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Historical backgrounds Egypt - The Great Pyramid
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There is no doubt that Pythagoras must have obtained some of his knowledge from the Egyptians. About two thousand years before Pythagoras lived (just as long ago as the Romans lived before us), the Egyptians already build pyramids with a geometrical accuracy that would even be difficult to copy in our time. To give an example; the pyramids were build on the North-South axes with such an accuracy that one could still calibrate a compass on it. This fact says much about their astronomical knowledge as well. Of all the great pyramids of Gizeh, the great pyramid, also called Cheops or Khufu possesses the most remarkable geometrical qualities. One of these regards the surface area of the vertical cut of the pyramid, which is equal to a quarter of the surface area of a surrounding circle.
A similar architectural achievement no less then 2650 years BC should be considered as very impressive. The measures that have been used for the calculations are the ones established by the famous Egyptologist Sir Flinders Petrie around 1880-81. Flinders Petrie accurate measurements are still used because of their reliability, although it must be said that slightly different measures like 147,8 meters are also used today. According to Flinders Petrie, the the pyramid is 148,2 meters high and at the basis 232,8 meter wide. This makes for a surface area of a vertically cut pyramid of 148,2 x 116,4 = 17250,48 square meters. The surface area of the surrounding circle with radius 148,2 meter is (πR2) 68999,55.. divided by 4 is: 17249,89. Compared with the surface area of the vertical cut, this creates a difference of only 0,59 square meters, equal to a deviation of only 0,0034% between both surface areas. Please remind that the Egyptians were not supposed to know the number pi (π ≈ 3,141.592.653..) and even if they would have known, it would have been almost impossible even with modern construction techniques to achieve such a high level of building accuracy, especially considering that the basis of the pyramid was fixed, just as the angle of the pyramid had to be fixed from the very first layers onwards, which makes it virtually impossible to modify the heights of a pyramid once the building has started. To obtain a total match between the surface areas of the vertically cut pyramid and its surrounding circle, the pyramid needed to have been 148,20254.. meters high. It should be made clear that a difference of only 2,5 millimeters does not even enter into the accuracy of Flinders Petrie's measurements which must have been no more precise then several centimeters instead of millimeters. In order to establish the standard deviation in the pyramid's measurements it is possible to compare compass orientations of the the fixed lengths of the basis.
North
side: 0o 2'
28''
south of
West
East
side: 0o 5'
30''
west of North As such the maximum deviation between two base lines of the pyramid is 0o 4' 25'', or ≈0,079o. Compared to the total length of the sides of the pyramid of 232,8 meters this results in a maximum drift of 0,322 meters, or ≈ 0,1385%. Compared to a tolerance of several centimeters, a difference of two millimeters should be considered coincidental (or deliberate coincidence) rather a proof of flawless building accuracy. Another geometrical quality of the great pyramid has to do with the existence of a "Golden Section" proportion in its side surfaces. The height of one of the pyramid's projected side surfaces (188,45mtr.) has a proportional relationship with ½ the width of the basis (116,4mtr.) of 1:1,618.958.844.. In the previous chapter it has already been explained that the golden section is a geometrical proportion that relates to 1 as (√5+1)/2 being approximately: 1,618.033.989, resulting in a difference of 0,057%. The illustration below explains this geometrical quality of the great pyramid.
