has two 6s and two 4s

right side

•

•

Latin square,

since numbers some

•

or

to make

Either start a

a

Latin

•

• • •

• •

v

•

•••

•

•••

s

iii.

•'

.'¦"

•

•

•

•

":9

m. •

•

•

•••

•>• •

• •

•

•

•

•••

••

•

•

•••

• •

•••

• •

• •

•

•

•

•

••• •• •

•

• • •

• •

278

TH E MAG IC O F MATH EMATI C$

THE CAMES MATHEMATICIANS PLAY THE CHAOS CAME

Ij label three points on a sheet

ofpaper as shown, (1,2); (5,6); (3,41 2) Pick a starting point at random. Role a die. 3) Rule: take 112 the distance between the starting point and the point which has the number on the die that

wasjust

rolled.

4) Continue step three indefinitely.

(1.2)

(5,6)

(3,4)

It Is not the first time that randomness turned out to conform to some

order.

18th

century French naturalist.

devised the Needle Problem and linked

favorable outcomes.

Similarly,

n

Count

to the

Buffon,

probability

of

in 1904 R. Chartres found that the

THE SPELL OF LOGIC, RECREATION 4. GAMES

probability prime

of two numbers, written at random,

6/jt2.

Is

randomness

Michael

as a

basis for

Barnsley

modeling

Invented The Chaos Came. There For

example,

coin

or

die

used to random

are

first

being relatively

considered

natural shapes. many ways to

using

Hence,

play

he

the game.

a

be

can

generate

a

occurrence.

The rules

are

also

flexible. The version above

uses

points, a

a

three fixed

random point, and

die. The

resulting design Is

astonishing. gives

A chaotic process

rise to order!

Another variation is to choose

a

point's location at random. Make up a

rule for when

heads and

(e.g.

one

tails.

a

move

up) and

tossed coin turns up two units to the one

Begin tossing and

In addition to

(following a

see

discovering

set of rules)

random method to

right

for when It turns up

what happens. that the limit of

a

random process

produced fractals, Barnsley also devised a

reproduce a given shape.

279

280


THE OLD

MATHEMATICAL CAME OF

Rlthmomachla

RITHMOMACHIA

mathematical game. levels

are

understanding strong strategy 11th

be

a

very

When advanced

played,

good

a

of number

theory

and

a

Rlthmomachla dates back to the

essential.

are

can

probably originating either in Byzantium or during the Middles Ages, that the intelligencia

century A.D.,

Alexandria. It

was

made Rlthmomachla the game of choice and considered It superior to chess.

Physical set-up ofgame The

playing board

triangles called as

or

even

odd.

pyramid.

Is

rectangular and

pieces used

units. The

pieces, and the other's

opponent has white pieces

are

black pieces and referred to

player has 8 circles, 8 triangles, 7 squares, and 1 pyramid has six faces, whose numbers total

the other with 1)

,

total 190, and is two squares

number 16. The can move.

with the number 4 and

triangles (one

two circles

(with

36 and

25), and two squares

has five faces whose numbers

(with 16 and 9). The black

piece

of circles, squares,

The white

91, and is composed of two

25),

composed of 8x16 squares

shapes

One

pseudopyramids.

Each

Is

In the

are

pyramid made-up of two triangles (with

(with numbers 64 and 49), and

shape of the piece dictates For

example,

the square

numbers 36 and

one

circle with the

the number of spaces

can move

a

4 consecutive

triangle moves 3, a circle moves 1. a circle, triangle or square pyramid depending on which face it is using. When beginning the game the pieces are set-up as illustrated In the diagram.

empty spaces and

in any

the

direction,

can

move

a

as

Object ofgame The

object

of the game is to

capture your opponent's pieces and

make certain number combinations which constitute

a

victory.

THE SPELL OF LOCK. RECREATION & GAMES

How to

capture your

opponent's pieces.

I

?

19£JJSZZJI»

by potential moves or

?i HTT

by actually moving.

^^F

Pieces

are

captured

The methods of

capture

are:

^m

? ¦Hi m

voooo T V 00o0

1) Siege capture Is done

by surrounding

an

opponent's piece

on

all four sides.

Then that piece is removed.

2) Meeting capture. If you

move a

required

piece the

number of

spaces and land an

on

opponent's piece,

then you take that f

opponent's piece without

actually

moving your piece. For example. If white

triangle #36

moves

3

0©©© AA©©0© A A R H A A A A <"> 153

spaces and lands

on

black circle#36. the

25

black circle #36 is removed without

~*T|

169

actually moving the white triangle

289

#36.

3) Assault capture. If a piece's number value ttmes the number of spaces in its

number

move

equaling

would land next to the

product,

an

then

opponent's piece with the the

player

takes

the

281

282


opponent's piece without actually doing the moving. a

white circle #4

removes

For

example,

black square #28 If 7 spaces lie between

the two pieces.

4) Ambuscade. If two pieces of a player an

opponent's piece and the For

example,

either side of

of these two pieces

sum

opponent's piece number, then the piece.

can move on

player

takes the

equals

the

opponent's

Ifblack

triangle #12 can be straddled by white black triangle #12 is removed without

circles #4 and #8, then

actually moving the pieces. It Is difficult to

capture

a

be used, unless it is done

#64) is captured when it threatened

by

one

pyramid, the number 91 and 190 must by siege or the base square (the #36 or

is

a

of the

base. If one of the

pyramid's

capturing methods,

collected in the form of an opponent's piece of equal value

as

or

be

can

acceptable

the threatened face.

Players decide before beginning the game, what victory

faces is

ransom

or

end the game. Here

game. Some

are

simple,

possible

are some

while others

are

will constitute

a

ways to end the

quite complicated and

involved.

Possible ways to end the game, i.e.

1) De corpore.

captured,

The

players agree

that will declare that

2) De bonis. Players agree

on a

target

3) De lite. Here

a

on

player

a

number, then that win

depends

a

number of pieces to be

the winner.

number value target. If the

the number value of the pieces exceed that

gain a victory.

both

player captures will player is the winner. on

the pieces and the total number of

total

of or

the total number value of of the pieces.

For

digits equaling only

6. So

digits

example, target value could be 160 with

sum

THE SPELL OF LOGIC RECREATION & GAMES

a

player with pieces 121,

9, 30 would win, while the

player having

56,64,28 and 15 would not (these pieces have 8 digits in all). 4) De honore. This win depends number value. For

exactly

example,

160 will win.

it

on

can

the number of pieces and their be

agreed that 5 pieces totaling players must be familiar

It is clear that the

with the number values of their pieces and their various

sums.

