Making a "Great Pyramid " Scale Model Dr N K Srinivasan Introduction The great pyramid of Giza is one of the most famous pyramids of Egypt. It was built by King Khufu,also called King Cheops by Greeks.It was around 2600 BC the pyramid was completed.It took ,perhaps, 40 years to build the pyramid; Nearly 40,000 workers were involved. Some say that they worked in shifts of 10,000 workers at a time. This pyramid is built to certain exact dimensions which we employ for the scale model you will build. The Egyptians used a measure of length called "CUBIT". We don't use it any more. The great pyramid has a square base of 440 cubits and a height of 280 cubits. May be, easy for them to remember! 1 cubit = 52.36 centimeter or 0.52 metre.= 20.16 inches. [One cubit is roughly equal to half a meter.] The basic dimensions of the great pyramid are: base = 230.4 metre, slant length = 186.4 metre height = 148.2 metre The height is a bit of variation: The original height was 148.2 metre, but the top was flattened later
,giving a height of 137 meters. Don't worry much if different authors give slightly different heights.
Scale Model dimensions We will build a small scale model of this pyramid. For convenience, we take a base of 15 cm length or 6 inches. Then the height becomes : H = 9.65 cm The slant length = L
= 12.1 cm
The length of the edge S = 14.3 cm Now we can proceed to cut the pieces required for building the model of the pyramid. Materials required: First let us collect the following materials and tools for this project.! 1 Cardboard ; this could be thin cardboard from a cereal box or thicker ones from the covers of a note book or regular cardboard sheet. You would need 3 pieces of roughly 20 cm by 30 cm or 8 in x 12 in. 2 Scissors 3 A rule and a compass or set square [from your Geometry box] 4 White glue 5 Adhesive "scotch" tape or sealing tape 6 Kid's paint [ poster or acrylic] 7 Play sand [color of your choice]
Procedure Step 1: Take a white sheet and draw a square of length 15 cm; transfer this drawing to the cardboard sheet and cut a square of 15 cm side with the scissors. Cut slowly in small cuts each time ["in small bite " ] so that you can get
sharp, close cut to
the
dimensions. Step 2: This is an important step. Do it carefully. You have to draw a isosceles triangle ABC, with the sides: Base BC= 15 cm,
AB = AC = 14.3 cm
[Since the two sides AB and AC are equal, we call it isosceles triangle.] Tips to draw this triangle: Method 1 : Draw a line of 15 cm length. This is the base BC. Find the midpoint P of BC. Draw a perpendicular to BC at this point P.You will use a set square for this; Call it PD. Mark point A on the line PD such that AP = 12.1
AP is
the slant length = L = 12.1 Join A to B and A to C. Now the triangle is done. Method 2: You will use a compass. Set the compass to a radius of 14.3 cm. Draw a line BC of length 15 cm. Placing the compass point at B, draw an arc ;placing the compass point at C,draw another arc. Let the two arcs cut each other. The intersecting point is the apex A. Join A to B .Join A to C. Now you have drawn the triangle ABC. Step 3:
Using the drawing of triangle ABC, transfer the drawing to the cardboard piece and make a drawing of ABC on the cardboard. You can use a tracing paper or mark the points A,B and C and join them. Cut the triangle ABC. Check the three sides of the traingle. You must make four (4) such triangles for the four sides of the pyramid. Step 4 Assemble the pyramid:
Paste the four sides to the
base with the tape. Also try to bring all the four sides to the sharp apex. Paste some tapes near the apex. Note: Don't worry if the four sides do not get postioned perfectly well and match . You may find some gap along the edges. Seal the edges with some paper strips and paste the edges. Now paste the edges between any two sloping sides. You may take the help of another student or friend to hold the sides together while pasting the sides. Step 5 Paint the sloping sides with color of your choice or use a light brown or yellow ochre color. Optional Step: You can also make a dilute paste of white glue [add a few drops of water to the glue and mix well].Using a brush, smear the sides with a light coat of the glue. Then spray the play sand over the sides till a thin layer of sand develops over the sides. Now the construction of Pyramid is over
----------------------------------------------------------------------------------------Some Calculations Let us do some calculations to understand the scale model and the magic of numbers underlying the geometry of this
pyramid:
What is the scaling we have used? base in the model = B = 15 cm = 0.15 m base in the pyramid = B' = 230.4 metre The scale factor = B/B' = 0.15 / 230 = 15/23000 = 1/1533 or approximately, our model is 1: 1500 model. Height in the model = H =9.6cm = 0.096 metre height of the Pyramid = H' = 148 metre The scale factor = H / H' = 0.096/148 = 9.6/14800 = 1/1541 Again the scale model = 1:1500 [approximately].