Calculating the deviation back to the basic dimensions of the pyramid, it should have been 14 cm lower in order to exactly fit the proportions of the Golden Section on its projected side surface. Such a small difference of only 0,09% remains well within the inherent inaccuracy of the building of 0,14% The question is which conclusions may be drawn from these remarkable geometrical peculiarities. Regarding the incredible accuracy between the surface areas of the pyramid and its surrounding circle it should be made clear that this accuracy is most likely due to favorable rounding-off of the pyramids measures by Flinders Petrie, who was most definitely aware of this remarkable coincidence. Should the Egyptians actually have build the pyramid with such a precision, this would have meant that in 2500 BC, they already knew the number pi with an accuracy of four digits behind the comma. Of coarse this can not be excluded, but at the same time it can also not be proven considering the inherent inaccuracy of the construction of 0,14% on the ground surface (which was the "easiest" part). A part of Flinders Petrie's measurements of the pyramid were based upon the expected thickness of the no longer existing surface layer. As such both theories ("surface area of surrounding circle" and "Golden Section") deserve equal attention. |
A closer examination of the
pyramids internal construction of tunnels, galleries and chambers reveals
some interesting information that suggests at least partial knowledge of
the Golden Section proportions. A first hint is provided by the fact that the system of tunnels inside the pyramid have a declining angle of 26o31' 23''. This is the angle that belongs to a Pythagoras triangle with straight angle lengths 1 and 2. The length of the oblique side of such a triangle is √5; the same √5 that also forms the basis of the Golden Section proportion. A second fact that suggest acquaintance with the Golden Section proportions can be found in the long descending tunnel that leads to the underground chamber below the pyramid. This tunnel has a total length of 105,5 meter. The point where the tunnel passes below the external surface of the pyramid is about 40,3 meter distant from the entrance and marked within the tunnel by a fissure. At this exact point the length of the tunnel can be divided in what was described by Euclid around 250 BC. as “a line divided in extreme and mean ratio”, better know as the proportion of the Golden Section.
Proof of systematic use of √5 based proportions inside the pyramid makes it highly likely that the same proportions were used in the external measures as well. There is one example of the use of a √5
based proportion within the great pyramid that almost looks like a
geometrical statement. The most upper room within the pyramid is the so
called Kings Chamber. The walls of this room are entirely constructed of black polished
marble and is build on a ground surface of 1:2. The length of the room is
about 10,48 meters, and the width 5,24 meters. By itself this is already
an interesting proportion because it is in conformance with the angle of
the main tunnels of the pyramid. The room does however have one other very
charming measure that really sets it apart, which is the height of the room
of 5,85 meters. This length relates to the width of the room as 2:√5
(with a precision of 99,73%). The
illustration below gives a good impression of the measurements of the
Kings Chamber. In this highly remarkable example of early Egyptian architecture, the length √5 has been used as one of the sides of the rectangle (blue), instead of one of the diagonals (red). This might very well be explain as an example of the deliberate use of √5 as a geometrical proportion within the architecture of the pyramid. The way in which √5 has been used to establish the height of the Kings Chamber seems almost a tribute to the phenomenon of irrational numbers. The angle of the tunnels within the pyramid could perhaps be explained by a decision to let the tunnels rise or decline at a 2:1 relation. Without vertical and horizontal reference points this does not automatically imply the deliberate use of √5 as a reference measure. This is explained in the illustrations below. By using however the length √5 in one of the straight angle sides of a Pythagoras triangle, the architect suggests that √5 had a particular meaning within the pyramid. Considering the important relation between √5 and the Golden Section, it is not impossible that the Golden Section was used in the projection of the side surface area.
There are countless theories regarding the construction techniques of Great Pyramid of Cheops and almost just as many mathematical theories about the mysteries that are hidden in its architecture. Some of these theories are highly likely and others not, but unless we dig up a 4.500 years old scroll titled "construction secrets of the great pyramid" we will probably never find out with certainty. In the context of this book it is sufficient to illustrate that geometry and mathematics were no merits of the Greek alone. Geometry and mathematics were highly developed already thousands of years before Christ. Greek philosophers from Pythagoras onwards most often merely described on paper what was already known for centuries in other cultures. This way many mathematical rules that got their names from Greek philosophers appear to have been unknown before the Hellenistic Heroes. The Egyptians, Phoenicians and Babylonians prove however that this is not true. Many Greek mathematicians from later centuries claimed to be followers of the great Pythagoras and directly attributed their findings to the legendary knowledge of their teacher. As such it is reasonable to say that Pythagoras was the greatest mathematician of antiquity thanks to the fact that he got credits for about every invention before and after his time. |
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