5) De honore liteque. This win requires three conditions be met— a number value,

specified specified The

a

specified number

number of digits of the

following

victories

are

of pieces, and

a

pieces. meant

for expert players

and

involve arithmetic, geometric, and harmonic progression. The pieces used

can

be from both players, but

one

must be

your opponents piece.

6) Victoria Magna.

Arrange three captured pieces

arithmetic, geometric,

example—2, 5, 8; example— 6, 8, 7)

Victoria

harmonic

Major.

Display

four pieces that

can

possible progression.

pieces with value 2,3, 4, and are

an

e.g arithmetic

12

form two out of the three

4 and the

in either

progression, geometric example—36, 49, 64; harmonic or

8

be combined to

For

example,

the

give the arithmetic progression 2, 3,

geometric progression 2, 4, 8. Note also that 2, 4, and 8

white pieces, while 3 is

a

black piece.

283

284

THE MAC IC OF MATH EMATI CS

THE GAME OF HEX The game of Hex

was

Invented In

1942byPletHeln. The game

board for hex Is made up of

hexagons. There

are

eleven

hexagons

In

each direction. As illustrated,

the first

In this particular game, the player with the black pieces was the winner, since that player was able to make apath to

Is

not

to

play

player

allowed his

or

her piece in any of the gray

the opposite side.

hexagons. one

can

After the first move, move

on

these

hexagons.

Players alternately

take turns

placing a piece down.

Object:

Each

player alternately

After thefirst gray

places trying

move, one can move on the

hexagons. his

or

her piece down,

to connect

opposite make

a

path

some

beginning

moves

path

to the

is the winner! This board shows

a

The first

side.

to the

player

to

opposite side

THE SPELL OF LOGIC, RECREATION <. GAMES

285

SOME MATHEMATICAL RECREATIONS

PLAYING WITH POLYHEXES

Polyominoes are formed

by joining congruent squaresmonomtno

domino

n

\ | | 1 [

|

trominoes

tetrominoes

Mill

flea p

pentominoes, and so on... another group of entertaining mathematical

Polyhexes

are

which

formed from joining congruent hexagons.

Here

are

are

the

configurations for up to 4-hexes (tetrahexes):

Which of the three each of the Have fun!

objects

seven

following shapes can be tessellated using one of

tetrahexes shown above?

286

THE MA6IC OF MATHEMATICS

LEWIS CARROLL

CHANCES "FOUR" TO "FIVE"

Mathematician Charles L. better

known

as

Dodgson, Carroll,

Lewis

created the game of doublets.

The

Fair

ran a

English magazine Vanity

number of contests in 1879

featuring

Carroll's game. The game

begins with

two words of the

length.

object the

is to

other

same

change word

change only

one

to

other

by forming

successive words of the same

You may

The

the first word

length.

letter at

a

time, trying to make the least number of transformations.

Here

are

some

others to try—

Cover EYE with UD Prove PITY to be GOOD

Get WOOD from TREE

Change OAT to

RYE

THE SPELL OF LOCK, RECREATION £. GAMES

287

POMINIZINC The kite and dart tiles

discovered

by

British mathematical Penrose In

physicist Roger

These two tiles tessellate in

a

special

very

1974.

plane

a

They

way.

phenomenally produce number of

PENROSE TILES

were

infinite

an

nonperiodic tilings (no

basic pattern in the

repeats Itself

on

when the eye of a

a

design regular basis,

moves

or

up

down)

plane. Playing a game of

dominoes with them would be both

and

challenge

produce

some

a

very

interesting designs. Construct the tiles in the following

way:

Each tile is divided two

triangles,

one,

two, three,

Now

try

out

a

by

which or no

a

line Into

can

have

dots.

game.

You may

want to increase the number of

dots used and

see

how the game

proceeds.

The mathematical formation Note: (p is the golden

mean

of the which

Penrose tiles.

equals

(l+^5)/2*>1.618..

288


MAKING A

HEXAFlexagons

rvA^^^k.1

TtTRAFLEXAGON

are a

wonderful math-

ematlcal recreation that aU ages '

can

desiring of

more

than

flexagons.

mere

recreation

enjoy.

can

Here Is how to make

a

In addition, those

delve Into the mathematics

hexatetraflexagon— It

has six

faces and four sides.

I

2

3

2

3

I

|6

4

4

5]

4 3

1

2

3

4

9

6

6|

9

would look Hke

2

2

6

__5 9 step

—

step CO Fold

S

2

•tep(l) Starting with the front aid*, fold adjacent j's unto each other. Do the same with adjacent X's. The resulting figure

9

step (l)

5

back side

front side

5

S

adjacent x's

unto each other.

This pattern of numbers will appear

on one

side.

6

along one edge. The resulting shape is

CO

6

step (3) Place the two 4's on top of each other, and tape

a

square with o's

on one

side, and s's on the other. Join the outer edges of • & 5 with a piece of tape. Turn over and flex the s's side along the center. Can you find all •

4 rzr

faces?

THE SPELL OF LOCK. RECREATION S. GAMES

289

GEOMETRIC DISSECTIONS &THE CURRY TRIANGLE Geometric dissection

be

well

dissection

can

Problems such

fascinating.

this

problems

known

PUZZLE

as

rectangle

which appears In Sam

puzzle,

Puzzles,

Loyd's Cyclopedia of

create

a

paradoxical type of

situation:

—:—^-:

** *» ta

*v

j

L

*., «

/

1

,

>-,.„

'-

1

*».^ •»

,; /

•*",

1

area= 13x5=65

1

Paul

that area=8x8-64 square units

with

a

hole Inside the

figure.

square units

Curry revised this paradox

so

cutting and rearranging the

parts resulted in

an

Identical

Here is how It works for this

figure

triangle.

Suppose one

side of

the

triangle

is shaded

and the other Is not. Cut it

apart. Turn

and

over

The

area of the original triangle is 60 square units. The revised triangle's area is 58 square units. Summing the area of each of the pieces gives a total of 59 square units. Therefore, the hole is 1 square unit

reassemble the shaded parts

as

shown.

triangle, yet It has a hole In the explanation? size

Supposedly It is the same

center. What's the

290


THE SQUARE

TRANSFORMATION

Determine how to cut this Into three

cuts,

so

pieces with

two

the three pieces

rearranged

<+*M»H IHW^

into

a

shape

straight can

be

square.

mni.il

in in ii

THE SPELL OF LOGIC RECREATION & GAMES

291

MACK SQUARES & OTHER RECREATIONS / make no

question but you will readily allow the square of 16 to be the most magically magical of any magic square ever made by any magician. —Benjamin Franklin

Imagine the formation of numbers

manipulative properties.

