Golden Ratio

Golden ratio This article is about the number. It is not to be confused with the with the pop music album or album or the  the calend calendar ar dates. dates .

is t o

as

ϕ =

√ 

1+ 5 2

= 1.6180339887 . . . .   A001622

golden den mean or golden Thegol The golde denn ratio ratio is also also call called ed the gol [1][2][3] section  (Latin:  sectio aurea ). Other names include [4] extre extreme me and and me mean an ratio ratio, medial section, divine divine proportion,  divine section  (Latin:   sectio divina ),  golden proportion , golden cut ,[5] and golden number .[6][7][8] Some twentieth-century twentieth-century artists  artists and  and architects  architects,, including Le Corbusie Corbusierr and Dalí  and Dalí,, have proportioned their works to approximate the golden ratio—especially in the form of the golden the golden rectangle, rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically be aesthetically pleasing.  pleasing. The golden ratio appears in some patterns some patterns in nature, nature, including the spiral the  spiral arrangement of leaves and leaves and other plant parts. Mathematicians since Mathematicians  since Euclid  Euclid have  have studied the properties properties of the golden ratio, including its appearance in the dimensions of a regular a regular pentagon and pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with with the the same same aspe aspect ct rati ratioo. The The gold golden en rati ratioo has has also also been been used to analyze the proportions of natural objects objects as well as man-made systems such as financial as  financial markets, markets, in some [9] cases based on dubious fits to data.

i s to

Line segments in the golden ratio

1 Calc Calcul ulat atio ionn Two quantities quantities a  and  b  are said to be in the  golden ratio φ if a+b a = =  ϕ. a b

A  golden rectangle (in rectangle (in pink) with longer side a  and shorter side adjacent to a square with sides of length a , will  b , when placed adjacent  produce a  similar  golden   golden rectangle with longer side a + b  and  b = ab ≡ ϕ . shorter side a. This illustrates the relationship a+ a

One method for finding the value of φ  is to start with the left fraction. fraction. Through simplifying simplifying the fraction fraction and substituting in b/a = 1/φ,

In mathematics In mathematics,, two quantities are in the golden ratio  if their ratio isthesameastheratiooftheir sum to the larger larger of the two quantities. quantities. The figure on the right right illustrates a + b = 1 + b = 1 + 1 . a a ϕ the geometric relationship. relationship. Expressed algebraicall algebraically, y, for quantities  a  and b  with a  >  b  > 0, Therefore, a+b a def = =  ϕ, a b

1+

1 ϕ

=  ϕ.

wher wheree the Greek Greek lette letterr phi ( ϕ or φ ) repr represe esents nts the golde goldenn ratio. Its value is: Multiplying by  φ  gives 1

2

2

HIST HISTOR ORY  Y 

ϕ + 1 =  ϕ 2

which can be rearranged to ϕ2

− ϕ − 1 = 0 .

Using the quadratic the quadratic formula, formula, two solutions are obtained:

√ 

1+ 5 = 1 .61803 6180339887 39887 . . . ϕ = 2

and

ϕ =

1

− √ 5 = −0.6180 33988 3398877 . . . 2

Because φ  is the ratio between positive quantities necessarily positive:

φ

is

√ 

1+ 5 = 1 .61803 6180339887 39887 . . . ϕ = 2

2 Histor tory Further information: Mathematics information:  Mathematics and art The golden ratio has been claimed to have held a spe-

Mathematician Mark Mathematician Mark Barr  proposed   proposed using the first letter in the name of Greek sculptor  Phidias   Phidias  , phi , , to symbolize symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes, the up percase form ( Φ  ) is used for the reciprocal  the reciprocal  of   of the golden ratio, [10] 1/ φ.

Michael Maestlin , first to publish a decimal approximation approximation of the  golden ratio, in 1597 

Renaissance astronomer  Johannes Kepler, Kepler , to present-day scientific figures such as Oxford physicist  physicist   Roger Penrose, Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined confined just to mathema mathematic ticians ians.. BioloBiologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. [12]

Ancient Greek mathematicians Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry.. The division geometry division of a line into “extreme “extreme and mean ratio” (the golden section) is important in the geometry cial fascination for at least 2,400 years, though without of regular pentagrams regular pentagrams and  and pentagons  pentagons..   Euclid's Euclid's  Elements  to Mario Livio Livio:: reliable evidence. [11] According to Mario (Greek Greek:: Στοιχεῖα) provides the first known written definition of what is now called the golden ratio: “A straight Some of the greatest mathematical minds line is said to have been  cut in extreme and mean ratio of all ages, from  Pythagoras and  and   Euclid in when, as the whole line is to the greater segment, so is ancient Greece, Greece, throu through gh the medi mediev eval al ItalItalthe greater to the lesser.” [13] Euclid explains a construcian mathematician Leonardo mathematician  Leonardo of Pisa and Pisa  and the tion for cutting (sectioning) a line “in extreme and mean

3 ratio”, i.e., the golden ratio.[14] Throughout the Elements , several propositions (theorems (theorems in  in modern terminology) and their proofs employ the golden ratio.[15] The golden ratio is explored in  Luca Pacioli's Pacioli's book De divina proportione  of 1509. The first first known known appro approxim ximati ation on of the (inve (inverse rse)) golde goldenn ratio by a decimal a  decimal fraction, fraction, stated as “about 0.6180340”, waswrittenin1597byMichae waswrittenin1597by Michaell Maestlin ofthe University of Tüb Tübinge ingenn in a lett letter er to his his forme ormerr stud studen entt Joha Johannes nnes Ke[16] pler.. pler Since the 20th century, the golden ratio has been represented by the Greek the Greek letter  φ  (phi  ( phi,, after Phidias after Phidias,, a sculptor who is said to have employed it) or less commonly by τ  (tau tau,, the first letter of the ancient the  ancient Greek root Greek  root τομή— [1][17] meaning cut ). ).

2.1 2.1 Time Timeli line ne

counter-clockwise were frequently two successive Fibonacci series.

•  Martin Ohm (1792–1872) Ohm (1792–1872) is believed to be the first

to use the term  goldener Schnitt  (golden  (golden section) to [21] describe this ratio, in 1835.

•  Édouard  ÉdouardLuca Lucass (1842–1891 (1842–1891)) gives gives the numeric numerical al sequence now known as the Fibonacci sequence its present name.

•  Mark Barr (20th century) suggests the Greek letter

phi (φ), the initial letter of Greek sculptor Phidias’s Phidias’s [22] name, as a symbol a symbol for  for the golden ratio.

•  Roger Penrose  Penrose   (b.

1931) disco discover vered ed in 1974 the Penrose tiling, tiling, a pattern that t hat is related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.[23] This in turn led to new discoveries about quasicrystals..[24] quasicrystals

Timeline according to Priya Hemenway:[18]

Applications ons and observati observations ons •   Phidias (490–430 Phidias (490–430 BC) made the Parthenon the  Parthenon statues  statues 3 Applicati that seem to embody embody the golden ratio.

•  Plato  Plato (427–347  (427–347 BC), in his  Timaeus , describes describes five

3.1 Aesth Aestheti etics cs

possible regular solids (the   Platonic solids: solids: the tetrahedron,,  cube tetrahedron  cube,,   octahedron, octahedron,   dodecahedron, dodecahedron, and See also:   History of aesthetics (pre-20th-century) and Mathematics cs and art icosahedron), icosahedron ), some some of which which are relate relatedd to the Mathemati golden ratio.[19] De Divina Divina Proporti Proportione one, a three-volume work by  Luca •   Euclid (c. 325– 325–c. c. 265 265 BC), BC), in his Elements , gave gave the the Pacioli Pacioli,, was publis published hed in 1509. 1509. Paciol Pacioli,i, a   Franciscan first recorded definition of the golden ratio, which friar friar,, was known mostly as a mathematician, but he was he called, as translated into English, “extreme and also trained and keenly interested in art.  De Divina Promean ratio” (Greek: ἄκρος καὶ μέσος λόγος).[4]  portione  explored the mathematics of the golden ratio. it is often said that Pacioli advocated the golden •   Fibonacci   (1170–1250) mentioned the   numerical Though ratio’s application to yield pleasing, harmonious proporseries now series  now named after him in his  Liber Abaci ; the tions, Livio points out that the interpretation has been ratio of sequential elements of the   Fibonacci setraced to an error in 1799, and that Pacioli actually actually advoquence approaches quence  approaches the golden ratio asymptotically. cated the Vitruvian the Vitruvian system  system of rational proportions.[1] Pareligious significance in the ratio, •  Luca Pacioli (1445–1517) Pacioli  (1445–1517) defines the golden ratio cioli also saw Catholic religious as the “divine proportion” in his Divina Proportione . which led to his work’s title.  De Divina Proportione  contains illustrations of regular solids by Leonardo by  Leonardo da Vinci, Vinci, •  Michael Maestlin   (1550–1631) publishes the first Pacioli’s longtime friend and collaborator. known approximation of the (inverse) golden ratio as a decimal a decimal fraction. fraction.

•   Johannes Johannes

Kepler   (1571–1 Kepler (1571–1630 630)) prov proves es that that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers,[20] and describes the golden ratio as a “precious jewel": “Geometry has two great treasure treasures: s: oneis one is the Theo Theorem rem of Pyth Pythagor agoras as,andthe ,andthe othe otherr the the divi divisi sion on of a line line into into extr extrem emee and and mean mean raratio; the first we may compare to a measure of gold, the second we may name a precious jewel.” These two treasures are combined in the Kepler the  Kepler triangle. triangle.

•  Charles Bonnet (1720–1793) Bonnet  (1720–1793) points out that in the

spiral   phyllotaxis   of plants going   clockwise and

3.2 Archit Architect ecture ure

Many of the proportions of the Parthenon the  Parthenon are  are alleged to exhibit  the golden ratio.

4 Further information: Mathematics information:  Mathematics and architecture The Parthenon The Parthenon's's façade as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. rectangles.[25] Other Other scho scholar larss deny deny that that the Greek Greekss had any aesthetic aesthetic association with golden golden ratio. For example, Midhat J. Gazalé says, “It was not until Euclid, however, however, that the golden ratio’s mathematical properties were studied. In the  Elements  (308  (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme extreme ratios. Its occurrence occurrence in regular pentagons and decagons was decagons  was duly observed, as well as in the dodecahedron (a regular (a  regular polyhedron whose polyhedron  whose twelve faces are regular pentagons pentagons). ). It is indeed indeed exempl exemplary ary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties.”[26] And Keith Devlin says, Devlin  says, “Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook  Elements , written around 300 BC, showed how to calculate its value.”[27] Later sources like Vitruvius like  Vitruvius exclusively  exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate mensurate as opposed to irrational proportions. proportions. A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals Kairouan  reveals a consistent application of the t he golden ratio throughout the design, according to Boussora and Mazouz. [28] They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret the  minaret.. The authors authors note, howeve however, r, that the areas areas where ratios close to the golden ratio were found found are not part of the original construction, and theorize that these elements were added in a reconstruction. The Swiss architect Swiss architect Le Corbusier, Corbusier, famous for his contributio butions ns to the mode modern rn inte internati rnational onal styl stylee, center centered ed his design philosophy on systems of harmony and proportion. Le Corbusier’s faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described described as “rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound resound in man by an organic inevitability, inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned.”[29] Le Corbusier explicitly used the golden ratio in his Modulor system Modulor  system for the scale the scale of  of architectural  architectural proportion. proportion. He saw this system as a continuation of the long tradition of Vitruvius of Vitruvius,, Leonardo da Vinci’s "Vitruvian " Vitruvian Man", Man", the work of Leon of  Leon Battista Alberti, Alberti, and others who used the proportions of the human body to improve the ap-

3

APPLICATION APPLICATIONS S AND OBSERVA OBSERVATION TIONS  S 

pearance and function of architecture of  architecture.. In addition to the golden ratio, Le Corbusier based the system on human on  human measurements,,  Fibonacci numbers, measurements numbers, and the double unit. He took suggestion suggestion of the golden ratio in human proportions tions to an extre extreme: me: he sectio sectioned ned his model model human human body’s body’s height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor the Modulor system.  system. Le Corbusier’s 1927 Villa Stein in Garches in Garches exemplified  exemplified the Modulor system’s application. The villa’s rectangular ground plan, elevation, and inner structure closely approximate golden rectangles. [30] Another Swiss architect, architect, Mario  Mario Botta, Botta, bases many of his designs designs on geometric figures. Several private private houses he designed in Switzerland are composed of squares and circles, circles, cubes and cylinders. cylinders. In a house he designed in Origlio,, the golden ratio is the proportion between the Origlio central section and the side sections of the t he house.[31] In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the Naqsh-e the  Naqsh-e Jahan Square and Square and the adjacent Lotfollah mosque. [32]

3.3 3.3 Pa Pain inti ting ng

The drawing of a man’s body in a pentagram a  pentagram suggests  suggests relation[2] ships to the golden ratio.