Check the dimensions used: 1 The triangle ABC is an isosceles triangle: Draw a perpendicular from apex A to the base BC. Let the midpoint of BC is P. Now APB is a right triangle;so using Pythogorian theorem: AP2 + BP2 = BA2 AP = 12.1 cm
BP = 7.5 cm
146.4 + 56.25 = 202.66 Then BA= 14.2
This checks with the dimensions of the sloping face or the pyramid. 2 Now taking the centre of the base of the pyramid, let us draw a perpendicular from apex A to this center D. Now AD is the height of the pyramid. Join P and D. Taking the right triangle ADP. we get PD = 7.5 cm
AP = 12.1 cm AP2 = PD2
+ AD2
146.25 = 56.25 + AD2 AD =
9.48
Pretty close to the height we took as H= 9.6 which is a more accurate value. Note: In constructing this pyramid and in the use of dimensions , the accuracy of + or - 1 mm would be involved. That is oKay!! The sloping angle of each face is an important parameter in constructing this pyramid. We use a bit of trigonometry. The sloping angle = tangent (angle) = AD/PD = 9.6/7.5 = 1.28 Using a trigonometric table or calculator ,you will find the angle is 52 deg. tan(52) = 1.28 The egyptologists say that the exact angle is 51 deg 51 min or 51.85 degrees. Our model is pretty close to that.
How did the Egyptians measure angles and slopes? The Egyptians used a right triangle
concept to measure angles
and fix the slopes. Take a right triangle ABC, with vertical leg AB and the horizontal leg BC and the sloping hypotenuse as AC. Take a right triangle ABC, with vertical leg AB and the horizontal leg BC and the sloping hypotenuse as AC.
These math wizards always kept the vertical leg AB as 1 cubit. They divided one cubit [about 53.2 cm] into 7 palms. [One cubit roughly resembles one elbow length and one palm would be roughly equal to the length of your palm or palm of an Egyptian in those days!.] They divided one palm into 4 digits. [ A digit is close to length of your digit or
finger!]
Keep this in mind: 1 CUBIT = 7 PALMS 1 PALM = 4 DIGITs Now construct a right triangle ABC with AB,vertical side equal to 1 cubit always. But you can vary the horizontal leg BC to get different slopes for the hypotenuse with respect to horizontal line. If BC=1 cubit= 7 palms , you get '7seked'
or the angle is 45 degrees.
The horizontal leg measured in palms they called SEKED or seqd. "SEKED" BECOMES THE MEASURE OF THE ANGLE OR SLOPE OR GRADIENT. Suppose you have 5palm and 2 digit as the horizontal leg BC . Now BC= 5.5 palms
AB= 1 cubit = 7 palms
Then the slope corresponds to the angle whose tangent(angle) = 1 cubit/5.5 palm = 7palms/5.5 palms = =7/5.5 =1.272 the angle ACB = 51.85 degrees.
The slope of Great Pyramid -- 5.5 seked This is the slope they used for the Great Pyramid. In other words, they used a right traingle with one leg as 7 units,the other leg as 5.5 units for slope measure. This gives the required slope. The Great Pyramid is then called a " 5.5 seked pyramid." This kind of taking measures must have been easy to use for making huge slabs for the sloping face or casing stones and so on. [Some pyramids used 5.25 seked or other sekeds too.] Therefore they used a unit "palm" which is 1/7 of a cubit. Why did they divide a cubit into 7 equal parts? There should be some good reasoning behind this. [Britishers divided a foot into 12 parts and called it an inch. We had problems in multiplying and dividing by 12 for various calculations as students. We did not use calculators then. There was no electronic calculator around. Here Egyptian boys and girls must have had problems multiplying or dividng by 7, a prime number at that!---for cubits and palms!! ]
Egyptians measure circumference of a circle Most probably, though they did not know about pi =3.1416, they used a simple ratio to convert diameter into circumference of a circle.That ratio is R= 22/7
We use 22/7 as a simple apporoximation for pi.! What is the circumference of a circle C then:
C= pi x
diameter = pi x D For Egyptians,
C = (22 x D) / 7
Since the denominator is 7 in this equation, it makes sense to use a measure of "7 parts" or 7palms for D. Suppose the diameter is 2 cubits length or 14 palms, then C= 22 x 14/7 = 44 palms. This becomes a convenient and easy way to calculate circles. Again taking the Great Pyramid example, the base was 440 cubits. This corresponds to a circle of diameter 140 cubits.! [440 = (22/7) x 140 ] Now 140 cubits is half the height of the pyramid.Height of the pyramid H= 280 cubits.The radius of this imaginary circle is then the height of the pyramid!!! [May be they drew a circle of this diameter for early measurements for the base.!] Now you see the connection why they divided the cubit into 7 palms for the angle or slope.!! Therefore the Egyptians did not know pi as such,but used the ratio of (22/7) to relate circumference of a circle to diameter or radius.A powerful approximation indeed.!! This also explains why they used base as 440 cubits and height as 280 cubits for the great Pyramid.