To many

of numbers seemed to possess

earliest record of

124

79

109

64

139

34

2200 B.C. when

by

a

3x3

the Emperor Yu

were

placed

and the forms In the 8th

square with

past

appears

special

these squares

or

powers. The

China around

In

magic number square, called lo-shu, on

the back of

the banks of the Yellow river.

blots

people

a

In the

19 941

49

seen

Into

magical qualities

magic square

a

PLAY I NO WITH

MAOIC SQUARES

on a

were

century

squares held the

sliver

read

In 9th

a

century

tray engraved

just

as

divine tortoise A.D.

with

a

was

along

Egypt, ink

magic

square

fortune tellers read tea leaves.

A.D. certain alchemists believed that these

key

to

converting metals

to

gold.

Islamic magic

squares often carried special meanings when the numbers of the

292

THE MAG IC O F MATH EMATICS

were

squares

to utilize their letter numerals. The Arabs

arranged

used different

arrangements of the

3x3

magic square to signify the

signs of fire, water, earth, and air. In addition, each planet had magic square associated with it, and magic squares their work with

astrology.

variety ways

placing

—

them

embroidering

them

the facade of

magic square

the

magic

one

used in

a

the womb of a

woman

garments of soldiers,

placing

over

for protection, wearing magic

the previous page

on

squares

used in

were

were

healing.

square with the year 1994

square, every was

on

building

a

square amulets for luck or

The

"mystical"

magic square

a

in labor, on

These

a

of its numbers

added to each number In the

was

transformed into

it.

Starting with

were

multiplied by

on

a

3x3

15.

a

magic

Then 4

resulting square.

Over the centuries many

properties and methods of construction

have been discovered and

developed.

Here

are some

1) Each same can

of these properties—

row,

column, and corner-to-corner

number.

Each

be obtained In

square has

magic

one

of the

a

total the

diagonal

magic

constant that

following ways:

(a) A magic square's order is determined by the number of rows or

columns it has.

numbers 1, 2, 3

where

n

is the

A magic square composed of the natural n% magic constant value is (l/2)(nfn^ +1)),

square's order. In general, ifiis the initial term

andfi the amount each term is increased, then the magic

constant=i*n+fi(n/2)(n2 -1). (b) Take any corner, sum

place

size

magic square and starting from the left hand

the numbers

sequentially along each

row.

The

of the numbers in either diagonal will be the magic

constant.

2) Any two numbers (In

equidistant

a

from Its center

row, are

column,

or

diagonal)

complements (2

that

are

numbers whose

THE SPELL OF LOCK, RECREATION & GAMES

This is Benjamin Franklin's super duper 16x16 magic square. It has all the properties of the regular magic square except its comer to comer diagonals do not total to its magic number, 2056. But as the diagram illustrates its

magic number pops up in

diagonals, broken 8-parallel

so

many other ways,

rows, any 4x4 square;

such,

and

as

—

broken 8-

perhaps

you can

In the 1952 Journal of the Franklin Institute, Albert Chandler contends that this magic square is not Franklin's original, but one that was set incorrectly by a printer.

find

more.

sum

equals

the

sum

of the

largest

and smallest numbers of the

square). 3) There Into

are

a new

many ways to transform

an

existing magic

square

magic square.

(a) Any number may be either added or multiplied number of a magic square, and

a new

to every

magic square will be created.

293

294


(b) If two

rows or

interchanged,

the

two columns,

equidistant from,

in

an even

(d) Interchange partial quadrants are

A few

are—

are

resulting square is a magic square.

(c) Interchange quadrants

There

the center

many methods

in

one can use

1) For odd order squares

—

order magic square.

an

odd order magic square.

to construct

magic squares.

a) the staircase method (by De la

Loubere), b) the pyramid method, c) the knights tour method for odd order

greater than 3x3, d) the Philip de la

Hire method

of

1705.

2) For

order squares

even

—

a) the diagonal replacement

method fora 4x4; b) 8x8 diagonal method.

3) For any order magic square

complicated

and involved.

developed by B.

Considering that

more

the

—

The methods that do exist

One such uses borders and

are

was

Frenicle.

lengthy history of magic

has been written

on

squares, It Is

no

surprise

magic squares than any other

mathematical recreation.

Some people have spent hours, others days, yet others months developing methods for creating magic squares, discovering their properties, developing new Ideas about

them, and playing with their magic. Among the vast list we find

Benjamin Franklin's square

super

magician*, who

can

starting number almost connecting the

designs

that

duper

16x16

magic square— the magic

create any 3x3

magic square with any

Instantaneously— magic

consecutive numbers of

are

also used to

—

classify the

a

lines'

(segments

magic square) dazzling

squares

as

symmetric

or

magic circles, cubes, hexagons, stars, spheres, magic domino squares, prime number magic squares, panasymmetric

—

diagonal magic

squares.

And the list continues to grow

people get hooked on playing with magic squares.

as more

THE SPELL OF LOGIC, RECREATION S. GAMES

•Anyone 3x3

can

be

a

magic

magic square. Challenge

location of

a

magician.

square

someone

Just memorize the basic

to

place

a

number In any

blank 3x3 square. For example, here the number 78

placed where the the 2 is located in the basic magic square. immediately create a new magic square by filling in the blanks, recalling that one of the

was

You

properties of a magic square is that a new one can

be created

by adding

constant to each number of

magic

an

a

old

square. In this case 76 is

added to each number of the basic 3x3

magic square.

84

77

79

81

80

85

82

?

?

?

?

?

78

The role of magic squares has centuries.

?

Today they

are

mathematical recreations.

changed significantly

viewed

as

over

the

fascinating, challenging

295

296


BENJAMIN FRANKLIN'S ..

»

Benjamin

^.- /'in/*I C

MAGIC CI KCLt

Franklin

square enthusiast.

he

was a

was

a

magic

In fact, while

clerk at the Pennsylvania

Assembly,

he often relieved the tedium of his work by making

magic circles.

For his

largest

magic circle, the numbers along a radius of the

circle all total the

same

amount,

fall In between Intertwined circles.

as

do the numbers that


297

LEONHARDEULER& THE KNIGHT'S TOUR This wonderful

created 18th

magic square

Leonhard Euler In the

by

century.

As In most

squares Its rows,

magic columns, and diagonals total the

In this case 260.

whose

was

and

rows

fascinating

Is how

In addition, there

columns each totals a

knight

chess

every number of the entire square

by just following

are

the

moves

same

number,

four squares within It, 130.

piece, starting

But at 1,

sequentially from knight.

1

even can

more

land

through

allowed of the

it ...

