June 25, 2012

The Unsolved Problems about Ancient Egyptian Pyramids P5

One of the problems that has exercised the minds of many people is the significance of the angle of elevation of the pyramids. Most of them rise at an angle of about 520; only the Red Pyramid and the upper part of the Bent Pyramid, both at Dahshur, are built at the same lower angle of 43 J°. We have mentioned that for the Khufu pyramid the angle (51 “52') leads to the ratio 1 /2TT with an accuracy that cannot comfortably be dismissed as fortuitous and has given rise to strange esoteric speculations which are mentioned in the Appendix to this book. Since this geometric relation was first noticed, more than a century ago, a number of very careful triangulations of the Giza plateau have been carried out.

From each of these measurements the ratio 1/27r emerged with increased accuracy. Since it is known that more than a thousand years after Khufu the Egyptians still did not know the ratio of circumference to diameter of a circle to a greater accuracy than 3, the accurate use of the factor i/27t in pyramid construction remained somewhat uncanny. A great number of mathematical explanations have been suggested and even one, made by a noted archaeologist, that the builders by accident used a ratio of 14/11, remains lamentably unconvincing. In spite of brilliant constructional skill and superb workmanship, we have no evidence that the Egyptians of the Old Kingdom had more than the most rudimentary command of mathematics. Any acceptable solution must therefore have a practical, rather than a theoretical, basis, and that suggested to me by an electronics engineer, T. E. Connolly, fulfils this condition.

The explanation is based on the assumption that the ancient Egyptians had not yet formed the concept of isotropic three- dimensional space. In other words, whereas to us measures of height and vertical distance are the same thing, namely a length for which we use the same unit, this may not have been regarded as natural by the pyramid builders. They used as height measure the royal cubit, based on the upward distance from elbow to 5 The astonishing accuracy with which the ratio of height to circumference of the Great Pyramid represents the squaring of the circle 1)271 was possibly due to the fact that the Egyptians may have measured long horizontal distances by counting the revolutions of a rolling drum. In this way they would have arrived at the transcendental number n = 3.141... without realising it fingertips, which in the Old Kingdom had already been standardised to a length of 52 centimetres. Since ropes of palm fibre tend to stretch, a much more accurate way of measuring long horizontal distances as, for instance, the base of a large pyramid, was required. One such method is to roll a drum and count the number of revolutions. The royal cubit, already used for height measurement, would immediately suggest itself as the standard diameter of the drum, and one revolution - we may call it a ‘rolled cubit’ - corresponds to the circumference of the drum stretched out horizontally.

It appears that, using this system of measurement, the Egyptian architects never did anything more sophisticated than to build pyramids according to the simple gradients of 4 :1 and 3:1. Taking the former first, the height of the pyramid will be 4x71 cubits, where n is the number chosen to determine its size. The horizontal distance from the centre of the building to its side will then have to be ixn rolled cubits or, in our mathematical terms, n-ir cubits. Since this distance is half the side length of the pyramid, the latter’s circumference comes to 8xmr cubits. Therefore, the ratio of height to circumference is 4 x n/8 x mr cubits or, by dividing this fraction by 4 xn cubits, simply 1 / 2ir. This is the mysterious ratio used in pyramid construction and which, at the same time, leads inevitably to an angle of elevation of 51 °52'. In other words, jjje pyramid builders discovered the transcendental number tr (3 141 • • •) without trying and without knowing. The explanation is made even more likely by the fact that using in this way the lower gradient 3 :1 instead of 4 :1, the angle of elevation comes out correctly as 434°.

Ancient Egyptian Pyramid :
The Unsolved Problems about Ancient Egyptian Pyramids P1
The Unsolved Problems about Ancient Egyptian Pyramids P2
The Unsolved Problems about Ancient Egyptian Pyramids P3
The Unsolved Problems about Ancient Egyptian Pyramids P4
The Unsolved Problems about Ancient Egyptian Pyramids P5
The Unsolved Problems about Ancient Egyptian Pyramids P6


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