The 16th-century philosopher Heinrich philosopher  Heinrich Agrippa drew Agrippa  drew a man over a pentagram a  pentagram inside  inside a circle, implying a relationship to the golden ratio.[2] Leonardo da Vinci's Vinci's illustrations of polyhedra of  polyhedra in  in  De divina proportione  (On the Divine Proportion ) and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate speculate that he incorporated the golden ratio in his paintings.[33] But the suggestion that his  Mona Lisa , for example, employs golden ratio proportions, is not supported by anything in Leonardo’s own writings.[34] Similarly, although the  Vitruvian Man

3.5 3.5

5

Desi Design gn

is often[35] shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios. [36] Salvad Sal vador or Dalí Dalí,, influ influen ence cedd by the the work workss of   Matila Ghyka,,[37] explicitly used the golden ratio in his masterGhyka piece,  The Sacrament of the Last Supper . The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus behind Jesus and  and dominates the composition.[1][38] Mondrian has Mondrian  has been said to have used the golden section extensively in his geometrical paintings,[39] though other experts (including critic Yve-Alain critic  Yve-Alain Bois) Bois) have disputed [1] this claim. A statistical study on 565 works of art of different great painters, performed performed in 1999, found that these artists had not used used the golde goldenn ratio ratio in thesi the size ze of their their canv canvase ases. s. The The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from from 1.04 (Goya) to 1.46 (Bellini).[40] On the other hand, Pablo Tosto listed over over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.[41]

3.4 3.4 Book Book desi design gn

3.5 3.5 Des Design Some sources claim that the golden ratio is commonly used in everyday design, for example in the shapes of postcards, playing playing cards, posters, wide-screen wide-screen televisions, televisions, [44][45][46][47][48] photographs, light switch plates and cars.

3.6 3.6 Mus Music Ernő Lendvai   analyzes   Béla Bartók's Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic the acoustic scale, scale,[49] though other music scholars reject that analysis.[1] French composer Erik composer Erik Satie used Satie used the golden ratio in several of his pieces, including  Sonneries  golden ratio ratio is also apparent apparent in de la Rose+Croix . The golden the organization of the sections in the music of Debussy of Debussy's's  (1st Reflets Re flets dans l'eau l'eau (Reflections  (Reflections in Water) , from Images  (1st series, 1905), in which “the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the t he main climax [50] sits at the phi position.” The musicologist Roy musicologist  Roy Howat has Howat  has observed that the forMer  r  correspond exactly to the mal boundaries of   La Me [51] golden section. Trezise finds the intrinsic evidence “remarkable,” but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions. [52] Pearl Drums positions Drums  positions the air vents on its Masters Premium models models based on the golden golden ratio. The company claims that this arrangement improves bass response and has applied for a patent a patent on  on this innovation.[53] Though Hei Heinz nz Bohl Bohlen en proposed the non-octave-repeating non-octave-repeating 833 cents scale based scale  based on combination on  combination tones, tones, the tuning features relations based on the golden ratio. As a musical interval intervalthe the ratio ratio 1.618... 1.618... is 833.090... 833.090... cents cents (   Play ).[54]

3.7 3.7 Natu Nature re Main article: Patterns article:  Patterns in nature

Depiction of the proportions in a medieval manuscript. According to Jan to Jan Tschichold : “Page proportion 2:3. Margin proportions  1:1:2:3. Text area proportioned in the Golden Section.” [42]

Main article: Canons article:  Canons of page construction construction According to Jan to Jan Tschichold, Tschichold,[43] There was a time when deviations from the truly beautiful beautiful page proportions 2:3, 1:√3, and the Golden Golden Section Section were were rare. Many Many books produced between 1550 and 1770 show these proporti proportions ons exactl exactly, y, to within within halfa half a millime millimeter. ter.

Adolf Zeisi Zeising ng,, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of parts such parts such as leaves and branches along the stems of plants and of veins of  veins in  in leaves. He extended his research to the skeletons the  skeletons of  of animals and the branchings ings of their their veins veins and nerve nerves, s, to the propo proporti rtion onss of chemchemical compounds and the geometry of crystals of  crystals,, even to the use of proportion in artistic endeavors. In these patterns these patterns in nature he nature  he saw the golden ratio operating as a universal law.[55][56] In connection with his scheme for goldenratio-based human body proportions, Zeising wrote in 1854 1854 of a unive universa rsall law law “in whic whichh is conta containe inedd thegro the ground und-principle of all formative striving for beauty and completeness pleteness in the realms of both nature and art, and which which permeates, as a paramount spiritual ideal, all  structures  structures,, forms and  and   proportions, proportions, whether cosmic or individual,

6

4

MATHE MATHEMA MATIC TICS  S 

4 Math Mathem emat atic icss 4.1 Irrati Irrationa onalit lityy The golden ratio is an irrational an irrational number. number. Below are two short proofs of irrationality: 4.1.1 Contradi Contradicti ction on from from an express expression ion in lowest lowest terms n–m

m

n

Detail of   Aeonium tabuliforme  showing the multiple spiral arrangement rangement ( parastich  parastichy y)

If φ were rational  were rational  , then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram) agram).. But But it would would also also be a ratio ratio of intege integerr sides sides of the smalle smallerr recta rectangl nglee (the (the rightm rightmos ostt portio portion n of the diagra diagram) m) obtai obtained  ned  by deleting deleting a square. square. The sequence sequence of decreasi decreasing ng integer side lengths formed by deleting squares cannot be continued indefinitely because the integers have a lower bound, so φ cannot be rational.

organic  or inorganic organic or  inorganic,,  acoustic  acoustic or  or optical  optical;; which finds its fullest realization, however, in the human form.”[57] In 2010, the journal journal Science repor reported ted that that the gold golden en ratio ratio Recall that: is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.[58] the whole is the longer part plus the shorter Since Since 1991, several several researche researchers rs have have proposed proposed conpart; nections between the golden ratio and  human genome the the whol wholee is to the the long longer er part part as the the long longer er part part DNA..[59][60][61] DNA is to the shorter part. However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard If we call the whole n  and the longer part m, then the to animal dimensions, dimensions, are fictitious. fictitious.[62] second statement above becomes

3.8 Optimi Optimizat zatio ionn

n is to  m  as  m  is to  n  −  m ,

The golden ratio is key to the golden the golden section search. search.

or, algebraically

3.9 Percep Perceptual tual studies studies

n m = . m n m

Studies Studies by psycho psychologi logists, sts, starting starting with  with   Fechner, Fechner, have have been devised to test the idea that the golden ratio plays a role in human perception of beauty of  beauty.. While While Fech Fechner ner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive. [1][63]

To say that  φ  is rational means that  φ  is a fraction  n /m where  n  and  m  are integers. integers. We may may take  n /m  to be in lowest terms and terms  and  n  and  m  to be positi positive. ve. But if n /m  is in lowest terms, then the identity labeled (*) above says m/(n  −  m ) is in still lower terms. That is a contradiction that follows from the assumption that φ is rational.

−

( )

∗

4.4

7

Alternati Alternative ve forms 

4.1.2 Derivation Derivation from from irrationality irrationality of of √5

Another Another short short proof—pe proof—perha rhaps ps morecomm more commonl onlyy known— known— of the irrationality of the golden ratio makes use of the closure of closure  of rational √  numbers under addition √  and multipli√ 5 1+ 5 1+ 5  1 = − cation. If 2 is rational, then 2 2 is also rational, which is a contradiction if it is already known that the square root of a non-square non-square natural number is ber  is irrational.

� �

4.2 Minimal Minimal pol polynom ynomial ial The golden ratio is also an algebraic an algebraic number and number and even an algebraic integer. integer. It has minimal polynomial x2

− x − 1

Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.

4.3 Golden Golden ratio ratio conju conjugate gate

Approximations to the reciprocal golden ratio by finite continued   fractions,  fractions, or ratios of Fibonacci numbers 

4.4 Alternati Alternative ve forms

The formula  φ  = 1 + 1/ φ can be expanded recursively to The conjugate root to the minimal polynomial x2 - x - 1 obtain a continued a continued fraction for fraction for the golden ratio: [64] is

− ϕ1 = 1 − ϕ =

√  1− 5 2

1

[1; 1, 1, 1, . . . ] = 1 + ϕ  = [1; =

 −0.61803 6180339887 39887 . . . .

The absolute value of this quantity quantity (≈ 0.618) corresponds corresponds to the length ratio taken in reverse order (shorter segment reciprocal: length length over over longer longer segment segment length, length, b/a), and is sometime sometimess and its reciprocal: [10] referred referred to as the  golden ratio conjugate . It is denoted here by the capital Phi (  Φ  ): [0; 1, 1, 1, . . . ] = 0 + ϕ−1 = [0; Φ=

1 ϕ

1

1+ 1+

1

 . 1 + ..

1 1

1+

=  ϕ −1 = 0 .61803 6180339887 39887 . . . .

1+

Alternatively, Φ  can be expressed as

1

 . 1 + ..

The convergents The  convergents of  of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) successive Fibonacci numbers. numbers. Φ =  ϕ −1 = 1.61803 39887 . . .−1 = 0.61803 6180339887 39887 . . . . are ratios of successive Fibonacci The equation  φ2 = 1 + φ likewise produces the continued This illustrates the unique property of the golden ratio square root, root, or infinite surd, form: among positive numbers, that 1 ϕ

=  ϕ

− 1,

or its inverse: inverse: 1  = Φ + 1. Φ

This means 0.61803...:1 = 1:1.61803....

ϕ  =

�  �  �  1+

1+

1+

√ 1 + · · ·.

An infinite series can be derived derived to express express phi: [65]

∞

13 ( 1)(n+1) (2n + 1)! + ϕ  = . 8 ( n + 2)!n!4(2n+3) n=0

∑−

8

4

MATHE MATHEMA MATIC TICS  S 

Also: C ϕ = 1 + 2 sin(π /10) = 1 + 2 sin 18◦

D

1 1 ϕ =  csc(π/10) =  csc 18◦ 2 2

1.

2. 3.

ϕ = 2 cos(π /5) = 2 cos 36◦

S

A

ϕ = 2 sin(3π /10) = 2 sin 54◦ .

These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram a  pentagram..

4.5 4.5 Geom Geomet etry ry

B

Dividing Dividing a line segment by interior interior division division accordin accordingg to the  golden ratio

2. Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D. 3. Draw an arc with center A and radius AD. This arc intersects the original line segment AB at point S. Point S divides the original segment AB into line segments AS and SB with lengths in the golden ratio.

1/ φ

φ-1

1

4.5.2 Dividing Dividing a line segment by exterior exterior division division 1    /    

1/ φ3

  φ

C

2  

φ Approximate and true golden true golden spirals . The green green spiral is made made  from  from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of  logarithmic   logarithmic  spiral . Overlappi Overlapping ng portions portions appear yellow yellow.. The length length of the side of one square divided by that of the next smaller square is  the golden ratio.

The number  φ  turns up frequently in geometry in  geometry,, particularly in figures with pentagonal symmetry pentagonal symmetry.. The length of a regular pentagon regular pentagon's's diagonal  diagonal is  is φ times its side. side. The vertices of a regular icosahedron regular icosahedron are  are those of three of three mutually  mutually orthogonal golden orthogonal  golden rectangles. There is no known general algorithm general  algorithm to  to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and area  and placing one node in each band at longitudes spaced by a golden section section of the circle, circle, i.e. 360°/φ ≅ 222. 222.5° 5°.. This This meth method od was was used used to arra arrang ngee the the 1500 1500 mirmirrors of the student-participatory student-participatory satellite  satellite Starshine-3. Starshine-3.[66] 4.5.1 Dividing Dividing a line segment by interior interior division division

3. 1. 2.

A

M

S

B

Dividing Dividing a line segment segment by exterior exterior division division according according to the  golden ratio

1. Construct on segment AS off the point S, a vertical length of AS with the endpoint C. 2. Do bisect segment AS with M. 3. The circul circular ar arc around M with the radius radius MC divides the extension AS in point B. Point S divides the constructed segment AB into line segments AS and SB with lengths in the golden ratio.

1. Havin Havingg a line line segme segment nt AB,co AB, cons nstruc tructt a perpe perpendi ndicu cular lar Appli Applica catio tionn examp example less you you can can see in the artic article less Pentagon BC at point B, with BC half the length of AB. Draw wi with th a give givenn si side de le length ngth,, Dec Decagon agon with giv given en circ circumc umcirc ircle le the hypotenuse the hypotenuse AC.  AC. and Decagon with a given side length. length.