1

48

31

50 33

16

63 18

62 19

14

35

15

34

17

64

4

45 20

61

36

13

44

25

56

40 21 60

28

53

8

41 24

57

12

37

43

6

55

26 39

10 59

22

54

27

42

7

30 51

46

3

47

2

49

32

52

29

5

9

58 23

38 11

on

64.

298


THE KONIGSBERG BRIDGE PROBLEM Tîe

iinnATC

UKPATt

°^

clt^

founded

by

1308,

and

K*nlgsberg

Teutonic

was

in

Knights

'

easternmost

power for

Kaliningrad

over

seven

was

called

naval base of the USSR

Today

is located between Lithuania and Poland. How do the

bridges

find the

largest

the

as

outpost of German

400 years. After World War II it

and became the

Konigsberg

served

of Konigsberg look

impossible path? First,

today?

Do

people

let's recap the

still

attempt

to

Konigsberg bridge

problem—

Konigsberg bridges in 1700s

"Guts-Giblets^ Rrw^l Br,d9e Bridge

Over the centuries,

wealth

of enjoyment

the

Konigsberg bridge problem has provided

back to the 1700s, The setting

along

the

Pregel

a

and mathematical interest. The problem dates was

the

city ofKonigsberg situated

River. The two islands in the river

city and connected to it

were

part of the

A

by seven bridges. delightful tradition had developed among the residents of Konigsberg of taking a Sunday stroll along the city banks and islands while attempting to discover a

path that

any

would transverse all

bridge. Although

it

was

seven

considered

bridges without recrossing an

amusing

entertainment

THE SPELL OF LOGIC RECREATIONS. GAMES

by

most

the time,

people of another

developed

twist

mathematician discovered and

a

this

to

recreational

pastime.

Swiss

mathematician Leonhard Eider (1707-1783) had learned of the in St.

Konigsberg bridge problem

Petersburg

while in the service

of

Catherine the Great ofRussia. In 1735 Euler

far

more

presented a paper to and

reaching

mathematics than

had

simply

a

the Russian Academy that more

substantial

the solution to the

was

Impact

on

He

bridge problem. topology. Unlike

set Into motion Ideas that launched the field of

Euclidean geometry, which deals with size,

shape and rigid

that studies properties of

objects, topology is geometry objects that remain unchanged even when the size and shape of the objects are distorted. For example, if a triangle is distorted Into a a

topology studies which of the object's properties unchanged. With the Konigsberg bridge problem Euler circle,

or a

square remain

transformed and simplified

design (called graph simplified the

the

city

to which the

assigned il ustrated each

vertex

a

bridge

concluded that the all

seven

back

was

lifting

each vertex

as

to

point.

being either

graph

that

tracing

either

without

an

odd

or even

an even or

was

starting

or

most 2 odd vertices— i.e. 0 if

beginning point

was

on

one an

e created and

by

either passing

ending your journey

the other hand,

ending point

traceable (no

and

point

by beginning

An odd vertex,

the

over

Euler identified

in and out of the vertex

at that

crossing

doubling

pencil

a

He noted that

point.

of

without

it up.

led he

He

an arc.

comparable

mathematical

and

problem

bridges

the network with ever

bridges

point

with

a

part of

For each

problem.

Into

physical setting

a

network) which encompassed and

or

was

created

by

of the journey. Thus any

doubling back)

could

only

have at

they were all even or 2 if one was a ending point. In addition, he figured

299

300


One

of the three original remaining bridges of Konigsberg.

out that If the one

graph

had

would have to lift the

an even

pencil

number of odd vertices, say 10,

half this number, that Is 5 times,

to trace the entire

graph. In his dissertation, Euler points out that the Konigsberg bridge problem seemed geometric In nature, but that Euclidean geometry did not seem to apply since the bridge problem did

not deal with

"magnitudes"

nor

could It be resolved

using "quantitative calculations". Instead the problem belonged the

"geometry of position",

as

topology

was

to

first described and

named by Gottfried Wilhelm von Leibniz. Thus, Euler's solution to the

Konlsgberg bridge

problem

acted

as

a

and

catalyst

Introduction to the field of topology. As the illustration

specific

probably

name

bridge. Today

indicates, each of the related to what

only three of the

seven

seven

was on

bridges had

that side of the

original bridges remain

Honey bridge, the High bridge, and the Wooden bridge.

bridgeway

was

a

constructed that connects the two banks

as

—

A

the

new

shown

THE SPELL OF LOCK, RECREATION S.CAMES

and

bypasses

the

Tour

completely.

island

guides

of

Kneiphof

often relate the

story of the Konigsberg bridge problem. Some

guides

unsolved. If

even

a

its

touched down more are

is

drawn,

appeal. If the

on

remains

updated the

new

as

important part

o

the island, the network would be somewhat

but the

its

new

bridge

interesting. Unfortunately, the

history,

easily

it

network of the

Konigsberg bridges problem loses

claim

this

seven

problem legacy bridges. Euler's brilliant

in the

development

bridges

left is not solution

of

Konigsberg

destroyed

as

remains

an

of topology.

High

Bridge

Acknowledgements: Special thanks for updated information and photograph to Art Cooley and to Rod Crittenden for bringing the information my attention.

to

301

30}


CHECKERBOARD

MANIA

I Thousands of years ago ancient

Egyptians r

played

checkered boards.

on

games However,

the

modern game of checkers dates back to the same

beginning

board

followed

as

along

of the 12th

century

In

Europe.

It used the

chess, playing pieces from backgammon, and

the line of the game

alquerque with regard

movement and number of pieces. The board Itself poses

of problems that have been

to

a

the

wealth

developed over the years.

COVERING CHECKERBOARD WITH DOMINOES If two a

opposite

corners

checkerboard

removed, checkerboard

are

the

can

be

dominoes?

of

covered

by

Assume that

each domino is the size of two

adjacent

squares of

the checkerboard. dominoes

placed

on

The

cannot

top

be

of each

other and must lie flat.

THE SPELL OF

LOGIC, RECREATION

TAKING A CHECKERBOARD APART WITH ONE FELL SWOOP

Using your visualization skills and folding lllllllllf techniques, determine

straight

cut to

a

way to make

one

|||||l|llll

separate the checkerboard 1|§1§1||||

Into 2x2 squares, such

as

this

one.

6. GAMES

303

304


THE SQUARES OF A CHECKERBOARD The 8

rows

and 8 columns of black and red squares composing

checkerboard

can

be

grouped

a

Into different sized squares. These

squares range In size from 8x8 to lxl. How many squares of various sizes

can

be found

on a

checkerboard?

THE SPELL OF

LOGIC RECREATION & GAMES

TWO OF SAM LOYDS CHECKERBOARD PUZZLES

Puzzllst Sam

Loyd

must devise

a

pieces which which the

puzzle was called A Battle

Royal.

