Winged disk image of David Furlong  Seked
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Keys to Temple Introduction
Keys to Temple Pt 1 - The British Pyramid
Keys to Temple Pt 2 - The British Pyramid
Sekeds and      Pyramid Geometry
666 - A magic number?
Whatever happened in 3100 bc?
Avebury's          Sacred Geometry
Who were the Elohim?
The Cotswold Circle
Marlborough Downs Long Barrow Mystery   (coming soon)
Main Selection
Sekeds and the Geometry of the
Egyptian Pyramids
article by David Furlong

A comparison between the angles generated by sekeds
and the angles of gradient of the pyramids

Part 4

 picture of The Great Pyramid of Egypt

Pyramid design

It can be seen from the seked's used that two ratios were adopted on more than one occasion. These are the sekeds 5.5 [5 palms and 2 digits] and 5.25 [5 palms and 1 digit]. These two sekeds can be shown to be based on simple ratios; the first being a height to base of 7:11 and the second a height to base of 2:3 (4:6). This latter ratio is the derived from the 3 : 4 : 5 triangle.

We might question why the Ancient Egyptians would have wished to choose these sekeds when, for practical construction purposes, more simple seked ratios of 5 or 6 palms would have been more preferable. The first produces an angle of slope of 54.46° and the second of 49.4°. There is no technical constructional reasons why either of these angles could not have been adopted yet the only surviving pyramid that uses either of these ratios, is the lower portion of the Bent pyramid. We are therefore forced to ask what was so special about the sekeds of 5.5 and 5.25?

Seked 5.25 (ratio 3:4:5 triangle)

 Pyramid of king Kaffre (Chephren)It has been claimed by a number of authorities that the Ancient Egyptians did not know the 3:4:5 Pythagorean ratio. For example T. L. Heath states in his book The Thirteen Books of Euclid's Elements, Vol 1:

"There seems to be no evidence that they (Egyptians) knew that triangle (3:4:5) is right-angled; indeed according to the latest authority (T. Eric Peet, The Rhind Mathematical Papyrus, 1923), nothing in Egyptian mathematics suggests that the Egyptians were acquainted with this or any special cases of the Pythagorean theorem."

This proposition is supported by Richard Gillings based on the known textual information. Set against this view there is considerable constructional and geometrical evidence to indicate that the Egyptians were well aware of the 3 : 4 : 5 ratio.

Firstly this triangle was used, or the ratios derived from it, in at least three pyramids including that of Khafre on the Giza plateau. Khafre's pyramid was measured by Petrie who gave the angle of slope as 53°-10' +/- 4'. We can therefore be fairly certain that the seked of 5.25, which produces an angle of 53°-7'-48", was intended in the construction.

It is clear from the cubit measures that have been recovered, that the Ancient Egyptians were quite capable of measuring to an accuracy of 1/16 of a digit. It is just not credible that they would never, whilst building or setting out their triangles, have bothered to measure the length of the hypotenuse of slope of the triangles based on their sekeds. Once a seked of 5.25 had been adopted, sooner or later someone would have measured the hypotenuse and discovered its relationship to the other two sides.

From other problems in the RMP it is clear that the Egyptians were quite capable of division. They would certainly have discovered that a triangle with sides of 6:8:10 (RMP 59) could be reduced to a 3:4:5 ratio. Indeed it could be argued that a whole number ratio of cubits, for the slope, base and height of the pyramid, was precisely why they chose to adopt a seked of 5.25. For this would considerably ease the technical problems of ensuring that the correct angle of slope was always maintained, in the final finishing of the casing stones. In support of this argument problems 57, 58 and 59 in the RMP are based on a seked of 5.25, which demonstrates its importance in pyramid design.

In addition there are two further items of corroborative evidence. The first stems from Pythagoras's exposure to Egyptian ideas during the ten years of his life that he spent in Egypt as part of the priesthood. Whilst it is very likely that he was the first individual to 'prove' the relationship between the sides of a right angled triangle, based on the square of its sides, we could also infer that he obtained a knowledge of the 3 : 4 : 5 triangle from his time in Egypt.

Secondly , whilst the problems in the RMP relate solely to the base and perpendicular sides of a right angled triangle we know that the hypotenuse was important in other calculations in relationship to areas. The unit of measure of a Remen, or more correctly double Remen is the diagonal of a square whose side is one cubit7. From this it is clear that the measure of the hypotenuse was an aspect of Ancient Egyptian mathematics and geometry which had practical application in the surveying of land.

Omitting all other evidence, that the seked of 5.25 occurs with such regularity both in the mathematic texts as well as in the practical construction of at least three pyramids lends substantial weight to the evidence that the 3 : 4 : 5 ratio was known to the Ancient Egyptians. Whether out of curiosity of intent the hypotenuse of a pyramid or triangle with this seked would have been accurately measured. This in turn would inevitably lead to a working knowledge of the 3 : 4 : 5 ratio.

Seked 5.5 (ratio 7:11 - height to base) - The Great Pyramid Ratio

We now need to question why a seked of 5.5 might have been used? This is not so obvious but an explanation could lie with a further understanding of Egyptian measures, particularly the relationship of the Royal cubit to the palm and the digit.

In modern times we are used to working with the metric system with its standard ten base ratios. Prior to this, in Britain, Imperial measures were used which incorporated a range of different ratios, 12 inches in one foot, 3 feet in one yard and so on. All ancient measures incorporate practical relationships to assist in the computation of lengths, areas and volumes. The relationship of the primary Ancient Egyptian measures has already been given. As a distance the cubit is generally reckoned to be 20.6 inches or 523 millimetres. For comparison the Sumerian cubit of 495 millimetres was divided into 30 digits as opposed to 28 digits in Egyptian measures. The Greeks also used a cubit of about 489 millimetres being divided into 24 digits. Both the Greeks and the Egyptians used 4 digits to equal a palm; giving 6 palms to one cubit in the case of Greek measures and 7 palms to one cubit in Egyptian measures.

In all ancient measures the division of the cubit into seven parts is, to say the least, very curious. As a measure it has no divisors being a prime number. It was probably for this reason that a short cubit was introduced of 6 palms, which could then be divided into halves and thirds; but not so the Royal cubit. It might be argued that the sevenfold division held some magical or numerological significance, such as a relationship to the 70 days between the rising and setting of the Dog Star, and that such a relationship has now been lost. There is however one practical reason why the Ancient Egyptians might have chosen to divide their cubit into seven parts.

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