mirrored file at http://SaturnianCosmology.Org/ For complete access to all the files of this collection see http://SaturnianCosmology.org/search.php ========================================================== Picture Seked [1]Home [2]Phi [3]Introduction [4]Purpose [5]Sitemap _The angle of the slope of the pyramid is accidental, and determined by its Seked_. [6][SoHoDataListIcon.gif] _Geometric Derivation of Phi_ A method of construction Phi by geometry [7][SoHoDataListIcon.gif] _Kings Chamber_ Presence of Phi within the Great Pyramid [8][SoHoDataListIcon.gif] _Great Pyramid_ Mathematical theories concerning the Great Pyramid [9][SoHoDataListIcon.gif] _Herodotus_ Phi presence and Herodotus [10][SoHoDataListIcon.gif] _Seked_ Use of Seked ratio to construct a pyramid [11][SoHoDataListIcon.gif] _Pi_ Presence or otherwise of Pi Picture The Seked is an ancient Egyptian term used to describe the run to rise ratio, or in modern terms the cotangent of angle _a_. The ancient Egyptians used the cubit as their principal measure, which certainly by the Middle Kingdom was divided up into seven palms of four digits, making 28 digits to the cubit. There is some question as to the derivation of the digit and the cubit, with strong evidence that the digit was in earlier times diagonally related to the cubit. For now however we will accept a straight linear relationship, and 28 digits to the cubit. So what was the purpose of a Seked? The answer suggested is straightforward. To build a pyramid you need to maintain the angle of incline. To do this you would use a Run to Rise ratio. Problem 56 of the Rhind Papyrus demonstrates the computation of a pyramid Seked. The pyramid is 250 cubits high, 360 cubits long. The answer is given as 5 and 1/25 palms. How this is computed isn't relevant, but to maintain the slope of a pyramid of height 250 cubits by base 360, the builder would simply move in 5 and 1/25th Palms then move up one cubit, then repeat the process. This is all very well in principle, but in reality has one glaringly obvious difficulty. To build a pyramid as in the previous example, it would certainly be convenient to measure 5 and 1/25_ palms_ in, then up a cubit, etc, to maintain the slope of the pyramid, but exactly how do you measure 5 and 1/25_ palms_? The Egyptians used cubits as their main measure, which were divided up into 7 palms of four digits, or 28 digits to a cubit. 5 and 1/25_ palms_ equals 20.16 digits. To measure 0.16 of a digit requires either that the artisans guess the length, or presupposes that the digit was subdivided into units of even smaller measure. A number of wooden cubit sticks have been found, with the earliest dating to around the Middle Kingdom. Some, but by no means all, have sub divisions into unit fractions. One that does is a measuring stick held in the Turin museum. Along one edge the stick is divided up into 28 digits, with each division being represented by a deity. For instance the first digit has the symbol for _Ra_, followed by, and in sequential order, the symbols for _Shou, Tefnout, Geb, Nout, Osiris_, etc, with _Osiris_ denoting the number _6_, and _Isis _and _Set_ denoting the numbers _7_ and _8_ respectively. The first fifteen of the digits have additional symbols referring to various fractions of a digit, up to and including the fractional number 16. Using one of these sticks it would therefore have been possible to measure fractions of a digit up to sixteenths, but not twenty fifths. One cubit is approximately equal to 20.6 inches, making the digit equivalent to 0.736 inches, or 18.68 millimetres. Roughly three-quarters of an inch. If the artisans rounded down to 20 digits, the slope of the pyramid would be wrong, and would give the wrong build height. It is possible to adopt the view that the example is merely a student's exercise, but one need only look at existing pyramids to see that the majority deliver a_ seked_ ratio that does not divide down into a whole number of digits. One that does give you a whole number is the _Great Pyramid_, or _Khufu_'s pyramid. This pyramid closely approximates to 280 cubits in height, and has base sides of 440 cubits. If you apply the same computation as in the Rhind Papyrus, the following applies: - Find ½ of 440, which is 220 and divide 220 by 280 giving 22/28. This breaks down to ½, + ¼, + 1/28. Multiply each of these numbers by 7 and you obtain, [(3 + ½), +(1, +1/2, +1/4), + (1/4)], or 5 + ½ palms. This equals 22 digits exactly. If the `_building_' hypothesis were correct, it would certainly be possible to construct the Great Pyramid by having a run of 22 digits and a rise of 28 digits. Hey Presto, you would end up with the right height and the right outside slope. However this in itself leads to an obvious problem that anyone who has ever stood on, or by the granite blocks that make up the Khufu pyramid will instantly recognise. Although by definition the pyramid has a_ seked_ ratio of 5 + ½, the individual blocks are much bigger than 22 digits in and 28 digits up. Therefore it is clear that if a seked ratio was used to build the pyramids, there would need to be a smaller unit of measure than the digit. Otherwise no granite blocks bigger than 1 cubit in height could have been used without loss of accuracy, and if nothing else is certain, nobody disputes that the Great Pyramid was built to incredible tolerances, making it one of the most accurately executed designs ever built. It can be seen then, that the ancient Egyptians had a method for computing the height to base ratio for a pyramid of given_ seked_. The_ seked_ found in Rhind papyrus problem 56, of 18/25, corresponds to a pyramid angle, or slope, of 54° 15'. Compare this slope to the main pyramids of the Giza plateau. That of Khufu, 51° 52', that of Khafre, 52°20', and that of Menkaure, 50° 47'. Although closely related it is clear that no uniform angle was chosen when building a pyramid. Far less obvious, with the exception of the Great Pyramid of Khufu, is that none of these angles gives a_ seked_ that corresponds to an exact number of digits. It is therefore unlikely that the pyramid slope was the governing factor when deciding on the design dimensions of a pyramid. The consensus of opinion is that pyramid angles were chosen as a compromise between aesthetic principles and the practicalities of construction. Whether or not the Seked was used to maintain the slope of a pyramid during construction, it is clear that it cannot have been the deciding factor, or we should expect all pyramid Seked's to compute to an exact number of digits. [12]The presence or otherwise of Pi? [[13]Home] [[14]Phi] [[15]Introduction] [[16]Purpose] [[17]Sitemap] Please contact the [18]Author with any questions or feedback Copyright 2000 Keith Squires. All rights reserved. References 1. http://www.gizagrid.fsnet.co.uk/index.html 2. http://www.gizagrid.fsnet.co.uk/html/phi.htm 3. http://www.gizagrid.fsnet.co.uk/html/introduction.htm 4. http://www.gizagrid.fsnet.co.uk/html/purpose.htm 5. http://www.gizagrid.fsnet.co.uk/html/sitemap.html 6. http://www.gizagrid.fsnet.co.uk/html/derivation.htm 7. http://www.gizagrid.fsnet.co.uk/html/kings_chamber.htm 8. http://www.gizagrid.fsnet.co.uk/html/great_pyramid_0.htm 9. http://www.gizagrid.fsnet.co.uk/html/herodotus.htm 10. http://www.gizagrid.fsnet.co.uk/html/seked.htm 11. http://www.gizagrid.fsnet.co.uk/html/pi.htm 12. http://www.gizagrid.fsnet.co.uk/html/pi.htm 13. http://www.gizagrid.fsnet.co.uk/index.html 14. http://www.gizagrid.fsnet.co.uk/html/phi.htm 15. http://www.gizagrid.fsnet.co.uk/html/introduction.htm 16. http://www.gizagrid.fsnet.co.uk/html/purpose.htm 17. http://www.gizagrid.fsnet.co.uk/html/sitemap.html 18. mailto:feedback at gizagrid.fsnet.co.uk