Rearrange the pieces

an

so

they form

8x8 checker

board.

Mrs.

Pythagoras' Puzzle. You

be

joined

Into

a

pattern

Into three

13x13 checkered square In

pattern and direction of the design

checkerboard

that

puzzle

method to cut this checkered

can

This Sam Loyd

8

called this

Is retained.

305

306

THE MAG IC O F MATH EMATI CS

AFEWOLPIES

SAM LOYD'S GET OFF THE EARTH PUZZLE

Sam

Loyd's

Get

Earth puzzle is the most

Off the one

of

popular

disappearing puzzles ever made. Loyd patented and sold

it in 1896, more

than 10

million copies. The puzzle with

a

was

made

movable

circular disk that rotated the Earth. Can you discover what

happens

to the 13th

warrior when the

sphere

is rotated from

NEtoNW?

THE SPELL OF

LOGIC, RECREATION 6. GAMES

SAM LOYD'S BATTLE OF THE FOUR OAKS PUZZLE Four trees

pieces

so

are

In

a

line

on a

square field. Divide the field Into four

that each piece Is the

same

size and

shape and has one

oak tree.

SAM LOYD'S WEIGHT OF A BRICK PUZZLE If

a

brick balances with three quarters of

a

brick and three

quarters of a pound, then how much does the brick weight?

TlTCAtoJlTXteCK

307

308


SAM LOYD'S CHEESE PROBLEM If the cheese round

were

cut

as

shown, how many wedges of

cheese would be produced?

1

WHATS LEFT IN THE CANDY BOXES ? Somehow the labels of these

boxes of candies got mixed up so

that

no

label

now

3 chocolates

indicates

the correct contents.

What is the least number of

candies you must test from which box(es)

to

determine

2 chocolates 1 creme

which box has what?

LEWIS CARROLL'S THE TWO CLOCK PUZZLE Which is better, that is year

clock

right only once a clock that is right

or a

twice

a

a

day?

THE SPELL OF LOGIC, RECREATION & CAMES

TARTAGLIA'S PUZZLE

16th for

Italian mathematician Niccold

century

discovering the solution

to the

He also wrote the three volume set Trattato misure

(Treatise

addition, he a

was

on

Numbers

Tartaglia

is famous

general cubic equation (1535).

generale

and Measures

di numeri et

(1556-1560).

In

the first to translate Euclid's The Elements into

modern Western

language (1543).

Here is

a

very famous

mathematical teaser he devised:

Three

newlyweds

come

to

a

river bank where

a

small boat is to take them across, but the boat hold

can

only

two

people

at

a

time.

Each

husband is the jealous type and very protective his

beautijul bride.

In

mellow, they decide that with

How do the

a man

order to no

woman

unless her husband is

couples get

minimum number of

of

keep things is to be

left

present.

to the other side of the river? What's the

trips?

309

SOLUTIONS

page 31. WHY ARE MANHOLES

page 273. THE IMPOSSIBLE

ROUND? SOLUTION: A circle by definition Is the set of all points In a plane that are equidistant from a given point. Therefore, a circular lid could not fall through a circular manhole, since all its diameters are equal. If the manhole were square, its lid could fall into the manhole by putting it through along the

POSSIBLE SOLUTION: The slit that is cut in the paper is placed into the hole. Then the buttons can be fed easily

square's diagonal. page 171. EULER'SMACIC FORMULA SOLUTION: For a rhomblcosldodecahedran— F=62 (30 squares, 20 triangle, and 12

pentagons); V=60; E=120; therefore F+V-E=2.

page 271. WHAT ARE THE SCENARIOS? SOLUTION: (1) Tom and Jerry are professional baseball players, but play on different teams. This particular day their teams were playing each other. Tom is a catcher who prevented Jerry from reaching

the slit and removed from the paper.

through

page 273. STACKING THE DIE SOLUTION: The trick to this problem is recalling that opposite faces of a die total 7. The first stack has 10 dice which would give 70 pips minus the face of the top die that is visible, gives 69 pips. The second stack has 7 dice, giving 49 pips, minus the top die's pips of 3 gives 46. The grand total is therefore 69+46=105 pips.

page 275. THE BLOCK LETTER PUZZLE SOLUTION:

home plate. (2) While walking down the street, Eric had a terrible attack of hiccups. He went into the bar to get a glass of see if it would help him.

water to

bartender, realizing Eric's problem, thought he would try to scare the hiccups out of Eric. It

The

workedl

(3) Mary's husband had died a number of years ago. She kept his ashes in an urn on her kitchen table. Also on her table was a bowl with goldfish. Mary had left her kitchen window open this particular day. Her neighbor's cat had come in, and in trying to get the goldfish, the cat knocked over the goldfish bowl and the urn. Her husband's ashes were spilled onto the floor. page 272.RECTANGULAR VOLUME ILLUSION SOLUTION: There are sixty-one 2x1x1 blocks in the structure.

page 276. THE MILKMAN'S PUZZLE SOLUTION:

lOqt.

lOqt 5qt.

10 5 5

10

0

5 1

0

9

10 10 10 10

1 0

0 1

4

10

5

1

4

10

2

4

8

10

2

0

8

6

2

10

6

2

4 2

9

4qt 0-starting 4

—ending

311

312

SOLUTIONS

page 277. IS IT POSSIBLE? SOLUTION:

Locate the 4 points on the sides of the angle as shown, and drawn In the two segments. The segments' point of Intersection and the vertex "of the angle determine the line that bisects the angle. Note: Showzls^2 by first showing AABCSAADE by SAS -?

zBs ZD-»ABEP3ADCP

by

SAA-+BPSDP

-?ABPASADPA

by

SSS-+Z1SZ2

page 277. CAN YOU MAKE A LATIN SQUARE? SOLUTION:

page 289. CURRY TRIANGLE EXPLANATION SOLUTION: When the other missing calculated using similar the results would actually create sections that would overlap In some places and leave gaps In others.

parts

are

triangles,

page 290. THE SQUARE TRANSFORMATION SOLUTION:

page 285. PLAYIN6WITH POLYHEXES SOLUTION: The triangle is the only figure that cannot be polyhexed.

page 302. COVERING THE CH ECKERBOARD WITH page 286. LEWIS CARROLL'S DOUBLETS SOLUTION: (1) EYE-LYE-UE-UD 12) pnv-prrs-Frrs-fins-findfond- FOOD- GOOD

(3) TREE-FREE-FLEE-FLED-FEEDWEED-WEU3-WOU3-WOOD (4) OAT-RAT-ROT-ROE-RYE

DOMINOES SOLUTION: It is not possible to cover the altered checkerboard with dominoes. A domino must occupy a red and black square. Since both corners removed were the same color, there will not be a compatible number of red and black squares left.