4.5 4.5

9

Geom Geomet etry ry

The both above displayed different   algorithms produce algorithms  produce geometric constructi constructions ons that  that divides a line a  line segment into segment  into two line segments where the ratio of the longer to the shorter line segment is the golden ratio. 4.5.3 Golden triangle, pentagon and pentagram A

X

φ

2

φ =1+φ

B 1

φ

φ

C Golden triangle

Let Let A and and B be midpo midpoin ints ts of the side sidess EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle of  DEF at C. |AB| |BC |

=

|AC | |AB|

=  φ

Golden triangle The golden The golden triangle can triangle can be charactercharacterized as an isosceles an  isosceles triangle ABC triangle  ABC with the property that bisecting the bisecting  the angle C produces a new triangle new  triangle CXB  CXB which two points, then these three points are in golden proporis a similar a similar triangle to triangle to the original. tion. This result is a straightforward consequence of the If angl anglee BCX BCX = α, then then XCA = α beca becaus usee of the the bise bisect ctio ion, n, intersecting chords theorem and theorem and can be used to construct and CAB = α because of the similar triangles; ABC = 2α a regular pentagon, a construction that attracted the atfrom the original isosceles symmetry, and BXC = 2α by tention of the noted Canadian geometer H. geometer  H. S. M. Coxsimilarity. The angles in a triangle add up to 180°, so 5α eter eter who  who published it in Odom’s name as a diagram in = 180, 180, givi giving ng α = 36°. 36°. So the the angl angles es of the the gold golden en trian triangl glee the American Mathematical Monthly Monthly  accompanied by the are thus 36°−72°−72°. The angles of the remaining ob- single word “Behold!"  [67] tuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°−36°−108°.

Suppose XB has length 1, and we call BC length  φ . Because of the isosceles isosceles triangles XC=XA and BC=XC, so these these are also length length φ. Length Length AC = AB, therefore therefore equals φ + 1. But triang triangle le ABCis ABC is simil similar ar to triang triangle le CXB, CXB, so AC/BC = BC/BX, AC/ φ = φ/1, and so AC also equals Thus φ2 = φ + 1, confirming that φ  is indeed the φ2 . Thus golden ratio. Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to  φ , while the inverse the inverse ratio  ratio is φ − 1. Pentagon In a regul regular ar pentag pentagon on therat the ratio io betwe between en a side side andadiagonalis Φ (i.e. 1/φ), while while intersec intersecting ting diagonal diagonalss section each other in the golden ratio.[8] Odom’s construction George Odom Odom has  has given a remarkably simple construction for φ  involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of

A pentagram colored to distinguish its line segments of different  lengths. The four lengths are in golden ratio to one another.

10

4

Pentagram The golden ratio plays an important role in the geometry of pentagrams of pentagrams.. Each intersection of edges sections other other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram’s center) is  φ, as the four-color illustration shows.

The pentagram includes ten isosceles ten  isosceles triangles: triangles: five acute five acute andfiv and fivee obtuse isosc isoscel eles es triang triangle les. s. In allof all of them, them, the ratio ratio of the longer side to the shorter side is  φ. The acute triangles are golden triangles. triangles. The obtuse isosceles isosceles triangles are golden gnomons. g nomons.

MATHE MATHEMA MATIC TICS  S 

4.5.5 Triangl Trianglee whose sides sides form form a geometric geometric proprogression

If the side lengths of a triangle form a geometric a  geometric progres2 sion and sion  and are in the ratio 1 :  r  : :  r  , where r  is  is the common ratio, then  r  must lie in the range φ−1 < r  <  < φ, which is a consequence of the triangle the  triangle inequality (the inequality  (the sum of any two sides of a triangle must be strictly bigger than the length length of the third side). side). If r  =  =  φ  then the shorter two sides are 1 and  φ but their sum is φ 2, thus r  <  <  φ. A similar calculation shows that  r  >  > φ−1. A triangle whose sides are in the ratio 1 : √ φ : φ is a right triangle (because 1 + φ =  φ2 ) known as a Kepler a Kepler triangle. triangle.[69] 4.5.6 Golden triangle, rhombus, and rhombi rhombicc triatriacontahedron

B

A

φ

a a

b b

b

C

1

a

D

One of the rhombic triacontahedron’s rhombi 

The golden ratio in a regular pentagon can be computed using Ptolemy’s Ptolemy ’s theorem theorem..

Ptolemy’s theorem The golden golden ratio properti properties es of a regular pentagon can be confirmed by applying Ptolemy’s applying  Ptolemy’s theorem to theorem  to the quadrilateral formed by removing one of its vertices. If the quadrilateral’s long edge and diagonals diagonals are b, and short short edges edges are a, then then Ptolemy Ptolemy’s ’s theorem theorem gives gives b2 =  a 2 +  ab  which yields

√ 

1+ 5 b = . 2 a

4.5.4 Scaleni Scalenity ty of triangle triangless

Consider a triangle a  triangle with  with sides of lengths a, b, and c  in All of the faces of the rhombic rhombic triacontah triacontahedro edron n are golden decreasing decreasing order. Define the “scalenity” “scalenity” of the triangle to rhombi  be the smaller of the two ratios  a/b and b/c . The scalenity scalenity is always less than  φ and can be made as close as desired A  golden rhombus is rhombus  is a  rhombus  rhombus whose  whose diagonals are in to φ .[68] the golden golden ratio. ratio. Therhomb The rhombic ic triacontahe triacontahedron dron is a convex

4.7

11

Symme Symmetri tries  es 

polytope  that has a very special property: all of its faces polytope that are golden golden rhombi rhombi.. In the  the   rhombic triacontahedron the triacontahedron  the F (n + a) dihedral angle between angle between any two adjacent rhombi is 144°, lim =  ϕ a , n→∞ F (n) which is twice the isosceles angle of a golden triangle and four times its most acute angle.[70] where above, the ratios of consecutive terms of the Fibonacci sequence, sequence, is a case when  a  = 1 . Furthermore, the successive powers of φ  obey the Fi4.6 Relation Relationshi shipp to Fibonacci Fibonacci sequence sequence bonacci recurrence bonacci  recurrence:: The mathematics of the golden ratio and of the Fibonacci the  Fibonacci sequence are sequence  are intimately intimately interconnected. interconnected. The Fibonacci sequence is: ϕn+1 =  ϕ n + ϕn−1 . 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, .... The closed-form The  closed-form expression for expression  for the Fibonacci sequence involves the golden ratio:

 ( n) = F  (

ϕn

n

− (1 √ − ϕ) 5

=

ϕn

− (√ −ϕ)− 5

n

.

This identity allows any polynomial in  φ to be reduced to a linear expression. For example: 3ϕ3

− 5ϕ2 + 4 = 3(ϕ2 + ϕ) − 5ϕ2 + 4 = 3[(ϕ + 1) + ϕ] − 5(ϕ + 1) + 4 =  ϕ + 2 ≈ 3.618.

The reduction to a linear expression can be accomplished in one step by using the relationship

The golden ratio is the limit the  limit of  of the ratios of successive ϕk =  F k ϕ + F k−1 ,

where  F k is the k th Fibonacci number. However, this is no special property of  φ, because polynomials in any solution  x  to  to a quadratic equation equation can  can be reduced in an analogous manner, by applying: x2 =  ax + b

for given coefficients  a , b  such that  x  satisfies the equation. tion. Even more more generall generally, y, any rational any  rational function (with function  (with rational coefficients) of the root of an irreducible nthA  Fibonacci spiral  which spiral  which approximates the golden spiral, using degree polynomial over the rationals can be reduced to Fibonacci sequence square sizes up to 34. a polynomial of degree  n  ‒ 1. Phrased in terms of field of  field theory,, if α is a root of an irreducible  n th-degree polyterms of the Fibonacci sequence (or any Fibonacci-like theory [20] nomial, then Q(α)   has degree n  over Q  , with basis sequence), as originally shown by by Kepler  Kepler:: {1, α , . . . , αn−1} . F (n + 1)

lim n→∞ F (n)

=  ϕ.

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than  φ, and converge on  φ  as the Fibonacci numbers increase, increase, and:

∞

∑|

n=1

F (n)ϕ

− F (n + 1)| =  ϕ.

More generally:

4.7 4.7 Symm Symmet etri ries es The golden ratio and inverse golden ratio ϕ± = (1 ± √  5)/2 have a set of symmetries that preserve and interrela relate te them. them. Theyare They are both both prese preserv rved ed by the frac fractional tional linear transformations  x, 1/(1 − x), (x − 1)/x, – this fact corresponds to the identity and the definition quadratic equation equation.. Further, Further, they are interchange interchangedd by the three maps  1/x, 1 − x, x/(x − 1) – they are reciprocals, symmetric about  1/2 , and (projectively) symmetric about 2. More deeply, these maps form a subgroup of the modular the  modular group PSL group  PSL(2, Z) isomorphic to the symmetric the  symmetric group on group on 3 letters, S 3 ,  corresponding to the  the   stabilizer of stabilizer  of the set

12

4

{0, 1, ∞} of 3 standard points on the projective the projective line, line, and the symmetries correspond to the quotient map S 3 → S 2 – the subgroup  C 3 < S 3  consisting of the 3-cycles and the identity ()(01∞)(0∞1) fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.

MATHE MATHEMA MATIC TICS  S 

4.9 Decim Decimal al expan expansi sion on The golden ratio’s decimal expansion can be calculated directly from the expression

√ 

4.8 Other Other prope properti rties es

1+ 5 ϕ  = 2

The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational rational number number (see Alternate forms  above). ). It is, for that that forms  above reason, one of the worst the  worst cases of cases  of Lagrange’s  Lagrange’s approxi approximamation theorem and theorem  and it is an extremal case of the Hurwitz the  Hurwitz inequality for inequality  for Diophantine  Diophantine approximations. approximations. This may be why angles close to the golden ratio often show up in phyllotaxis (the phyllotaxis  (the growth of plants).[71] The defining quadratic polynomial and the conjugate conjugate relationship lead to decimal values that have their fractional part in common with  φ :

with √5 ≈ 2.2360679774997896964   A002163. A002163. The square root of 5 can 5  can be calculated with the  Babylonian method,, starting with an initial estimate such as  xφ  = 2 method and iterating and iterating

xn+1  =

(xn + 5/ xn ) 2

for n = 1, 2, 3, ..., ..., until until the differe difference nce between between xn and xn₋₁ becomes zero, to the desired desired number of digits. The Babyl Babyloni onian an algor algorith ithm m for √5 is equi equival valent ent to ϕ2 =  ϕ + 1 = 2.618 . . . 2 Newton’s method for method for solving the equation x  − 5 = 0. In 1 its more general form, Newton’s method can be applied =  ϕ − 1 = 0.618 . . . . ϕ directly to any algebraic any algebraic equation, equation, including the equation The sequence of powers of φ   contains these values  x 2 − x − 1 = 0 that defines the golden ratio. This gives an 0.618..., 0.618..., 1.0, 1.618..., 1.618..., 2.618...; 2.618...; moregene more generall rally, y, anypow any power er iteration that converges converges to the golden ratio itself, of φ is equal equal to the sum sum of the two immedi immediate ately ly prec precedi eding ng powers: ϕn =  ϕn−1 + ϕn−2 =  ϕ

2  + 1 xn xn+1  = , 2xn  1

−

· F  + F −1 . n

n

As a result, one can easily decompose any power of φ for an appropriate appropriate initial estimate xφ  such as  xφ  = 1. 1. A into a multiple of φ  and a constant. constant. The multipl multiplee and slightly faster method is to rewrite the equation as  x  −  − 1 the constant constant are alway alwayss adjac adjacent ent Fibonac Fibonacci ci numbers. numbers. This This − 1/ x  =  = 0, in which case the Newton iteration becomes becomes leads to another property of the positive powers powers of  φ : If ⌊n/2 − 1⌋ =  m , then: ϕn =  ϕ n−1 + ϕn−3 + ϕn

− ϕ −1 = ϕ n

−2 .

n

· · · + ϕ −1−2 n

m

+ ϕn−2−2m

xn+1  =

2  + 2 xn xn . 2  + 1 xn

These iterations all converge all  converge quadratically; quadratically; that is, each step roughly roughly doubles the number number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision. precision. The time needed to compute  n  digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. numbers. This is considerably considerably faster faster than known algorithms for the transcendental the transcendental numbers π and e. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F   25001 and  F   25000 , each over 5000 digits, yields over 10,000 significant significant digits of the golden ratio.

