By Prof. L. Kaliambos (Natural Philosopher in New Energy)

January 14  ,  2016

Historically, the Egyptians used the  Phi = Φ = 1.6180339887…) in the design of the Great Pyramids and they thought that the golden ratio was sacred.  Therefore, they used the golden ratio when building temples and places for the dead.  The Egyptians were aware that they were using the golden ratio Φ, but they called it the "sacred ratio."

There is debate as to the geometry used in the design of the Cheops Pyramid in Egypt. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with absolute certainty. The outer shell remains though at the cone, so this does help to establish the original dimensions.

Golden Section Phi = Φ = (1 + 50.5)/2 is obtained by dividing a line into two parts (α and β) such that the square of the first part is equal to the product of the whole segment (α+β) and the second part.

That is  α2 = (α +β )β   or  (α +β )/α = α/β = Φ

One method for finding the value of Φ  is to write α = Φ and β = 1 (unit length).

Therefore

(Φ+1)/Φ = Φ/1    or   Φ + 1 = Φ2

which can be rearranged to   Φ2 - Φ  -1 = 0.

Using this quadratic formula the solution is obtained as

Φ = (1 + 50.5)/2 = 1.6180339887….

The Greeks, who called  Phi = Φ  the Golden Section, based the entire design of the Parthenon on this proportion. Phidias (500 BC - 432 BC), a Greek sculptor and mathematician, studied Φ and applied it to the design of sculptures for the Parthenon. Plato (circa 428 BC - 347 BC), in his views on natural science and cosmology presented in his "Timaeus," considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos.  Moreover  Dinocrates for the two Caryatids in the Hephaestion tomb in Amphipolis used the same golden section. ( See my “Discoveries in Amphipolis”).

The Egyptians used the  Phi = Φ = 1.6180339887…) in the design of the Great Pyramids and they thought that the golden ratio was sacred.  Therefore, they used the golden ratio when building temples and places for the dead.  The Egyptians were aware that they were using the golden ratio Φ, but they called it the "sacred ratio."

There is debate as to the geometry used in the design of the Cheops Pyramid in Egypt. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with absolute certainty. The outer shell remains though at the cone, so this does help to establish the original dimensions.

On the other hand one should ask how the ancient  Egyptians were able to find the solution of the quadratic formula Φ2  - Φ  -1 = 0 . The study of this called algebra goes to the antiquity. Recent discoveries have shown that  Babylonians  and Egyptians solved problems in algebra , although they had no symbols for variables. They used only words to indicate such numbers, and for that reason their algebra has been referred to as theoretical algebra. The Ahmes Papyrus, an Egyptian scroll going back to 1600 BC has a number of problems in algebra, in which the unknown is referred to  as a hau, meaning “a heap”. Also practically the so -called  Pythagorean theorem (6th century BC) was well known to Babylonians and Egyptians. Thus writing

1 + Φ = Φ2   as (1)2 + (Φ0.5)2 = Φ2

one sees that the 1 (unit length) should be the radius r of a cone pyramid, while Φ0.5 = h   (height) and Φ = L (slant height). In this case the circumference C =2π  because r = 1. In other words such a cone pyramid was believed to be a sacred pyramid because it includes the mystic numbers Phi = Φ and Pi = π .  Also the great pyramid of Cheops was believed to be a sacred square pyramid, because a theoretical cone pyramid was inscribed in the square pyramid.

According to Wikipedia, the Great Pyramid  as a square pyramid has a base a square with a side length α = 230.4 meters  and an estimated original height h = 146.5 meters. If indeed exactly α = 230.4 meters then a perfect golden ratio would have a height of 146.5367. In this case when a circle of a circumference C = 2πr inscribed in this square the diameter  d = 2r is equal to the side length α = 230.4 m. Then the slant height L of the inscribed cone pyramid is found by using the Pythagorean theorem as

L2 = h2 + r2 . Then writing L= Φr,  and  h = Φ0.5r

one gets Φ2r2  = Φr2 + r2  or Φ2 = Φ +1 or  Φ2 - Φ -1 = 0

in which Φ = (1 + 50.5)/2  =  1.618034 or  Φ0.5 = 1.2720196

Since h = Φ0.5r and r = α/2 we get h = 1.2720196(230.4/2) = 146.537 m

In addition to this relationship of the pyramid’s geometry to Φ it’s also possible that the pyramid was constructed  by using the ratio C/d = C/α = π of the inscribed circle in the square pyramid. Before the construction of the pyramid they determined the side length (α) of the square pyramid on the ground and the inscribed circle of a circumference C = πd = πα. Thus measuring carefully the C and the α they were able to approximate the value of π close to π = 3.1415927.    It was John Taylor (1859) who first proposed the idea that the number π = 3.1415927 might have been intentionally incorporated into the design of the Great Pyramid of Khufu at Giza. He discovered that if one divides the half perimeter (P/2) of the Pyramid by its height one obtains a close approximation to π. Indeed using the dimensions of the great pyramid and the value of Φ0.5 I discovered that the ratio P/2 to h cannot give the true value of π. In this case dividing  P/2 = 2α = 460.8 m  by its height h = 146.537 m, one obtains

2α/h = 460.8/146.537 = 3.1445983 which is a close approximation to π = 3.1415927.

Then it is possible to formulate a formula relating the constant π = C/d = C/α  to Φ = (1+50.5)/2

Since  h = Φ0.5( α/2)  we may write

2α/h = 4/Φ0.5 = 3.1445983 > π = 3.1415927

Nevertheless this formula is very useful because it can give us the difference

4/Φ0.5 - C/α =  3.1445983 -3.1415927 =  0.003

Under this condition  in such a construction of the pyramid we formulate the following relationship

π = 4/Φ0.5 - 0.003

Here we see that the mystic numbers 3 and 4 are the numbers of another sacred cone pyramid in which r = 3, h = 4, and L = 5. It is surprising that this formula gives a value π = 3.1416 existing between the 223/71 and the 22/7. Note that much more later Archimedes (250 BC) using polygons upper and lower bounds of π and by calculating the perimeters of these polygons, he proved that 223/71< π < 22/7.

To conclude we see that a careful analysis of the dimensions of Cheops great pyramid led to my formulation of  π = 4/Φ0.5 - 0.003 used for the construction of the great pyramid by relating the mathematical constant π = C/d to the golden section Φ = (1 + 50.5)/2 = (α+β)/α = α/β.   Here the difference  4/Φ0.5 - C/d = 0.003 could be found under a detailed measurement of the circumference C of the inscribed circle  in the square of the square pyramid before the construction of it. Surprisingly, this formula includes the three mystic numbers π, Φ, and 3 used by Dinocrates for the construction of the mathematical cone pyramid in Amphipolis for the divine hero Hephaestion. It means that in Spring of 323 BC Alexander the Great received from the oracle of Amon such mystic numbers. In other words the mathematical tomb of hero Hephaestion (320 BC) includes mystic numbers  not only of Alexandria (331 BC) but also of the Great Pyramid of Cheops (2560 BC). We conclude also that this formula should be known to Dinocrates who determined the perimeter of the surrounding wall

C = π = 3.1416= 157.5 = 494.8 m.

From such a determination he calculated the diameter of the tomb d = 1 stade = 157.5 m . (See my “PLAN OFAMPHIPOLIS TOMB”).