SOLUTIONS

page 303. TAKING A CHECKERBOARD APART

page 305. A BATTLE ROYAL SOLUTION:

SOLUTION:

StcP

Step 1

2

StcpS Step

Step

3

4 I / I Cut ^—'

the

along diagonal

dotted line.

page 307. THE FOUR OAKS PUZZLE SOLUTION:

page 304. THE SQUARES OF A CHECKERBOARD SOLUTION: There are: 1 square of size 8x8 4 square of size 7x7 9 squares of size 6x6 16 squares of size 5x5 25 squares of size 4x4 36 squares of size 3x3 49 squares of size 2x2 64 squares of size lxl for a total of 204 squares

page 305. MRS. PYTHAGORAS' PUZZLE SOLUTION:

1 -

»

A

1 ^

i

-$'

¦

*

"

y

<>

a

«¦

~

*

•.

%

ii.

r

**

.

s_: _

"

"¦

_,

f

_:_ ::

*

>

*

»

^

_:_n-

.:.ir; *

zjz jizz.z

.

r -~ t~-

T:::7:,-Er-::

::::-î::?i: *—^y zizizi-—--.M-i'i'i'tv1^ T--^-if'

n

hi

:::-::£:EB

313

314

SOLUTIONS

attend.

page 30"7. THE WEIGHT OF A BRICK PUZZLE

good,

SOLUTION: One way to approach the problem Is to add three more bricks to the left side of the scale. To

all, and the other loses

balance this quadruple the items on the right side. This means 4 bricks on the left balance a total of 3 bricks and 3 pounds. This produces the

equation 4B=3B+3. Solving for B (brick's wight pounds), we get B=3. Another

approach

one

doesn't go at a

minute

every day; which would you prefer? "The losing one," you answer, a doubt" Now observe: the which loses a minute a day has

"without one

to lose twelve hours

or seven

hundred and twenty minutes, before it is right again, consequently it is only right once in two years, whereas the other is evidently right as often as the time it points to comes around, which happens to be Lewis Carroll twice a day.

in

is to write

now

I have two clocks:

an

equation from the first scale's balance, namely B=(3/4)B+(3/4)

—

and solve for B.

page 308. TARTAGLIA'S PUZZLE

page 308. THE CHEESE PROBLEM

SOLUTION: The rule that no woman is to be left with a man unless her

SOLUTION: According to Sam Loyd, the first cut divides the cheese into two pieces. The second cut makes pieces. The third makes 8; the

4

fourth 15; the fifth 26; and the sixth 42.

page 308. WHAT'S LEFT IN THE CANDY BOXES? SOLUTION: You only have to test one candy from the box labeled 2

chocolates & 1 cream. If the candy you test is chocolate, you know it cannot have 2 chocolates & 1 cream because the box is mislabeled. Therefore, it must have 3 chocolates. Since all boxes are

mislabeled, this would

mean

the box

labeled 3 creams must have 2 chocolates & 1 cream and the box labeled 3 chocolates has 3 creams. If the candy you test is a cream, you know that the box has 3 creams. This would mean the box labeled 3 creams must have 3 chocolates, and the box labeled 3 chocolates has 2 chocolates & 1 cream.

page 308. LEWIS CARROLL'S CLOCK PUZZLE SOLUTION: Which is better, a clock that is right only once, or a clock that is right twice everyday? "The latter," you

reply, "unquestionably." Very

husband is present makes

11

crossings necessaiy. Notation:

husband-one=Hl;

wife-one-Wl etc.

1) HI takes crossings

Wl

over

and returns— 2

2)W3 takes W2 over and crossings

returns— 2

3) H2 takes HI over and returns with W2- 2 crossings 4) H3 takes H2 crossings 5)

H3 takes W3

to

pick

over

and returns— 2

over and H2 returns up his wife.— 3 crossings.

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organic

243-263

spacefilling solids 254-256 Mary's Cathedral 249, 260 251, 252

A-life

Frank

40

buckmlnsterfullerene

253

253

buckyballs

Bufano, Benlamlno 261-263

Lloyd

Buffon, Count

68

278

191

Achilles & the tortoise

44

chess board addition

22

All At One Point, Italo Cavlno's

60

camera

207

44

Arlandes, Marquis d'

6

projective geometry M.C. Escher

85-87

79-82

Cardan, Jerome 41 Carroll, Lewis 22-23, 286 Calvlno, Italo 60 Sir

B 191

mathematical concepts 147, 148 number system 40, 147-150 square root

approximations

147-150

Babylonians

and the

Pythagorean

theorem 148, 149, 150 Bamsley, Michael 279

Bellaschi, Pietro Bernoulli, Daniel

bicycles, m 18 binary numbers Napier

260 6

& 193-195

47

chaos,

flight of birds

game

Babbage, Charles Babylonian

6

George

Cesaro, Ernesto the

210

reality

16

45, 47

Cantor set 47

Cayley,

art,

69

and mathematics

Cantor, Georg

162

Archimedes spiral 158 area o f circle

artificial

194-195

Calder, Alexander

44, 156

Archimedes

193

chess board

multiplication

Anaxagoras 44 Andre, Carl 69 Andrews, Patricia

Aristotle

calculator,

85

Leon Battista

Alice in Wonderland

John

61

216-217

abacus

Albert!,

Luis

Brancroft, Thomas 65 Brooks, Robert 57

261

tensegrity

170

Bolzano, Bernhard 45 Bondone, Giotto di 85 Book of Sand,The 61 Borelll, Giovanni 6 Bouvet, Pere Joachim Bragdon, Claude 65

St.

Wright,

156

Janos

Borges, Jorge 262

250. 252

Dymaxlon House

Bolyal,

Chartres, R. Chen

128-129

278-279

278

Jing-run

161

Chinese, musical scales

181

piling squares proof Pythagorean theorem Chou Pel

152-153 153

152

Chudnovsky, George V. and David V.

Chuquet, circle,

215

Nicholas

area

41

158

Coates, Robert M.