When the golden ratio is used as the base of a  numeral system (see system  (see Golden  Golden ratio base, base, sometimes dubbed   phinary  or  φ-nary), every integer has a terminating representation, despite φ  being irrational, but every fraction has a non-terminating representation. representation. The golden ratio√ is a fundamental a  fundamental unit of unit  of the algebraic the  algebraic number field Q( 5) and is a Pisot–Vijayaraghavan a Pisot–Vijayaraghavan num√  √  L +F  5 [72] n ber.. In the field Q( 5)  we have ϕ = ber , 2 where  Ln  is the n  -th Lucas  -th Lucas number. number. The golden ratio also appears in hyperbolic in hyperbolic geometry, geometry, as the maximum distance from a point on one side of an ideal triangle to triangle  to the closer closer of the other two sides: sides: this distan distance, ce, the side side length length of the equila equilateral teral triangle formed The deci decimal mal expan expansi sion on of the golde goldenn ratio ratio φ ( by the points of tangency of a circle inscribed within the A001622 A001622)) has been calculated to an accuracy of two trilideal triangle, is  4 log(ϕ) .[73] lion (2×1012 = 2,000,000,000,000) digits. [74] n

n

5.2

13

Egypti Egyptian an pyrami pyramids  ds 

5 Pyra Pyram mids

minutes).[78] The The slant slant heig height ht or apoth apothem em is 5/3or 5/3 or 1.666. 1.666..... times times the semi-base. semi-base. The Rhind Rhind papyrus papyrus has another another pyramid problem as well, again with rational slope (exFurther information: mathematics information:  mathematics and art did not Both Both Egy Egypti ptian an pyram pyramid idss and the regul regular ar square pyramid pyramidss pressed as run over rise). Egyptian mathematics [79] include the notion of irrational numbers, and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional conventional units of palms per cubit) was used in the building of pyramids. pyramids.[77] Another mathematical pyramid with proportions almost identical to the “golden” one is the one with perimeter h a equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the Kepler Kepler triangle. This pyramid relationship √ relationship b corresponds corresponds to the coincidental the coincidental relationship ϕ ≈ 4/π . Egypti Egy ptian an pyram pyramid idss very very close closein in propo proporti rtion on to these these mathmathematical pyramids are known.[78] A regular square pyramid is determined by its medial right trian gle, whose edges are the pyramid’s apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Math √   1 : ϕ : ϕ  and  3 : 4 : 5 and  ematical proportions b:h:a of  1 1 : 4/ π  : 1.61899 are of particular interest in relation to Egyptian pyramids.

that resemble them can be analyzed with respect to the golden ratio and other ratios.

5.1 Mathemati Mathematical cal pyrami pyramids ds and triangles triangles A pyramid in which the apothem (slant height along the bisector of a face) is equal to  φ times the semi-base (half the base width) is sometimes called a   golden pyramid . The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem.  √  The height of this pyramid is ϕ  times the semi-base  √  (that is, the slope of the face is ϕ ); the square of the height is equal to the area of a face,  φ times the square of the semi-base. The medial right medial  right triangle of triangle  √  of this “golden” pyramid (see diagram), with sides  1 : ϕ  : ϕ is interesting in its own right, demonstrating demonstrati ng via the Pythagorean the Pythagorean theorem the retheorem the  √  √  2 lationship ϕ  = ϕ − 1 or  ϕ  = 1 + ϕ . This “Kepler triangle”[75] is the only right triangle proportion with edge lengths in geometric in  geometric progressi progression on,,[69] just as the 3– 4–5 triang triangle le is the only only right right triang triangle le propor proportio tionn with with edge edge lengths in arithmetic in arithmetic progression. progression. The angle with tangent tangent √ ϕ corresponds to the angle that the side of the pyramid makes with respect respect to the ground, g round, 51.827... degrees (51° 49' 38”).[76] A nearly similar pyramid shape, but with rational proportions, is described in the   Rhind Mathematical Papyrus (the pyrus  (the source of a large part of modern knowledge of ancient Egyptian ancient  Egyptian mathematics), mathematics), based on the 3:4:5 [77] triangle; the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8

√ 

5.2 Egypti Egyptian an pyrami yramids ds In the mid-nine mid-nineteen teenth th century, century, Röber Röber studied studied various various Egyptian Egyptian pyramid pyramidss includi including ng Khafre, Khafre, Menkaure Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other other authors authors identifi identified ed as the Kepler Kepler triangle triangle;; many many other other mathematical mathematical theories of the shape of the pyramids have also been explored. [69] One Egyptian pyramid is remarkably close to a “golden pyramid”—the Gre Great at Pyr Pyrami amidd of Gi Giza za (als (alsoo know knownn as the the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the “golden” pyramid inclination of 51° 50' and the π-based pyramid inclination inclination of 51° 51'; other pyram pyramid idss at Giza Giza (Chep (Chephre hren, n, 52° 20', 20', andMy and Myce cerin rinus, us, 50° [77] 47') are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains open to speculation.[80] Several Several other Egyptian tian pyram pyramid idss arevery close close to therat the ratio ional nal3:4 3:4:5 :5 shape shape..[78] Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Pyramid,  Eric Temple Bell, Bell , mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, exce except pt in the case case of the3:4 the 3:4:5 :5 pyram pyramid id,, sinc sincee the3:4 the 3:4:5 :5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, theorem, nor any way to reason about irrationals such as π or  φ .[81] Michael Rice[82] asserts that principal authorities on the history of   Egyptian architecture   have argued that the Egyptians were well acquainted with the golden ratio and that it is part of mathematics of the Pyramids, citing Giedon (1957).[83] Historians of science have always debated whether the Egyptians had any such knowledge or not, contending rather that its appearance in an Egyptian building is the result of chance. [84] In 1859, the pyramidologist the  pyramidologist John Taylor claimed Taylor  claimed that, in

14

8

the Gre Great at Pyr Pyrami amidd of Giz Gizaa, thegol the golde denn ratio ratio is repre represen sented ted by the the rati ratioo of the the lengt lengthh of the the face ace (the (the slop slopee heig height ht), ), ininclined at an angle θ angle  θ to  to the ground, to half the length of the side of the square base, equivalent to the  secant  secant of  of [85] the angle θ. The above two lengths were about 186.4 and 115.2 meters respectively. respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. measurements. Similarly, Similarly, Howard Vyse, Vyse, ac[86] cording to Matila Ghyka, reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.

6 Disput Disputed ed observ observati ation onss Examples of disputed observations of the golden ratio include the following:

REFEREN REFERENCES CES AND FOOTNO FOOTNOTES  TES 

7 See also

•  Golden angle •  Section d'Or • List of works designed with the golden ratio •  Plastic number •  Sacred geometry •  Silver ratio 8 Refere References nces and footnotes ootnotes [1] Livio, Mario (2002).  The Golden Ratio: The Story of Phi, York: BroadBroadThe Wo World’ rld’ss Mo Most st Asto Astonish nishing ing Numb Number  er . New York: way Books. ISBN Books.  ISBN 0-7679-0815-5. 0-7679-0815-5. [2] Piotr Sadowski Sadowski (1996). (1996). The knigh knightt on his quest: quest: sy symm-

•  Historian John Man states that the pages of the Gutenberg Bible were Bible were “based on the golden section shape”. However, according to Man’s own measurements, the ratio of height to width was 1.45. [87]

•  Some specific proportions[88][89] in the bodies of many

animals (including humans ) and parts of the [3] shells of mollusks are often claimed to be in the golden golden ratio. There There is a large variation variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.[88] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[89] The nautilus The nautilus shell,  shell, the construction of which proceeds in a logarithmic a  logarithmic spiral, spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is proportioned by the golden ratio relative to the previous one;[90] howev however, er, measurement measurementss of nautilus nautilus shells shells do [91] not support this claim.

•  In investing, some practitioners of technical of  technical anal-

bolic patterns of transition in Sir Gawain and the Green Univers ersity ity of of Delaw Delaware are Pres Press. s. p. 124. 124.   ISBN Knight . Univ

978-0-87413-580-0.. 978-0-87413-580-0 [3] Richar Richardd A Dunla Dunlap, p, The Golde Golden n Ratio Ratio and Fibona Fibonacc ccii NumNumbers , World Scientific Publishing, 1997 [4] Euclid, Euclid, Elements , Book 6, Definition 3. [5] Summerson Summerson John, John, Heavenly Mansions: And Other Essays  on Architecture (New York: W.W. Norton, 1963) p. 37. “And the same applies applies in architec architecture, ture, to the rectangles the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful fruitful and suggest the rhythms of modular design.” [6] Jay Hambidge, Hambidge,  Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University University Press, 1920 [7] William William Lidwel Lidwell,l, Kritina Kritina Holden, Holden, Jill Butler,   Universal Principles of Design: A Cross-Disciplinary Refe Reference rence, Gloucester MA: Rockport Publishers, 2003 [8] Pacioli, Luca.  De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice. [9]   Strogatz, Steven (September Steven (September 24, 2012). “Me, 2012).  “Me, Myself, and Math: Proportion Control”. Control”.  New York Times . [10] Wei eiss sste tein in,, MathWorld .

Ericc W. “G Eri “Gol olde denn Ra Rati tioo Co Connju juga gate” te”..

ysis  use the golden ratio to indicate support of a ysis use price price leve level,l, or resis resistan tance ce to price price incre increase ases, s, of a stock stock [11] Markows Markowsky, ky, George George (January (January 1992). 1992).   “Misconceptions or commodity; after significant price changes up or about the Golden Ratio” (PDF). Ratio”  (PDF).  The College Mathematdown, new support and resistance levels are supposics Journal .  23  (1). edly found at or near prices related to the starting Livio,The Golden Golden Ratio: Ratio: The Story Story of Phi, Phi, The price via the golden ratio.[92] The use of the golden [12] Mario Livio, World’s World’s Most Astonishing Number , p.6 ratio in investing is also related to more complicated patterns described by Fibonacci by  Fibonacci numbers (e.g. numbers (e.g. [13] "῎Ακρον "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, λέγεται, Elliott wave principle and principle  and Fibonacci  Fibonacci retracem retracement ent). ). ὅτανᾖὡςἡὅληπρὸςτὸμεῖζοντμῆμα,οὕτωςτὸμεῖζον Howev However, er, other other market market analysts analystshav havee publis published hed analπρὸς τὸ ἔλαττὸν" as translated in Richard Fitzpatrick yses suggesting that these percentages and patterns (translator) (2007).  Euclid’s Elements of Geometry.  ISBN 978-0615179841.,., p. 156 978-0615179841 are not supported by the data.[93]

15 [14] Euclid, Euclid,

[30] Le Corbusie Corbusier, r,  The Modulor , p. 35, as cited in Padovan, Richard,   Proportion Proportion:: Science, Science, Philoso Philosophy phy,, Architec Architecture ture (1999), p. 320. Taylor & Francis.  ISBN 0-419-22780[15] Euclid, Euclid, Elements , Book 2, Proposition 11; Book 4, Propo6: “Both the paintings and the architectural designs make sitions 10–11; Book 13, Propositions 1–6, 8–11, 16–18. use of the golden section”.   [http:/.aleph0.clarku.edu/~{}djoyce/java/  elements/toc.html Elements], Book 6, Proposition 30.

Urwin, Simon. Simon. Analysing Analysing Architecture (2003) pp. 154-5, [16]  “The Golden Ratio”. Ratio”.  The MacTutor History of Mathemat- [31] Urwin, ISBN 0-415-30685-X ics archive. Retrieved 2007-09-18. [32] Jason Elliot Elliot (2006). (2006).   Mirrors of the Unseen: Journe Journeys ys in Iran. Macmillan. pp. 277, 284.  ISBN 978-0-312-301910. [18] Hemenway, Hemenway, Priya (2005).  Divine Proportion: Phi In Art, ork: Sterli Sterling. ng. pp. 20–21. 20–21. [33] Leonardo da Vinci’s Polyhedra Nature, and Science. New York: by George W. Hart Polyhedra, by George ISBN 1-4027-3522-7. 1-4027-3522-7 . [34] Livio, Livio, Mario. “The golden golden ratio and aesthetics” aesthetics”.. Retrieved 2008-03-21. [19]   Plato. Plato.   “Timaeus”. “Timaeus”. Translated by Benjamin Benjamin Jowett. Jowett. The Internet Classics Archive. Retrieved 30 May 2006. [35] “Part of the process process of becoming becoming a mathemat mathematics ics writer is, it appears, learning that you cannot refer to the golden [20] James Joseph Tattersall (2005).  Elementary number theratio without following the first mention by a phrase that   (2nd ed.).   Cambridge University ory in nine chapters  (2nd goes something like 'which the ancient Greeks and others Press.. p. 28. ISBN Press 28.  ISBN 978-0-521-85014-8. 978-0-521-85014-8 . believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the [21] Underwoo Underwoodd Dudley Dudley (1999). (1999). Die Macht Macht der Zahl: Zahl: Wa Was  s  lines of 'Leonardo Da Vinci believed that the human form die Numerologie uns weismachen will . Springer. p. 245. displays displays the golden ratio.' There is not a shred of evidence ISBN 3-7643-5978-1. 3-7643-5978-1 . to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others [22] Cook, Theodore Theodore Andrea (1979) [1914]. [1914].   The Curves of  in a similar vein, live on.” Keith Devlin (May 2007).  “The New York: Dover Publications. Publications. ISBN  ISBN 0-4860-486-2370123701Life. NewYork: Myth That Will Not Go Away”. Away”. Retrieved September 26, X. 2013. [23]   Gardner, Martin (2001), Martin  (2001),  The Colossal Book of Mathe[36] Donald Donald E. Simanek. Simanek.  “Fibonacci Flim-Flam”. Flim-Flam” . Retrieved matics: Classic Puzzles, Paradoxes, and Problems : NumApril 9, 2013. [17] Weiss Weisstein, tein, Eric W. “Golden Ratio”. Ratio”.  MathWorld .

ber Theory, Algebr Algebra, a, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational  Company, y, p. 88, ISBN 88, ISBN Mathematics , W. W. Norton & Compan

9780393020236.. 9780393020236

[37] Salvador Salvador Dalí (2008).  The Dali Dimension: Decoding the  (DVD). Media 3.14-TVC-FGSD-IRLMind of a Genius  (DVD). AVRO.

