Mysteries of the Egyptian Pyramids

The infinite, uniform sea of sand, infrequent dried bushes of plants, hardly noticeable tracks from an elapsed camel are swept with a wind. The incandescent sun of wasteland ... And it seems by dull, as if is covered with a fine sand.

And suddenly, as if a mirage, before the amazed look there arise pyramids (Fig.1), fancy rock figures directed toward the Sun. By their vast sizes, perfection of the geometric form they strike our imagination. According to many descriptions, these gigantic monoliths had earlier other view than presently. They shined on the Sun by a white glaze of the polished calcareous tables on the background of many-pillar adjacent temples. Near to pharaoh's pyramids there were the pyramids of the wives and members of pharaoh's family.

Complex of pyramids in Gisa
Figure 1. Complex of pyramids in Gisa.

Pharaoh's authority in the Ancient Egypt was huge, the divine's honors were given to him, the pharaoh is called the "Great God". The God-Pharaoh was the Promoter of country, the Judge of people fates. The cult of the died pharaoh gained a huge importance in the Egyptian religion. The gigantic pyramids are constructed for preservation of the pharaoh body and his spirit and for extolling his authority. And not without reason these works of human hands fall into one of seven miracles of the World.

The assigning of pyramids was multifunction. They served not only burial vault of pharaohs, but also were attributes of majesty, power and riches of country, monuments of culture, storehouses of the country history and items of information on life of the pharaoh and people.

It is clear, that the pyramids had steep "scientific contents " embodied in their forms, sizes and orientation on terrain. Each part of a pyramid, each element of the form was selected carefully and should demonstrate a high level of knowledge of the creators of pyramids. They were constructed on millennia, "for all time". And not without reason the Arabian proverb claims: "All in World frights of time. The time frights of pyramids".

Among the gigantic Egyptian pyramids the Great Pyramid of the pharaoh Cheops is of special interest. Before to begin the analysis of the form and sizes of Cheops pyramid it is necessary to remind the Egyptian measure system. The Ancient Egyptians used three measure units: "elbow" (466 mm) equaling to 7 "palms" (66,5 mm), which, in turn, was equal to 4 "fingers" (16,6 mm).

Let's make the analysis of the sizes of Cheops pyramid (Fig.2), following to reason given in the remarkable book of the Ukrainian scientist Nickolai Vasutinski "The Golden Proportion" (1990).

Geometric model of Cheops pyramid
Figure 2. Geometric model of Cheops pyramid.

The majority of the researchers believe that the length of the side of the pyramid basis for example, GF is equal to L = 233,16 m. This value corresponds almost precisely to 500 "elbows". The full fit to 500 "elbows" will be, if the length of "elbow" to consider as equal to 0,4663 m.

The altitude of the pyramid (H) is estimated by the researchers variously from 146,6 m up to 148,2 m. And depending on an adopted pyramid altitude all ratios of its geometric elements change considerably. In what is the cause of distinctions in an estimation of the pyramid altitude? Strictly speaking, Cheops pyramid is truncated. Its topic platform today has the size approximately 10 ´ 10 m, but one century back it was equal 6 ´ 6 m. Apparently, that the top of the pyramid was dismantled and it does not fit to the initial pyramid.

Estimating the pyramid altitude, it is necessary to take into consideration such physical factor, as "shrinkage" of construction. For a long time under effect of enormous pressure (reaching 500 tons on 1 2 of a undersurface) the pyramid altitude decreased in comparison to its initial altitude.

What was the initial altitude of the pyramid? This altitude can be reconstructed if to find the main "geometrical idea" of the pyramid.

In 1837 the English colonel G. Vaise measured the inclination angle of the pyramid faces: it appeared equal a = 51°51'. The majority of the researchers recognizes this value and today. The indicated value of the inclination angle corresponds to the tangent equal to 1,27306. This value corresponds to the ratio of the pyramid altitude to the half of its basis CB (Fig.2), that is, AC / CB = H / (L / 2) = 2H / L.

And here the researchers expected a large surprise! If we take the square root of the "golden" proportion we get the following outcome = 1,272. Comparing this value with value tg a = 1,27306, we can see that these values are very close among themselves. If to take the angle a = 51°50', that is, to decrease it by one arc minute the value of tga will become equal to 1,272, that is, will be equal to the value of . It is necessary to note, that in 1840. G. Vaise repeated his measurements and corrected the value of the angle a =51°50'.

These measurements resulted to the following rather interesting hypothesis: the ratio AC / CB = = 1,272 was put in the basis of the triangle of the Cheops pyramid!

If to designate now the lengths of the triangle ABC sides through x, y, z, and also to take into consideration that the ratio y / x = , then according to the Pythagorean theorem the length z can be computed as the following:

(1)

If to take x = 1, y = , then

'Golden' right triangle
Figure 3. "Golden" right triangle.

The right triangle, in which the sides are in the ratio t :: 1, is called the "golden" right triangle.

Then if to accept for the basis the hypothesis, that the "golden" right triangle is the main "geometrical idea" of Cheops pyramid, from here it is possible to compute the "design" altitude of the Cheops pyramid. It is equal:

H = (L/2) ´ = 148,28 m.

Let's deduce now some other relations for the Chops pyramid resulting from the "golden" hypothesis. In particular let's find the ratio of the external area of the pyramid to the area of its basis. For this purpose we take the length of the leg CB for the unit, that is: CB = 1. But then the length of the side of the pyramid basis GF = 2 and the area of the basis EFGH will be equal to SEFGH = 4.

Let's calculate now the area of the lateral face of Cheops pyramid. As the altitude of AB of the triangle AEF is equal to t, the area of each lateral face will be equal to SD = t. Then the common area of all four lateral faces of the pyramid will be equal to 4t, and the ratio of the common external area of the pyramid to the area of its basis will be equal to the "golden" proportion! This also is the main geometrical secret of Cheops pyramid!

The analysis of other Egyptian pyramids demonstrates, that the Egyptians were always aimed to embody in the pyramids some relevant mathematical knowledge. In this respect the Chefre pyramid is rather interesting. The measurements of the pyramid shown that the inclination angle of the lateral faces is equal to 53°12', that corresponds to the leg ratio of the right triangle: 4:3. Such leg ratio corresponds to the well-known right triangle with the side ratio: 3:4:5; this one is called "perfect", "sacred" or "Egyptian" triangle. According to historians testimony the "Egyptian" triangle had a magic sense. Plutarch wrote, that the Egyptians compared the nature of the Universe to the "sacred" triangle; they symbolically assimilated the vertical leg to the husband, the basis to the wife, and the hypotenuse to the child who is born from both.

According to the Pythagorean theorem for the triangle 3:4:5 we have: 32 + 42 = 52, Possibly just this theorem the Egyptian priests wanted to perpetuate, carrying up the pyramid based on the triangle 3:4:5? It is difficult to find a more successful example for demonstration of the Pythagorean theorem, which was well known for the Egyptians long before its discovering by Pythagor.

Thus, the ingenious designers of the Egyptian pyramids were aimed to strike of their far offsets by depth of their mathematical knowledge, and they have reached this by selecting of the "golden" right triangle as the main "geometrical idea" of Cheops pyramid and of the "sacred" or "Egyptian" triangle as the main "geometrical idea" of Chefre pyramid.