14

codes, communications of bees

123

322

INPEX

codes, continued

De

228 sequence music Julius Caesar's substitution code 200

genetic

molecular National

225, 226

Security Agency

complex numbers complexity 14-15

201

42

computation, addition, early English

Egyptian division 155 Egyptian multiplication 155 multiplication, Italy 1400s 192 computers and A-llfe art

261-217

206-207

fuzzy logic

219-221

DES

encryption formula

dice,

ancient

Arithmetica

Discobolus

68

189 -221

language

monitoring

DOrer, Albrecht conic section

215

geometric

196-198

pixel 162

cube

69

203

77,78, 171

cuboctahedron-truncum

99

Curry, Paul 289 cyberpunks 203 cyberspace 208-211 cycloid curves and

ocean waves

85, 22-23 90

constructions of 91

91

E

Egyptian division 155 Egyptian fractions 40 Egyptian multiplication

counting numbers 41 Crandall, Richard E. 202

cryptography

162

112

Gothic letters

191 45

155

ellipse 19 elliptic geometry 37-38 elliptic geometry's name 170 elliptlpool and mathematics 19 encryption formula, equation equation

DES

of time

29-30

of time chart

equiangular spirals

30

133,136

Eratosthenes 202 P

Dali, Salvador Dante

Escher, M.C. 65

woodcuts

60

Danzig, George

Eublides B.

11

Euclid

200

184, 185

Epidaurus 139, 140

157

77

Dubner, Harvey

Nicomedes

57

226

duplicating a cube

computers/calulators,

Cristo

Adrien

square, Plato's Conan 200

duals

218

the past 191-195 the present 196-207

of

155

doubling the

Doyle,

208-221

countabiliry

163

60

Comedy.The

Douady,

215

optical generated

conchoid

43

Diophantus'

double helix

204-205

conservation

Little Fermat mathematics

obsolete

44

dimensions

Divine

200

154

226, 227 dodecahedron 77,78, 171 Dodgson, Charles L. 22-23, 286

irrigation & water

tree

167

Descartes, Rene

DNA

forest fires

the Younis

Denneau, M.M. 215 Desargues, Girard 87

division, Egyptian

92-95

the future

44

Democritus

Dichotomy 192

40

Progressione Dyadica Dedekind, J.W.R. 45

69, 79-82 59

220, 221

35

INDEX

Euler's F+V-E=2

double helix

171

Euler, Leonhard 161, 164, 171, 202, 297, 298-301

exponential

156

ancient

experimentation,

genes

function

and instruments

177

genetic bases genetic codes

230-231

knots & DNA

238

lipids

F

230-232

230-233 230-232

230

macromolecules

F+V-E=2

f-stop

171

and mathematics

230, 233 nucleotides 230, 231 proteins 230, 231, 232, 233 RNA 229, 230,233

16

57

Fatou, Pierre

Helaman 70, 72 Fermat, Pierre de 154, 162 Fermat's Last Theorem 162,

Ferguson,

163-164 Fermat numbers

202,215

Fermat, 10th number 202 Fibonacci numbers 133,134,135

Hexagons flying and

230

nucleic acids

288

starch

geodesic

230

domes

251-252

geometric dissections 289 geometric fractal 47 geometric worlds 37-38 geometries 167-169 evolution time chart

mathematics

5-8

time line

166

169

Fourier, John 177,178,186 fractal geometry 37-38

Geometry Supercomputer

fractal worlds

Gertner, Franz

47-57

fractals 47-57, 133 fractals and the coastline

Goldbach's 131

graphs

85

Greeks and

180,183

42

the gnomon 159 method for approximating 150, 151 square roots musical scales

games, mathematical,

trisecting

hex

tiles

286

287

an

181

angle

159

Green, Michael 240, 241 Greene, David 256

284

Rithmomachia

280-283

the chaos game Galileo 13, and

278-279

Dialogues Concerning Sciences

paradoxes

Two

44, 46

genetic engineering and 229-233

genectic engineering, chaos

theory

DNA

229-233

helices

H.

232

128

13

133, 136

Heuring, Vincent

165

mathematics

H

Happner, Frank

headlights.m

165

proportion

159

161

conjecture 112,

C

doublets

angle

Gold Bug, The 200

fractlonal dimensions Francesca, Piero della

domlnlzlng Penrose

an

Goldbach, Christian

132-

frequency and music Frey, Gerhard 164 fuzzy logic 219-221

139

gnomon trisects

130-131

nature

241

Project 238,

P.

hexagon and the honeycomb nature

122

122, 124-147

hexagrams hexahedron

40

77,78

hexatetraflexagon Hsuan-thu

218

152

288

161

323

324

INPEX

Hubbard, John

57

L

human

body & proportions hyperbolic geometry 37-38 hyperbolic paraboloid 249, hyperbolic plane 164 hyperbolic world 58, 59 hypercube 43, 65, 67 hypertext 212-214

260

icosahedron

77,78, 171

infinite worlds

/2

Large Magellanic Cloud Law.The 14

44-46

44-46

134, 136

curves

Isozaki, Arata

256

125,126

Little Fermat compu ter 215 Lobachevsky, Nikolai 170

Lowe, Danny 186 Loyd, Sam and puzzles

289,305, 306, 307, 308 M

magic circle of Benjamin Franklin

J

Jefferson, Thomas, cipher wheel 200 La Jangada 200

Julian sets

knight's

11 12

29-30, 228

170

tour &

magic

Koch, Helge

Matelskt, J. Peter

57

curve

embroidery stitching 75

and

mathematical ideas In stamps 21 mathematical models & chemistry

problems

solved &

mathematical mystery tales 267-271

54

mathematical stars

snowflake

curve

snowflake

curve, area

47, 48, 127

Konigsberg bridge problem 160,162, update 298-301 Rummer, Ernst E.

122

202

unsolved, famous 160-162

238

von

293

mathematical

mathematical

237, 238

knots &DNA

297

234

squares 297 knot theory 236-241

knots

tour

49, 50, 56,57 Mandlebrot controversy 57 manholes and mathematics 31

algorithm

Johannes

Klein, Felix

knight's

Mandelbrot, Benoit

K

Karmarkar

the

Manasse, Mark

57

Karmarkar, Narendra

296

magic squares 291-295 and

Benjamin Franklin magnetic fields & bees

Jordan, Harry F. 218 Julia, Gaston 57

Kepler,

126

Lee, John 260 Lees, John 186 Leibniz, Gottfried Wilheim 40,45,191,300

hexagon

41, 42

interactive compu ting 212 interactive literature 213

Involute

59

lines of symmetry of a

170

imaginary geometry imaginary numbers infinities

Madeline

Lenstra, Arjen 202 Lilienthal, Otto 6

40

IChing

L'Engle,

ladder method for approximating 151 the

170

origin

name

235

55

74

mathematical stories, A Mathematical Visit

24-28

The parable 163

164

ofn 109-110 The parable of the Fractal 51-53

The parable 101-105

of the quaternion

INDEX

mathematical wavelets 199 mathematical worlds 34-61

formation

M6bius, August 212,234 Mobot Lab, MIT9 217

35-36

In literature

modular arithmetic

58-61

mathematicians chart,famous 146 mathematics and the arch 257-259

designs & art 74-78 early container design elllptlpool 19 everyday things 4-31 f-stop/camera 16 fingerprints 199 flying 5-8 the garden 133-137