Herndon and Gilkey, Gilkey, Susan Nicodemus. Nicodemus. [24] Jaric, Jaric, Marko V. (2012), (2012),  Introduction to the Mathematics  [38] Hunt, Carla Herndon   pp. 44, 47, ISBN 47,  ISBN 1Teaching Mathematics in the Block  pp. Elsevier,, p. x, ISBN x,  ISBN 9780323159470, 9780323159470, of Quasicrystals , Elsevier 883001-51-X Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions functions had been known for [39] Bouleau, Charles, The Painter’s Secret Geometry: A Study nearly sixty years, it was the mathematics of aperiodic of Composition in Art  (1963) pp.247-8, Harcourt, Brace Penrose tilings, mostly developed by Nicolaas de Bruijn, & World, ISBN World, ISBN 0-87817-259-9 that provided the major influence on the new field. [40] Olariu, Olariu, Agata, Agata,   Golden Golden Section and the Art of Painting Painting [25] Van Mersbergen, Mersbergen, Audrey M., “Rhetorical Prototypes in Available online Architecture: Measuring the Acropolis with a PhilosophTosto, Pablo, Pablo, La composición áurea en las artes plásticas  ical Polemic”,  Communication Quarterly, Vol. 46 No. 2, [41] Tosto, – El número de oro, Librería Hachette, 1969, p. 134–144 1998, pp 194-213. [42]   Jan Tschichol Tschicholdd. The Form pp.433 Fig Fig Form of of the Book  Book , pp.4 4. “Frame “Framewor workk of ideal ideal propo proporti rtions ons in a medie medieva vall manuscript without multiple multiple columns. Determined by Jan Tschichol Tschicholdd 1953. Page Page proportion proportion 2:3. margin margin proporpropor[27] Keith Keith J. Devlin  The Math Instinct: Why You're You're A Mathtions 1:1:2:3, Text area proportioned in the Golden Secematical Genius (Along With Lobsters, Birds, Cats, And  tion. The lower outer corner of the text area is fixed by a 108. New York: Thunder’s Mouth Press, 2005, Dogs), p. 108. diagonal as well.” ISBN 1-56025-672-9

[26] Midhat J. Gazalé ,  Gnomon, Princeton University Press, 1999. ISBN 1999.  ISBN 0-691-00514-1

[43]  Jan Tschichold, Tschichold,  The Form of the Book , Hartley & Marks [28] [28] Bousso Boussora, ra, Kenza Kenza andMazouz, andMazouz, Said, Said, TheUse of theGolden theGolden (1991), ISBN (1991), ISBN 0-88179-116-4. 0-88179-116-4. Section in the Great Mosque of Kairouan , Nexus Network [44] Jones, Jones, Ronald (1971). (1971). “The golden golden sectio section: n: A most reJournal, vol. 6 no. 1 (Spring 2004), markable measure”.   The Structurist . 11: 44–5 44–52. 2. Who Who [29] Le Corbusier, Corbusier, The Modulor  p.   p. 25, as cited in Padov Padovan, an, would suspect, for example, that the switch plate for sinRichard,   Proportion gle light switches are standardized in terms of a Golden Proportion:: Science, Science, Philosop Philosophy hy,, Architec Architecture ture (1999), p. 316, Taylor and Francis, ISBN Francis, ISBN 0-419-22780-6 Rectangle?

16

8

REFEREN REFERENCES CES AND FOOTNO FOOTNOTES  TES 

[45] Art Johnson (1999).   Famous problems and their math- [61] Perez, Perez, J.-C. (September (September 2010). “Codon populat populations ions in Librarie iess Unli Unlimi mited ted.. p. 45.   ISBN 978-1single-s single-strand tranded ed whole whole human genome DNA are fracfracematicians . Librar 56308-446-1.. The Golden Ratio is a standard feature of 56308-446-1 tal and fine-tuned by the Golden Ratio 1.618”. Inmany modern designs, from postcards and credit cards to terdisciplinary Sciences: Sciences: Computational Computational Life Life Science. 2 posters and light-switch light-switch plates. (3): 228–240. doi doi::10.1007/s12539-010-0022-0. 10.1007/s12539-010-0022-0 .   PMID 20658335..  PDF full text 20658335 [46] [46] Alexe Alexeyy Stakho Stakhov; v; Scott Scott Olsen; Olsen; Scott Scott Anthon Anthonyy Olsen Olsen [62] Pomme Pommersh rshei eim, m, James James E., TimK. Marks, Marks, and Eri Erica ca L. Fla Fla-(2009). The mathe mathematic maticss of harmo harmony ny:: from from Eucl Euclid id to con- [62] pan,, eds. 2010. “Number Theory: A Lively Introduction pan temporary mathematics and computer science. World Sciwith Proofs, Applications, and Stories”. John Wiley and entific. p. 21.  ISBN 978-981-277-582-5. 978-981-277-582-5 . A credit card Sons: 82. has a form of the golden rectangle. aesthetics,, by Mario Livio. [47] [47] Simon Simon Cox (2004) (2004)..  Cra  Cracki cking ng the Da Vi Vincicode: ncicode: the una unauu- [63] The golden ratio and aesthetics thorized guide to the facts behind Dan Brown’s bestselling Barnes & Noble Books. ISBN Books. ISBN 978-0-7607-5931novel . Barnes

[64] [64] Max. Max. Hailpe Hailperin rin;; Barbar Barbaraa K. Kaiser Kaiser;; Karl Karl W. Knight Knight (1998).   Concrete Abstractions: Abstractions: An Introduction to Com8. The Golden Ratio also crops up in some very unlikely Co. ISBN  puter Science Using Scheme. Brooks/Cole Pub. Co. ISBN places: widescreen televisions, televisions, postcards, credit cards and 0-534-95211-9.. 0-534-95211-9 photographs all commonly conform to its proportions. Roselle, “Golden  “Golden Mean Series” [48]   “THE “THE NEWRAPI NEWRAPIDE DE S : De Desi sign gn””. The ‘Gold ‘Golden en Ratio’sits Ratio’sits [65] Brian Roselle, at the heart of every Aston Martin. [66]  “A Disco Ball in Space”. Space” . NASA. 2001-10-09. Retrieved 2007-04-16. [49] Lendvai Lendvai,, Ernő (1971).   Béla Bartók: An Analysis Analysis of His  His  Music . London: Kahn and Averill. [67] Chris and and Penny. Penny. “Quandaries  “Quandaries and Queries”. Queries”. Math Central. Retrieved 23 October 2011. [50] [50] Smith, Smith, Peter Peter F. The Dyna Dynamics mics of Delig Delight: ht: Architectu Architecture re and    (New York: Routledge, 2003) pp 83, ISBN 83,  ISBN 0Aesthetics  (New 415-30010-X [51] Roy Howat (1983). Debu Debussy ssy in Proporti Proportion: on: A Musical  Musical  Cambridgee Univers University ity Press. Press.   ISBN 0-521Analysis . Cambridg 31145-4.. 31145-4

[68] American Mathematical Mathematical Monthly, pp. 49-50, 1954. [69] RogerHerzRoger Herz-Fisc Fischler hler (2000). (2000).  TheShapeof theGrea theGreatt Pyr PyraaWilfrid Laurier University University Press.   ISBN 0-88920mid . Wilfrid 324-5.. 324-5

[70] [70] Koca Koca,, Mehm Mehmet et;; Koca Koca,, Nazi Naziffe Ozde Ozdes; s; Koç, oç, RaRamazan mazan (2010) (2010),, “Catal “Catalan an solid solidss deri derived ved from threethreedimensional-root systems and quaternions”,   Journal of  043501,,   arXiv: arXiv:0908.3272 , Mathematical Physics , 51: 043501 [53]  “Pearl Masters Premium” Premium”.. Pearl Corporation. Retrieved doi::10.1063/1.3356985 doi 10.1063/1.3356985.. December 2, 2007.

[52] Simon Trezise (1994). Debu Cambridge dge Debussy ssy:: La Mer  Mer . Cambri University Press. p. 53. ISBN 53.  ISBN 0-521-44656-2. 0-521-44656-2 .

[54] "An "An 833 Cen Cents ts Scal Scale: e: An experim experiment ent on har harmon monyy", Huygens-Fokker.org. Accessed December 1, 2012.

[71]  Fibon  Fibonacc accii Numbers Numbers and Nature Nature - Par Partt 2 : Why is the Golden section the “best” arrangement?, arrangement?, from  from   Dr. Dr. Ro Ronn Knott’s Fibonacci Numbers and the Golden Section, Section, retrieved 2012-11-29.

[55] Richard Padovan Padovan (1999).   Proportion. Taylor & Francis. pp. 305–306. ISBN 305–306.  ISBN 978-0-419-22780-9. 978-0-419-22780-9 . [72]  Weisstein, Eric W. “Pisot Number”. Number”.  MathWorld .

[56] Padovan, Padovan, Richard Richard (2002). “Proportion: Science, PhilosoPhiloso- [73] Horocycl Horocycles es exinscrits : une propriété hyperbolique remarphy, Architecture”.  Nexus Network Journal . 4 (1): 113– quable, cabri.net, retrieved 2009-07-21. quable, 122. doi 122.  doi::10.1007/s00004-001-0008-7 10.1007/s00004-001-0008-7.. [74] Yee, Yee, Alexande Alexanderr J. (17 August August 2015).   “Golden Ratio”. Ratio”. [57] Zeising, Adolf (1854).  Neue Lehre van den Proportionen Independent computati computations ons done by Ron numberword.org . Independent des meschlischen Körpers  Körpers . preface. Watkins and Dustin Kirkland. [58]   “Golden ratio discovered in a quantum world”. world”. Eu- [75] Radio, Astraea Web (2006).  The Best of Astraea: 17 Arrekalert.org. 2010-01-07. Retrieved 2011-10-31. ticles on Science, History and Philosophy. Astrea Web Radio. ISBN dio.  ISBN 1-4259-7040-0. 1-4259-7040-0. [59] [59] J.C.Perez J.C.Perez (1991) (1991),, “Chaos DNA and NeuroNeuro-compu computers: ters: A Golden Link”, Link”, in  Speculations in Science and Technology [76] Midhat Midhat Gazale,   Gnomon: Gnomon: From From Pharaohs Pharaohs to Fractals  Fractals , vol. 14 no. 4, ISSN 4, ISSN 0155-7785. 0155-7785. Princeton Univ. Press, 1999 [60] [60] Yamagis amagishi, hi, Mich Michel el E.B., E.B., and Shimab Shimabuku ukuro, ro, Alex Alex I. [77]  Eli Maor, Maor,  Trigonometric Delights , Princeton Univ. Press, (2007), “Nucleotide (2007),  “Nucleotide Frequencies in Human Genome and 2000 Fibonacci Numbers”, Numbers”, in Bulletin of Mathematical Biology, ISSN 0092-8240 (print), 0092-8240 (print), ISSN  ISSN 1522-9602 (online). 1522-9602  (online). PDF  PDF [78]   “The Great Great Pyrami Pyramid, d, The Grea Greatt Disco Discovery very,, and The Great full text Coincidence”.. Retrieved 2007-11-25. Coincidence”

17 [79] Lancelot Hogben,  Mathematics for the Million, London: Allen Allen & Unwin, Unwin, 1942, p. 63., as cited by Dick Dick Teresi, Teresi, Lost Discoveries: The Ancient Roots of Modern Science—   from  from the Babylonians Babylonians to the Maya Maya, New York: Simon &

Schuster, 2003, p.56 [80] Burton, David David M. (1999).  The history of mathematics: an (4 ed.). ed.). WCB McGraw-Hi McGraw-Hill. ll. p. 56.   ISBN introduction  (4 0-07-009468-3.. 0-07-009468-3 [81] [81] Bell, Bell, Eric Eric Templ Templee (1940) (1940)..  The Deve Developm lopment ent of Ma Mathema themattics . New York: Dover. p. 40. [82] Rice, Rice, Michael, Michael,   Egypt’s Legacy: Legacy: The Archetypes of WestWest  pp. 24 Routledge, 2003, ern Civilisation, 3000 to 30 B.C  pp. ISBN 0-415-26876-1 [83] S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Mellon Lectures in the Fine Arts, 457, as cited in Rice, Michael,  Egypt’s Legacy: The Archetypes of Western  pp.24 Routledge, 2003 Civilisation, 3000 to 30 B.C  pp.24 [84] Markows Markowsky, ky, George George (January (January 1992). 1992).   “Misconceptions about the Golden Ratio”   (PDF).   College College Mathematics  Mathematics  Mathematical Association of America. America. 23  (1): Journal . Mathematical 2–19. doi 2–19.  doi::10.2307/2686193 10.2307/2686193..  JSTOR 2686193. 2686193. [85] Taylor, Taylor,  The Great Great Pyramid: Pyramid: Why Why Was It Built and Who Built It? , 1859 [86] Matila Ghyka The Geometry of Art and Life, New York: Dover, 1977 [87] Man, John,  Gutenberg: How One Man Remade the World  (2002) pp. pp. 166–167, 166–167, Wiley, Wiley,  ISBN 0-471with Word   (2002) 21823-5.. “The 21823-5 “The half-f half-foli olioo page page (30.7 (30.7 × 44.5 44.5 cm)was made made up of two rectangles—the whole page and its text area— based on the so called 'golden section', which specifies a crucial relationship relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.” [88] Pheasant, Stephen (1998).   Bodyspace. London: Taylor & Francis. ISBN Francis. ISBN 0-7484-0067-2. 0-7484-0067-2 . [89] van Laack, Laack, Walter Walter (2001). (2001).  A Bette Betterr Histor History y Of Our World:  orld:  Volume 1 The Universe. Aachen: van Laach GmbH. [90]  Ivan Moscovich, Moscovich,  Ivan Moscovich Mastermind Collection:  The Hinged Square & Other Puzzles, New York: Sterling, 2004 [91] Peterson, Ivars. “Sea Ivars.  “Sea shell spirals” spirals”..  Science News .