Advanced Math Fun Mysteries of The Great Pyramid The Great Pyramid holds several mysteries. Some are architectural, some mathematical! Did you wonder why they chose 51.85 deg or say 52 degrees for the angle between the sloping surface and the base. Why it is not just 45 degrees? Many may even presume it is 45 degrees. This mystery was solved by Johannes Kepler, the astronomer, the same guy who studied the solar system!
Let us consider again the right triangle made with the apex, the centre of the base area D, and the midpoint of any one side ,P. This triangle ADP, has three sides : the slant length AP, the height AD, and half of the base length PD. Let us find the ratios of these sides: AP/AD = 12.1 / 9.6=1.28 AP/PD = 12.1/7.5 = 1.613 Hey! This ratio is close to 1.618 which is "Golden Ratio". You know from Geometry class what this golden ratio or phi means? You know that the Greeks used this ratio for width/height of Parthenon to lend aesthetic beauty to that building. We will say more about this later. So, the Egyptians used this angle of 52 degrees so that the ratio of the edge to half of base would be the Golden ratio. Since ADP is a right triangle, the other side must follow the Pythogorian formula; Yes, indeed, if PD
=1, and if
AD = √1.618 , then AP = √2.618
= 1.618 Let us check this out: PD=1 Therefore
AD=
1.272
AP = 1.618
PD:AD:AP = 1 : 1.272 : 1.618
Such a right triangle is called "Kepler Triangle.", in honor of Joahnnes Kepler. Did you notice another interesting thing in Kepler's triangle: Using Pythagoras theorem: 1 + 1.618 = 2.618 = (1.618)2
--------(equation 1)
This relation looks interesting! Let us call x=1.618 Then equation (1) can be rewritten quadratic equation: 1+ x = x2
in the familiar form of
Or
x2
- x - 1 =0
Let us solve this using the Quadratic formula : One solution is:
x = [1 + √ 5] / 2 = (
1 + 2.236)/2 = 1.618 So, the Golden Ratio is the solution for this wonderful quadratic equation---equation (1)! We can also write this quadratic equation like this: x = 1 + 1/x { if you are interested, study about continued fraction at this point.} Fibonacci Sequence: Let us tie up everything with Fibonacci sequence; Apparently , rabbit breeding with each generation follows this sequence: Start with 0 and 1, add the two previous numbers:0, 1,1,2,3,5,8,13,21,34,55,89,144,..... Take the ratio of any two consequent numbers in this sequence. 55/34 =1.6176
89/55 = 1.6181818
144/89=
1.6179775 Yes,
the ratios tend to the limit of Golden ratio.
So, the Golden ratio is at the base of all these interesting stuctures, including the Great Pyramid of Giza. [Note: There are some mathematicians who maintain that the Egyptians did not know such advanced algebra and math and did not follow these things. May be they were familiar with Pythagorian theorem...one doubts that too.!]Yet the fact remains that they used the sloping angle as 51.85 degree, not any simple number such as 45 deg or 50 deg or 60 degrees . One possibility is that they thought in terms of 15 degrees or multiples &fractions of this. Then possibly they wanted this slope angle to 15x 3 + 15/2 = 52.5 deg. No ,that is not close to 51.85 degrees.The mystery remains, unless we use the Golden ratio argument given here.