India

the musical

225-227

Napier's rods

& games

nature &

175,176

119-141

179-183

156, 184-187

sculpture 68-73 telephone call

the

waves

9-12

138-141

Mazur, Barry

164

Menaechmus

13

Mersenne, Marin 202 Mersenne prines 202

17

13

122,124-147

278

negative numbers

179-183

Qsound 186, 187 the recycling numbers

130-131

hexagons

Needle Problem

score

calculating

191

nature & fractals

31

parabolic electric heaters parabolic reflectors 13 pool tables 18

sound

191

numbers & 193-195

binary

mathematics and nature

181 68

N

174-183

the octave

Myron

Napier, John

144-171

body

musical scales

181

Persia

265-309 music

181

Greece

121-123

manholes

177

shapes

musical scales and mathematics 179-183 musical scales, China 181

13

historical notes

logic, recreation,

spheres, Kepler's

182

their 166

genetic code music 228 genetic engineering 229-233

honeybees

174-183

176

music, artifacts 181,182 musical instruments and

189-221

headlights

22-23

music & mathematics

79-82

18

computers

the human

Tale, The

Mouse's

215

56

curves

music of the

64-95

the art of M.C. Escher

bicycles

monster

music and ratios

architecture 243-263 art

68

Michelangelo

41

Nervi, Pier Luigi 256, 260 networks 133, 134 Noguchi, Isamu 68, 69 non-Euclidean geometries 37-38, 167-169

non-periodic tiling

67

number system chart & overview

144

number worlds

numbers &

39-42

recycling,

17

m

O

octahedron

77,78, 171

octave & mathematics

179 -183

optical computers 218 optical illusions 208-209

325

326

INPEX

P

problems of antiquity, impossbile 160, 162, 163 projective geometry & art 85-87

pangeometry 170 Paoli, Luca 235

the human and music

13

&

Are all wheels round?

bisecting an angle the block letter the

Peano

68

square transformation

persepctive 85, 86 Phene, Claude 204, 205

stacking the die Tartaglia's puzzle

309

Q

Plato

quadratrix of Hippias

157

quipu

Poisson, Simeon

R

276

Polybius' square cipher polyhedra 77, 78, 251, property

200

ragas, music of India 181 random number generator

253

ratios & music

171

remailers

285

m

honeycombs

primes

202

probability

112

154

256

12

Rlbet, Kenneth A. Robbin, Tony 67 robots

217

Romans and 202

171

164

robots, Attila 217 rod numerals, Chinese

202

problems 160-161 and encoding messages primes

73

rhombicosidodecahedron

Fermat primes 202 Germain primes 112 Mersenne

203

rhombic dodecahedron

18

Pozzo, Andrea del 87 prime numbers 45, 112,

154

176

Resch, Ronald Dale

Poncare, Henri 58,59 Poncelet, Jean Victor 87

pool tables,

162

191

77

Poe, Egar Allan cipher 201

polyhexes

186, 187

Qsound

157

Platonic solids

twin

290

273

Pilatre de Rozier, Jean 6 pine cone numbers 135

invariant

272

135

phyllotaxis

Meno

308

palls 277 rectangular volume illusion SamLoyds 306,307,308

181

Perry, Charles

273

the milk

Penrose, Roger 287 pentatonic, Chinese musical scale

302-305

puzzles

Latin square 277 Lewis Carroll's two clock

48

curve

308

Curry triangle 289 impossbile possible

56

curves

274

277

275

candy box

checkerboard

Parallel Postulate, Euclid's 5th postulate 160,162,163, 167,170 Pascal, Blaise 41,87,154,191

pathological

176

Pythagoreans 156,182 puzzles & problems.

218

fuzzy logic

235

body 182

Pythagoras

Achilles and the tortoise 44

paradoxes

165

proportions

Paquette, Leo 234 parabola 13 parabolic celings 185, 186 parabolic electric heaters, m parabolic reflectors, m 13 paraboloids 13 paradoxes 44, 46 Dichotomy 44, 46

area

Russell, Bertrand

Ryan, Paul

260

191

the of circle 158

220, 221

INPEX

tessellations 122 and bees

$

Sand Reckoner, The Schartz, John 240

Scherks, Joel

44

scientific

L.

20

43,58,59 tetrachords, Greek musical scales

161 ancient

laboratory,

156

201

scytale 200,

84

advertizing

tesseract

Schlegel diagrams 78 Schlegel, Viktor 78 Schnlrelmann,

from the past In

240

TOE

sexagimal number system,

181

tetrahedron 77,78,171 Theon of Smyrna 156

CThe Theory of Everything)

239-241

Sumerian/Babylonian numbers

triple junction 125, 126,134,137 trisecting an angle 159,162

163

conjecture

Sierpincki, Waclaw Sierpinski triangle Sierpinski gasket simplex method 11 sinusoidal

curves

&

56

truncated octahedron

56

U

Uccello, Paolo

squaring

circle

a

V

Vaillancourt, Louis

69

Verne, Jules 200 Vinci, Leonardo da 85, 94, 95, 235

5,64,69, 70,

Vinogradov, Ivan M. 161 reality 208-211

162

srutis, music of India 181 St. Peter's Cathedral 184

virtual

Star Trek- The Next Generation 61,211

Voltaire, q

Stldel, Michael line

Vitruvius

41

stone age numbers

76

wave

height

140

wave

length period

140

wave

135, 137

Wright, Frank Lloyd

Wright,

call and 10

Z

Zadeh, Lofti

mathematics

Zeno

44

Zeno's

paradoxes

simplex

9-12

method

tessellatlng rectangle

11

135, 137 a

plane by modifying 83

curves

a

220 44

163

261-263

Orville and Wilbur

linear programming 11 Karmarkar algorithm 12

tessellate

163, 164

Elliptic

Fermat's Last Theorem

Taniyama, Yuktaka 164 Tartaglia, Niccolo 309 numbers

138-141

139, 140

Wiles, Andrew J. Wiles, Modular

T

telephone binary

126

and mathematics

65

29

symmetry

58

waves

Sumerian numbers 40

sundials

235

Wang, Llfan

39, 40

spirals

Strauss, Charles

166

waves

76

spirals

256

49,56

139, 140 Soleri, Paoli 249, 255, 256 sound 156,174, 180 sound, Qsound 186,187

straight

45

transflnite numbers

40

Shirnura-Taniyama-Weil

6

&

317

319

Mathematics teacher and consultant Theoni B.A. from the

University

her M.A. from Stanford to

demystifying

Pappas

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