9 Furth Further er read readin ingg

•  Doczi, György (2005) [1981].   The Power of Lim-

its: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala Publications.  ISBN

1-59030-259-1.. 1-59030-259-1

•  Huntley, H. E. (1970).

The Divine Divine Proportio Proportion: n: A York: Dover Dover Study in Mathematical Beauty . New York:

Publications. ISBN Publications.  ISBN 0-486-22254-3. 0-486-22254-3 .

•  Joseph, George G. (2000) [1991].  The Crest of the

Peacock: Peacock: The Non-European Non-European Roots of MathematMathemat  (New ed.). Princeton, NJ: Princeton University ics  (New

Press. ISBN Press.  ISBN 0-691-00659-8. 0-691-00659-8 .

• Livio, Mario Mario (2002) [2002]. The Golden Ratio: The

Story of PHI, the World’s Most Astonishing Number 

(Hardback ed.). NYC: Broadway Broadway (Random House). ISBN 0-7679-0815-5. 0-7679-0815-5.

• Sahlqvist, Leif Leif (2008).  Cardinal Alignments and the

Golden Section: Principles of Ancient Cosmography Cosmography and Design  (3rd Rev. ed.). Charleston, SC: Book-

Surge. ISBN Surge. ISBN 1-4196-2157-2. 1-4196-2157-2.

•  Schneider, Michael S. (1994).   A Beginner’s Guide

to Const Construc ructin tingg the Univer Universe: se: The Mathe Mathemat matica ical  l  Archetypes of Nature, Art, and Science . New York:

HarperCollins. ISBN HarperCollins. ISBN 0-06-016939-7. 0-06-016939-7 .

•  Scimone, Aldo (1997).

La Sezione Sezione Aure Aurea. a. Storia Storia culturale di un leitmotiv della Matematica . Palermo:

Sigma Edizioni. ISBN Edizioni.  ISBN 978-88-7231-025-0. 978-88-7231-025-0.

•  Stakhov, A. P. (2009).

The Mathematics Mathematics of Harmony: mony: From From Euclid Euclid to Contempor Contemporary ary Mathematics  Mathematics  and Computer Science . Singapore: World Scientific

Publishing. ISBN Publishing.  ISBN 978-981-277-582-5. 978-981-277-582-5.

•  Walser, Hans (2001) [Der Goldene Schnitt  1993]. The Golden Section .

Peter Peter Hilton Hilton trans. trans. Washi Washingngton, DC: The Mathematical Association of America. ISBN ica.  ISBN 0-88385-534-8. 0-88385-534-8 .

10 Exte Extern rnal al link linkss

[92] [92] For insta instance nce,, Osler Osler writes writes that that “38.2 “38.2 percen percentt and 61.8 61.8 perpercent cent retrac retraceme ementsof ntsof recentrises recentrises or decli declinesare nesare common common,” ,” in Osler, Carol (2000).  “Support for Resistance: Technical Analysis and Intraday Exchange Rates”  (PDF).   Federal Reserve Bank of New York Economic Policy Review . 6 (2): 53–68.

•   Hazewinkel Hazewinkel,, Michiel, Michiel, ed.

(2001),  “Golden ratio”, ratio”,  Springer,, ISBN 978Encyclopedia of Mathematics ,  Springer 1-55608-010-4

[93]  Roy Batchelor and Batchelor and Richard Ramyar, "Magic "Magic numbers in the Dow,” Dow,” 25th International Symposium on Forecasting, 2005 2005,, p. p. 13, 13, 31. 31. "Not since the 'big is beautiful' days have giants looked better", better", Tom Stevenson,   The Daily Telegraph,, Apr. 10, 2006, and “Technical Telegraph “Technical failur failure”, e”, The  The Economist,, Sep. 23, 2006, are both popularEconomist popular-pres presss accounts of Batchelor and Ramyar’s research.

• Gold Golden en Section in

•  “Golden Section” by Section”  by Michael Schreiber,   Wolfram Demonstrations Project, Project, 2007.

Photography: Photograp hy: Gold Golden en Ratio, Golden Triangles, Golden Spiral

•  Weisstein, Eric W. “Golden Ratio”. Ratio”.  MathWorld . • “Res “Researc earcher her exp explain lainss my myster steryy of gol golden den rati ratio” o”.. PhysOrg. December 21, 2009..

18

10

•  Knott, Ron. “The Ron.  “The Golden section ratio: ratio: Phi” Phi”.. Information and activities by a mathematics professor.

•   The The

Pentagram Penta gram & The Gol Golde denn Ra Ratio tio.. Green, Thomas M. Updated June 2005. Archived Archived November 2007. Geometry Geometry instructi instruction on with problems problems to solve.

• Schne Schneid ider, er,

Robert Robert P. (2011). (2011). “A Golden Golden Pair Pair of Identi ntitie ties in the the Theo Theory ry of Num Numbers bers”. ”. arXiv::1109.3216 [math.HO arXiv math.HO]. ]. Prov Proves es formu ormu-las that involve the golden mean and the   Euler totient and totient  and Möbius  Möbius functions. functions.

• The Myth That That Will Not Go Awa Awayy, by Keith by Keith Dev Devlin lin,, addressing addressing multiple allegations about the use of the golden ratio in culture.

EXTE EXTERN RNAL AL LINKS  LINKS 

19

11 Te Text xt and and image image sources, sources, contributo contributors, rs, and and licen licenses ses 11.1 11.1 Text •

 https://en.wikipedia.org/wiki/Golden_ratio?oldid=744090812 Contributors:  Contributors:  AxelBoldt,  AxelBoldt, WojPob, Mav, Bryan Derk  Golden ratio  Source:  https://en.wikipedia.org/wiki/Golden_ratio?oldid=744090812 sen, Zundark, The Anome, Tarquin, Josh Grosse, Youssefsan, XJaM, Arvindn, Heron, Lightning~enwiki, Olivier, Olivier, Patrick, Michael Hardy, Paul Barlow, Dominus, Nixdorf, Kku, Gabbe, SGBailey, Menchi, Wapcaplet, Ixfd64, Eliah, Tango, Sannse, Seav, GTBacchus, Ppareit, Looxix~enwiki, ArnoLagrange, Ellywa, Ahoerstemeier, Cyp, Haakon, Ryan Cable, William M. Connolley, Angela, DropDeadGorgias, Mark Foskey, Glenn, Whkoh, Gisle~enwiki, Dod1, Jacquerie27, Smaffy, Rob Hooft, Tobias Conradi, Raven in Orbit, Etaoin, Ideyal, Stephenw32768, Feedmecereal, Crusadeonilliteracy, Alex S, Charles Matthews, CecilBlade, Dysprosia, Prumpf, Tpbradbury, Furrykef, Hyacinth, AndrewKepert, Kwantus, Johnleemk, Finlay McWalter, Frazzydee, Owen, Denelson83, Phil Boswell, Donarreiskoffer, Robbot, KeithH, Jakohn, Fredrik, Chris 73, Mayooranathan, Mayooranathan, Gandalf61, Sverdrup, Academic Challenger, Challenger, Rursus, Timrollpickering, Timrollpickering, Alan De Smet, Sheridan, Miles, HaeB, Jleedev, Mattflaschen, Snobot, Weialawaga~enw Weialawaga~enwiki, iki, Giftlite, Gene Ward Smith, Tom harrison, J abra, MSGJ, Herbee, Fropuff, Peruvianllama, Anton Mravcek, Everyking, Gus Polly, Mcapdevila, Alison, JeffBobFrank, Sunny256, WHEELER, Thierryc, Daveplot, Guanaco, Eequor, Matthead, Jay Carlson, Bobblewik, Architeuthis, Utcursch, Mike R, Jackcsk, Noe, Antandrus, GeneMosher, GeneMosher, OverlordQ, Jossi, Girolamo Savonarola, Savonarola, DragonflySixtyseve DragonflySixtyseven, n, Secfan, Icairns, Karl-Henner, Sam Hocevar, DanielZM, Immanuel Goldstein~enwiki, Goldstein~enwiki, Neutrality, Uaxuctum~enwiki, Uaxuctum~enwiki, Klemen Kocjancic, Kocjancic, Chmod007, Kousu, Trevor MacInnis, Grunt, Eisnel, ELApro, Flex, Gazpacho, Fls, Frankchn, Mormegil, AAAAA, Freakofnurture, Imroy, Discospinster, Brianhe, Rich Farmbrough, Rhobite, Guanabot, FranksValli, Vsmith, Max Terry, Deelkar, Paul August, Joblio, Bender235, ESkog, Kaisershatner, Jnestorius, Mattdm, Ground, Elwikipedista~enwiki, TruthSifter, El C, Workster, Shanes, RoyBoy, Triona, Andrewpmack~enwiki, Bobo192, Reinyday, C S, Cmdrjameson, NickSchweitzer, Daf, Apostrophe, DCEdwards1966, Manu.m, Nsaa, Perceval, Mareino, Hyperdivision, Sigurdhu, Osmosys, Alansohn, Jic, Free Bear, Tek022, Jesset77, Andrewpmk, Arvedui, Anittas, Jnothman, Jamiemichelle, MarkGallagher, BryanD, Sligocki, Sligocki, PAR, Wdfarmer, Hu, Yolgie, DreamGuy, Hohum, Pianoplayerontheroof, Snowolf, Gloworm, ReyBrujo, Suruena, Uffish, Evil Monkey, Lokeshwaran~enwiki, Jheald, Grenavitar, Mikeo, Woodstone, Kouban, Oleg Alexandrov, Mnavon, Gmaxwell, Roylee, Angr, Velho, Simetrical, Linas, Ormy, Mindmatrix, Blumpkin, RHaworth, Shreevatsa, Scriberius, Spettro9, StradivariusTV, Benbest, Ruud Koot, WadeSimMiser, MONGO, Brunopostle, CS42, Thruston, Macaddct1984, M412k, Noetica, Reddwarf2956, Wikipedian231, Dysepsion, Tslocum, Sin-man, Energizerrabbit, Graham87, Magister Mathematicae, Taestell, Keknehv, Keknehv, Kbdank71, FreplySpang, Padraic, AllanBz, Icey, Enzo Aquarius, Jorunn, Rjwilmsi, Nightscream, Bill Cannon, Red King, TheRingess, Salix alba, SMC, Nneonneo, HappyCamper, AlisonW, MapsMan, Bfigura, Yamamoto Ichiro, FayssalF, Nzguy2004, [email protected], RobertG, Doc glasgow, Mathbot, Tumble, GünniX, Nivix, RexNL, Gurch, Scythe33, Glenn L, Stephantom, Chobot, DVdm, Volunteer Marek, Korg, Digitalme, E Pluribus Anthony, Algebraist, Banaticus, The Rambling Man, YurikBot, Hairy Dude, Jimp, Pip2andahalf, Phantomsteve, Jtkiefer, SpuriousQ, CanadianCaesar, RadioFan2 (usurped), Stephenb, Rsrikanth05, Wimt, JohanL, Jaxl, Nad, JocK, Nucleusboy, Brandon, Dimabat, DeadEyeArrow, JMBrust, Tachyon01, TUSHANT JHA, Imgregor, Tetracube, Mike Serfas, 2over0, Ali K, Chesnok, Arthur Rubin, Charlik, StealthFox, BorgQueen, JoanneB, Nae'blis, HereToHelp, HereToHelp, Willbyr, Moonsleeper7, Appleseed, Persept, JDspeeder1, GrinBot~enwiki, Mejor Mejor Los Indios, DVD R W, Finell, That Guy, From That Show!, NetRolller 3D, Attilios, SmackBot, RDBury, Looper5920, Mmernex, Unschool, Adam majewski, Jerdwyer, IddoGenuth, Nihonjoe, Tomer yaffe, InverseHypercube, KnowledgeOfSelf, Melchoir, McGeddon, CopperMurdoch, Unyoyega, Pgk, Rokfaith, Jagged 85, Jdmt, Tr0gd0rr, Delldot, Jihiro, Eskimbot, Kaimbridge, BiT, Wittylama, Xaosflux, Richmeister, Youremyjuli Youremyjuliet, et, Skizzik, Cabe6403, ERcheck, Afa86, Durova, Cowman109, Kaiserb, Chris the speller, Pdspatrick, Pdspatrick, Persian Poet Gal, B00P, Tree Biting Conspiracy, Silly rabbit, BALawrence, ERobson, Adpete, Octahedron80, Nbarth, Baa, ACupOfCoffee, Zven, Gracenotes, Can't sleep, clown will eat me, Joerite, Jamse, J amse, Jahiegel, Tamfang, Scray, TheGerm, Berland, LouScheffer, LouScheffer, Addshore, Alton.arts, Rrc2002, ConMan, Wen D House, Flyguy649, Napalm Llama, Lhf, Angellcruz, M jurrens, Attasarana, Illnab1024, DMacks, Lisasmall, Wizardman, Just plain Bill, Xiutwel, Kukini, Drunken Pirate, John Reid, Will Beback, Pinktulip, Zchenyu, Lambiam, Esrever, Phi1618~enwiki, ArglebargleIV, Rory096, OliverTwist, BorisFromStockdale, JzG, Kuru, Titus III, Scientizzle, Kumarsenthil, Park3r, Adj08, Aroundthewayboy, Eikern, Minna Sora no Shita, Mgiganteus1, NongBot~enwiki, Jbonneau, Phancy Physicist, Ckatz, MarkSutton, Stwalkerster, George The Dragon, Alethiophile, Mr Stephen, Childzy, Dicklyon, Cxk271, Jaipuria, Waggers, AdultSwim, Ryulong, MTSbot~enwiki, Inquisitus, Xionbox, Iridescent, Kencf0618, Madmath789, Andreas Rejbrand, Lenoxus, GrammarNut, Tibbits, Tawkerbot2, Gco, Pi, JRSpriggs, Hum richard, Emote, Lavaka, Owen214, The Haunted Angel, JForget, Tanthalas39, Asteriks, Aypak, Ale jrb, Sjmcfarland, Stmrlbs, Woudloper, CBM, Eric, JohnCD, Collinimhof, Runningonbrains, Runningonbrains, Dgw, Tac-Tics, Joelholdsworth, Joelholdsworth, Emesghali, MrFish, Myasuda, Cydebot, Scorpi0n, Mblumber, Reywas92, Carifio24, Gogo Dodo, Alanbly, Islander, Muhandis, Julian Mendez, Gtalal, Tawkerbot4, Bsdaemon, Dblanchard, Gimmetrow, KamiLian, Billywestom, Billywestom, Thijs!bot, Epbr123, Wikid77, David from Downunder, Herbys, Headbomb, Marek69, Ronbarton, West Brom 4ever, John254, Frank, James086, Jmelody, Dfrg.msc, Dfrg.msc, CharlotteWebb, CharlotteWebb, Heroeswithmetaphors, Escarbot, Kazrian, Kazrian, Stannered Stannered,, Maxhawkin Maxhawkins, s, Mentifist Mentifisto, o, Somnabot, Somnabot, John Smythe, Smythe, AntiVan AntiVandalBo dalBot,t, Nimo8, Nimo8, Luna Luna Santin, Santin, Seivad, Seivad, Clf23, Clf23, Seaphoto, Seaphoto, Opelio, QuiteUnusual, Czj, Quintote, DomainUnavailable, Pixiebat, Ellie57, NSH001, Manushand, LibLord, Minhtung91, Andrew Parkins, Spencer, Moonraker0022, ChrisLamb, Dhrm77, Gökhan, Michael Tiemann, Leuko, Curmi, Aheyfromhome, Jelloman, MER-C, Ricardo sandoval, Alpinu, PhilKnight, Twospoonfuls, Twospoonfuls, Acroterion, Prof.rick, Bensonchan, Bongwarrior, VoABot II, 12398745 6, JNW, Swpb, Ling.Nut, PeterStJohn, PeterStJohn, Richard Bartholomew, Bartholomew, Midgrid, Catgut, Animum, SSZ, Erichas, Powerinthelin Powerinthelines, es, Shocking Blue, Virtlink, Allstarecho, David Eppstein, Styrofoam1994, Madmanguruman, Gomm, Martaclare, DerHexer, JaGa, GhostofSuperslum, Lelkesa, A2-computist, EduardoAndrade91, Hbent, TimidGuy, GravityWell, AVRS, Tentacles, MartinBot, Lord Ibu, Bbi5291, ExplicitImplicity, Poeloq, Axlq, Math Lover, CommonsDeli CommonsDelinker, nker, Pbroks13, Pbroks13, The Anonymous Anonymous One, J.delanoy, J.delanoy, Pharaoh of the Wizards, Svetovid, Svetovid, Wmjohn6217, Wmjohn6217, Personjerry, Personjerry, Eliz81, WarthogDemon, AVX, Bluesquareapple, Vanished user 342562, Deborah A. Becker, Laurusnobilis, Ixaciyelx, Johnbod, 5theye, Tarotcards, Jacksonwalters, Aupoverq, AntiSpamBot, (jarbarf), Kurney, Chiswick Chap, FamicomJL, NewEnglandYankee, Kit Cloudkicker, kicker, In Transit, Fibonaccimaster, Fibonaccimaster, Policron, Bobianite, Cobi, 83d40m, 8 3d40m, Hanacy, Ha nacy, Cmichael, Phillipkwood, WJBscribe, Donut50, Jevansen, Agrofe, Mmoople, Natl1, Bonadea, Underthebridge, Fjackson, Sand village, Ja 62, Pushups, Xiahou, Black Kite, Lights, Deor, Timotab, Caspian blue, 28bytes, Zen Lunatic, VolkovBot, ABF, Julia Neumann, Pleasantville, Science4sail, Jeff G., JohnBlackburne, Kyle the bot, Philip Trueman, TXiKiBoT, Jh559, Amitkrdce, Vipinhari, Eylenbosch, Hobe, Oconnor663, Sparkzy, Rei-bot, Z.E.R.O., Anonymous Dissident, Bigyaks, Captain Wikify, Arnon Chaffin, Clark Kimberling, Qxz, Someguy1221, Voorlandt, Ocolon, Lradrama, Clarince63, Mbasit, Don4of4, Yomcat, Ripepette, Borealis9, PDFbot, Gautamkaul, Cuddlyable3, Ilyushka88, RKThe2, Kızılsungur, Kpedersen1, Kmhkmh, Insightfullysaid, Luqqe, Graymornings, Philipmarkedwards, Insanity Incarnate, Brianga, Jamrb, Chenzw, Sattar82, PAntoni, FlyingLeopard2014, S8333631, Ccartmell, Dinesh Menon, Serprex, Kbrose, The Random Editor, Arjun024, Gaelen S., GamesSmash, SieBot, Jediwizardspy, FergiliciousFilipino, Tresiden, Gmeisner, Dedeche, Euryalus, CWDURAND, Xluffyoox, Spyderxskierx4, Caltas, Erier2003, Yintan, Vanished User 8a9b4725f8376, Grimblegrumble85, Keilana, Cole SWE, Phiman91, Tiptoety, Radon210, Tucapl~enwiki, Larfi, Rexpilger, Oxymoron83, Steven Crossin, ShadowPhox, Darius X, Svick, Coldcreation, Garies, Dstlascaux, Ward20, Prekageo, Prekageo, Sauron1495, Dust Filter, Mr. Stradivarius, Stradivarius, Nic bor, Struway2, Denisarona, Xandras, Athenean, Ministry of random random walks,

20

11

TEXT TEXT AND IMAG IMAGE E SOURCES, SOURCES, CONTRI CONTRIBUTO BUTORS, RS, AND LICENSES  LICENSES 

Jamesfranklingresham, Jamesfranklingresham, Elassint, ClueBot, Redeyez114, Jbening, Binksternet, Justin W Smith, F cooper 8472, The Thing That Should Not Be, Matdrodes, Plastikspork, JuPitEer, Bhuna71, Drmies, Jessie1994, TheOldJacobite, TheOldJacobite, Timberframe, Niceguyed Niceguyedc, c, Blanchardb, LizardJr8, Rotational, Alindsey, 11quintanarq, Vanillagz, Puchiko, Bonzai273, Chimesmonster, Fox2030, John Pons, Excirial, Jusdafax, Pabbom, Dagordon01, Naveeyindren Naveeyindren 1618, Brews ohare, NuclearWarfare, NuclearWarfare, Arjayay, Arjayay, Jotterbot, SirXaph, Psinu, Wprlh, 7&6=thirteen, Enoch Wong, 20-dude, GlasGhost, TruthIsStrangerThanFiction, Calor, Thingg, Nibi, 7, MelonBot, Theunixgeek, DOR (HK), Adamfinmo, Johnuniq, Lethalstrike5, Lethalstrike5, Vanished user uih38riiw4hjlsd, uih38riiw4hjlsd, Jerker L., DumZiBoT, Pooptarts, Jean-claude perez, XLinkBot, Marc van Leeuwen, Leeuwen, Fastily, Spitfire, Wertuose, Hannah434, Ost316, Avoided, Nicolae Coman, Virginia-American, Virginia-American, JKelly1808, SkyLined, Infonation101, Kbdankbot, HexaChord, Tayste, Addbot, Willking1979, Some jerk on the Internet, RobinClay, DOI bot, Fgnievinski, TutterMouse, RAC e CA12, Fieldday-sunday, Briandamgaard, CanadianLinuxUser, Cst17, LaaknorBot, SoxBot V, Glane23, Favonian, LinkFA-Bot, 5 albert square, IOLJeff, Toddles9, Numbo3-bot, Ehrenkater, Thom1555, Tide rolls, Lightbot, Megaman en m, Legobot, Luckas-bot, Yobot, Philglenny, Pink!Teen, TaBOT-zerem, Msadinle, Paepaok, THEN WHO WAS PHONE?, Bradym80, Knownot, IW.HG, Aswxmike, Synchronism, AnomieBOT, Miccospadaro, Rubinbot, Jim1138, Deke omi, Hydreptsi, Kingpin13, Ulric1313, Flewis, Laksdjfosdkf Laksdjfosdkfj, j, Materialscientist, Materialscientist, Kool Aid Relic, Citation bot, StrontiumDogs, GB fan, Frankenpuppy, ArthurBot, Ladyburningrose, Xqbot, Sathimantha, The sock that should not be, Capricorn42, 22over7, Crookesmoor, Nokkosukko, Nokkosukko, Grim23, Br77rino, Isheden, Gap9551, Vmt164, Shillu123, GrouchoBot, RaWrMonSter, RaWrMonSter, Miesianiacal, Omnipaedista, Davidhoskins, Doulos Christos, Wiiman222, E0steven, Prezbo, Screamoguy, FrankWasHere, SirEbenezer, Sushiflinger, Coffeerules9999, Coffeerules9999, Captain-n00dle, Canton Viaduct, Komitsuki, LucienBOT, Riventree, Mark Renier, Lagelspeil, Lagelspeil, Tinyclick, Mootown2, Insectscorch, Lunae, D'ohBot, Ktbbabe, Silverhammermba, Tavernsenses, Alphobrava, Drew R. Smith, Robo37, Asheryaqub, Citation bot 1, Tkuvho, DrilBot, Emilu18, Pinethicket, Honeymancr12, Foothiller, Hamtechperson, Achim1999, A8UDI, MasterminderBS, RedBot, MastiBot, Flashharry9, Marsal20, Toolnut, FoxBot, Kapgains, Ambarsande, TobeBot, Pollinosisss, Fox Wilson, Mileswms, Mileswms, Xx3nvyxx, Zvn, BeebLee, , Gzorg, Tbhotch, FKLS, Minimac, Magic cigam, Spencerpiers, Spencerpiers, Berg.Heron, RjwilmsiBot, Bento00, MrRight425, Fiboniverse, Regancy42, Balph Eubank, Going3killu, Salvio giuliano, WikitanvirBot, Broselle, Whalefishfood, Nerissa-Marie, EddyLevin, RA0808, Dooche101, 8bits, Ryan c chase, Kris504, Tommy2010, Wikipelli, Dcirovic, Misscmoody, Slawekb, Slawekb, Chricho, Benchdude, Destiney Kirby, ZéroBot, Jargoness, John Cline, Fæ, Shuipzv3, Fg=phi, Chharvey, Matthewcgirling, Matthewcgirling, Cobaltcigs, cigs, H3llBot, H3llBot, Aughost Aughost,, SporkBot, SporkBot, Aknicho Aknicholas, las, Gz33, Didi42bro Didi42brown, wn, Tolly4b Tolly4boll olly, y, Vanish Vanished ed user fijw983kj jw983kjaslk aslkekfh ekfhj45, j45, Sbmeiro Sbmeirow, w, Num Ref, Mayur, Arkaever, Donner60, Smartie2thaMaxXx, DeltaQuad.alt, Dragfiter234, JanetteDoe, GrayFullbuster, GrayFullbuster, DASHBotAV, Nikolas Ojala, Davesteadman, ClueBot NG, Fridakahlofan, Wcherowi, MelbourneStar, Nepanothus, Frietjes,  half-moon bubba, O.Koslowski, KirbyRider, Widr, Kant66, Devsinghing, Anon5791, Oddbodz, Helpful Helpful Pixie Bot, Bxzooo, Rosetheprof, Calabe1992, Vagobot, Gasberian, MusikAnimal, Metricopolus, Wikimpan, Davidiad, FrostBite683, EspaisNT, Sparkie82, Brad7777, Gdmall88, Klilidiplomus, Irung4, Rob Hurt, Tejasadhate, Tejasadhate, Bill.D Nguyen, Gmoney123456789, Boeing720, WebFlower1, WebFlower1, Cyberbot II, Mauricio1994, Mauricio1994, Maxronnersjo, Maxronnersjo, Petrus3743, , YFdyh-bot, Giufra9396, Afonfbg, Kennethdjenkins, Dexbot, Luckimg2, Hmainsbot1, Iivanyy89, Webclient101, Lugia2453, Jamesx12345, NealCruco, Limit-theorem, Moony22, RPFigueiredo, D. Philip Cook, Ultimatesecret12, Mre env, Melonkelon, Tentinator, Bzavitz, Goss is super, Ugog Nizdast, Mynameisrichard, Evensteven, MrBearHugger, Dvorak182, Jianhui67, Paul2520, Fortok1, Xenxax, Amrik singh nimbran, AnonymousAuthority, ThatRusskiiGuy, Monkbot, Tunisie98, BethNaught, Tk plus, 400 Lux, KH-1, Loraof, Gabrielcwong, , KasparBot, Martin Peter Clarke, BU Rob13, Lemondoge, Skyllfully, Sgr ganesh, GreenC bot and Anonymous: 1633

11.2 11.2 Imag Images es •

•

•

•

•

•

•

•

•

•

•

•

•

•

File:7234014_Parthenona File:7234014_Parthenonas_(cropped).jpg s_(cropped).jpg   Source:    https://upload.wikimedia.org/wikipedia/commons/c/c8/7234014_Parthenonas_ Contributors:  Own %28cropped%29.jpg License:  CC0  CC0 Contributors:   Own work  Original artist:  C   C messier https://upload.wikimedia.org/wikipedia/commons/4/4f/Aeonium_tabuliforme.jpg nium_tabuliforme.jpg License:  CC  CC  File:Aeonium_tabuliforme.jpg Source:  https://upload.wikimedia.org/wikipedia/commons/4/4f/Aeo BY-SA 3.0   Contributors:  Own work  Original artist:  Max  Max Ronnersjö

 https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License:  CC-BY-SA-3.0  CC-BY-SA-3.0 Contribu  File:Commons-logo.svg  Source:  https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg tors:  ?  ?  Original artist:  ?  ? File:FWF_Samuel_Monnier_d File:FWF_Samuel_Monnier_détail.jpg étail.jpg   Source:    https://upload.wikimedia.org/wikipedia/commons/9/99/FWF_Samuel_Monnier_d% C3%A9tail.jpg License:  CC  CC BY-SA 3.0 Contributors:  Contributors:  Own  Own work (low res file)  Original artist:  Samuel   Samuel Monnier  File:FakeRealLogSpiral.svg   Source:  https://upload.wikimedia.org/wikipedia/commons/a/a5/FakeRealLogSpiral.svg License:  CC-BYSA-3.0  Contributors:   FakeRealLogSpiral.png Original artist:  FakeReal FakeRealLogSpiral.png LogSpiral.png:: Pau File:Fibonacci_spiral_34.svg File:Fibonacci_spiral_34.svg   Source:  https://upload.wikimedia.org/wikipedia/commons/9/93/Fibonacci_spiral_34.svg License:   Public domain   Contributors:  Own work using: Inkscape  Original artist:  User:Dicklyon  User:Dicklyon

 https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Ico s://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg n.svg License:  Cc-by  Cc-byFile:Folder_Hexagonal_I File:Folder_Hexagonal_Icon.svg con.svg  Source:  http sa-3.0  Contributors:  ?  Original artist:  ?  ?  CC0   File:Free-to-read_lock_75.svg Source:  https://upload.wikimedia.org/wikipedia/commons/8/80/Free-to-read_lock_75.svg License:  CC0 Contributors:  Adapted from 9px|Open_Acce 9px|Open_Access_logo_PLoS_white ss_logo_PLoS_white_green.svg _green.svg   Original artist:  artist:    This version: version:Trappist_the_monk (talk talk)) (Uploads Uploads)) File:Gold,_silver,_and_bronze_rectangles_vertical.png   Source:    https://upload.wikimedia.org/wikipedia/commons/c/c1/Gold%2C_ File:Gold,_silver,_and_bronze_rectangles_vertical.png silver%2C_and_bronze_rectangles_vertical.png License:  CC  CC BY-SA 4.0 Contributors:  Contributors:  Own  Own work  Original artist:  Hyacinth   Hyacinth File:Gold,_square_root_of_2,_and_square_root_ File:Gold,_square_root_of_2,_and_square_root_of_3_rectan of_3_rectangles.png gles.png   Source:  https://upload.wikimedia.org/wikipedia/commons/3/ Contributors:  Own work  Original  34/Gold%2C_square_root_of_2%2C_and_square_root_of_3_rectangles.png License:  CC BY-SA 4.0 Contributors:  artist:  Hyacinth   Hyacinth

https://upload.wikimedia.org/wikipedia/commons/5/58/GoldenRhombus.svg hombus.svg License:  CC  CC BY-SA 3.0  File:GoldenRhombus.svg Source:  https://upload.wikimedia.org/wikipedia/commons/5/58/GoldenR Contributors:  en:File:GoldenRhombus.png Original artist:  Zom-B  Zom-B (Original); Pbroks13 (Original); Pbroks13 (Derivative)  (Derivative) https://upload.wikimedia.org/wikipedia/commons/b/b7/Golden_mean.png _mean.png License:  CC  CC BY-S BY-SA A 3.0 Con File:Golden_mean.png Source:  https://upload.wikimedia.org/wikipedia/commons/b/b7/Golden tributors:  Own work  Original artist:  Adam  Adam majewski  File:Golden_ratio_line.svg   Source:  https://upload.wikimedia.org/wikipedia/commons/4/44/Golden_ratio_line.svg License:  Public domain Contributors:  Contributors:  en:Image:Golden  en:Image:Golden ratio line.png  Original artist:  Traced  Traced by User:Stannered by User:Stannered

11.3 11.3

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

Conte Content nt licens licensee

 File:Golden_triangle_(math).svg   Source:    https://upload.wikimedia.org/wikipedia/commons/a/a5/Golden_triangle_%28math%29.svg License:  Public   Public domain   Contributors:  Own work  Original artist:  Jheald  Jheald File:Goldener_Schnitt_(Äußere_Teilung File:Goldener_Schnitt_(Äußere_Teilung).svg ).svg   Source:    https://upload.wikimedia.org/wikipedia/commons/f/fe/Goldener_Schnitt_ %28%C3%84u%C3%9Fere_Teilung%29.svg  License:    CC-BY-SA-3.0   Contributors:    This   vector image  was created with  Inkscape  Inkscape.. Original artist:  Wolfgang  Wolfgang Beyer (Quelle in PNG) & Konstantin1996 (PNG → SVG)

https://upload.wikimedia.org/wikipedia/commons/1/1c/Goldener_Schnitt_Konstr_ er_Schnitt_Konstr_ File:Goldener_Schnitt_Konstr_beliebt.svg File:Goldener_Schnitt_Konstr_beliebt.svg Source:  https://upload.wikimedia.org/wikipedia/commons/1/1c/Golden beliebt.svg  License:   CC-BY-SA-3.0   Contributors:  als SVG selbst erstellt, basiert auf PNG-Version von Wolfgang Beyer  Original artist:  Daniel Seibert  https://upload.wikimedia.org/wikipedia/commons/8/8a/Loudspeaker.svg License:  Public   Public domain Contrib File:Loudspeaker.svg  Source:  https://upload.wikimedia.org/wikipedia/commons/8/8a/Loudspeaker.svg utors:  New version of Image:Loudspeaker.png of  Image:Loudspeaker.png,, by AzaToth by  AzaToth and  and compressed by Hautala by Hautala  Original artist:  Nethac DIU, DIU, waves corrected by Zoid  File:Mathematical_Pyramid.svg   Source:   https://upload.wikimedia.org/wikipedia/commons/3/32/Mathematical_Pyramid.svg License:  Public domain Contributors:  Contributors:  Own  Own work  Original artist:  Dicklyon at  Dicklyon at English  English Wikipedia File:Medieval_manuscript_framew File:Medieval_manuscript_framework.svg ork.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/f/f4/Medieval_manuscript_ framework.svg License:  Public   Public domain   Contributors:  Jan Tschichold, Tschichold, The Form of the Book Original artist:  jossi  File:Michael_Maestlin.jpg   Source:  https://upload.wikimedia.org/wikipedia/commons/6/6a/Michael_Maestlin.jpg License:  Public doContributors:  http://de.wikipedia.org/wiki/Bild:Michael_Maestl main Contributors:   ht tp://de.wikipedia.org/wiki/Bild:Michael_Maestlin.jpg in.jpg Original artist:  original   original uploader  File:OEISicon_light.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/d/d8/OEISicon_light.svg License:    Public domain Contributors:  Own work  Original artist:  Watchduck   Watchduck (a.k.a. Tilman Piesk)   File:Odom.svg  Source:  https://upload.wikimedia.org/wikipedia/commons/b/ba/Odom.svg License:  CC BY 3.0   Contributors:  Own work Original artist:  Kmhkmh   Kmhkmh   File:Pentagram-phi.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/3/30/Pentagram-phi.svg  License:    Public domain Contributors:  Transferred from en.wikipedia from en.wikipedia to  to Commons.  Original artist:   Jamiemichelle at Jamiemichelle at English  English Wikipedia File:Pentagram_and_human File:Pentagram_and_human_body_(Agrippa).jpg _body_(Agrippa).jpg   Source:    https://upload.wikimedia.org/wikipedia/commons/d/dc/Pentagram_and_ human_body_%28Agrippa%29.jpg   License:    Public domain  Contributors:    From Heinrich Cornelius Agrippa’s   Libri Libri tres de occulta occulta  philosophia. Scanned by Jörgen by  Jörgen Nixdorf; Nixdorf; originally at en:Image:Pentagram3.jpg at en:Image:Pentagram3.jpg..  Original artist:  Heinrich   Heinrich Cornelius Agrippa Contributors:   File:Phi_uc_lc.svg   Source:  https://upload.wikimedia.org/wikipedia/commons/c/c2/Phi_uc_lc.svg License:  Public domain Contributors:  Own work  Original artist:  Dcoetzee,  Dcoetzee, F l a n k e r https://upload.wikimedia.org/wikipedia/commons/5/52/Ptolemy_Pentagon.svg _Pentagon.svg License:  Public do File:Ptolemy_Pentagon.svg  Source:  https://upload.wikimedia.org/wikipedia/commons/5/52/Ptolemy main Contributors:  Contributors:  Own  Own work by en:User:Dicklyon by  en:User:Dicklyon  Original artist:  en:User:Dicklyon  en:User:Dicklyon File:Rhombictriacontahedron.svg File:Rhombictriacontahedron.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/2/2c/Rhombictriacontahedron.svg License:  CC-BY-SA-3.0 Contributors:  Vectorisation  CC-BY-SA-3.0 Contributors:    Vectorisation of Image:Rhombictriacontahedron.jpg of  Image:Rhombictriacontahedron.jpg Original artist:  User:DTR  User:DTR File:SimilarGoldenRectangles.svg File:SimilarGoldenRectangles.svg   Source:    https://upload.wikimedia.org/wikipedia/commons/8/8e/SimilarGoldenRectangles.svg License:  Public   Public domain   Contributors:  Own work  Original artist:  Ahecht (Original);   Ahecht (Original); Pbroks13  Pbroks13 (Derivative  (Derivative work); Joo. work); Joo. (Editing)  (Editing)  File:Whirling_squares.svg   Source:  https://upload.wikimedia.org/wikipedia/commons/2/22/Whirling_squares.svg License:  CC BY-SA 3.0  Contributors:  Own  Own work  Original artist:  Dicklyon  Dicklyon

11.3 11.3 Conten Contentt licen license se •

21

Creativee Commons Attribution-Share Creativ Attribution-Share Alike 3.0