The Egyptian equivalent of our pi was 3.1605, NOT 3.1416. The Egyptians simply did not know our value of pi, so they could hardly have featured it intentionally in the Great Pyramid.
(see K. Mendelssohn, “The Riddle of the Pyramids” p. 73.)
Connolly's rolling drum idea was an ingenious explanation of why our pi should appear in the G.P. and not the Egyptian pi. It also gives a good explanation of the change in angle of slope of the Bent Pyramid. But on the other hand, it doesn’t fit other pyramids (e.g. Chephren) very well.
The conservatism of the Egyptians and the practical nature of their mathematics suggest that the techniques of pyramid design and construction were standard. It would seem desirable that any theory of pyramid design should apply also to other pyramids.
Again, although the phi theory holds well for the G.P. it doesn’t appear to hold for other pyramids.
Also Egyptian maths was very cumbersome, so that the Egyptians would not have been able to calculate phi at all accurately. If they had known the theorem of Pythagoras, and had they known even the rudiments of theoretical geometry (as opposed to practical) then they might have been able to accurately construct phi. But (R.J. Gillings “Mathematics in the Time of the Pharaohs” Appendix 5) there is no evidence that the Egyptians knew the Pythagoras theorem. In fact it appears that Egyptian Geometry never actually reached the heights later attributed to it by the Greeks.
So like the pi theory before it, the phi theory is inconsistent with what is known of Egyptian mathematics, and it would appear likely that both pi and phi are unintended side-effects superimposed on an innocent pyramid by essentially modern imaginations.
(Note: the pi theory implies the phi theory as a side-effect, and vice versa, so we are not claiming here a double accident – merely an interesting single one. This follows from the known coincidence of numbers that 4/π ≈ √φ.)
It is frequently stated that the area of a square on the height of the G.P. equals the area of a G.P. face. This is known as the Equation of Herodotus, because it is commonly believed that Herodotus said this (Tompkins “Secret of the G.P.” p. 190.). In fact, Herodotus didn’t say this – it was John Taylor who ‘interpreted’ what he said in this way, and it is this interpretation which has been repeatedly quoted as the Equation of Herodotus. The commonest translation of Herodotus II:124 reads: “It is a square eight hundred feet each way, and the height is the same …”.
How and Wells in their “Commentary on Herodotus” (1912), vol. 1, p. 228, claim that height in this passage refers to slant height. This agrees with a conclusion reached independently by myself: Herodotus believed that the G.P. faces were equilateral triangles.
The Equation of Taylor (as opposed to Herodotus) would, intentionally or otherwise, have given rise to the pi and phi properties attributed to the G.P. But as I say, this is the Equation of Taylor, and not, as is so commonly claimed, the Equation of Herodotus.
Gillings (above) reports that there is no evidence at all that the Egyptians were familiar with the Pythagorean Triangle – not even the simplest 3:4:5.
There is a reference in Plutarch’s “Of Isis & Osiris”, but this is open to a charge of Greek ‘intrusion’ into Egpytian maths.
Two commonly alleged uses of the 3:4:5 are: a) in the slope of Chephren’s Pyramid at Giza (this was believed by Petrie: Pyramids & Temples of Gizeh. 1883) and b) in the King’s Chamber of the G.P. (Tompkins p. 101; Ivimy p. 125–6.)
b), in accordance with the principle of “in our attempts to decide how the Egyptians decided upon the design & proportion of the G.P., we should take care not to go beyond the known limits of Egyptian mathematics”, I would tentatively reject, in view of Gillings. In fig 1 I give a very much simpler explanation of the Kings Chamber design based solely on the use of the double-square.
Fig. 1. Proposed Construction of Kings Chamber.
The floor was laid out in the form of a double square. The height of the chamber was then taken to be half the floor diagonal, thus making the diagonal cross-section of the chamber a double square also.
In the diagram, ABCD is a double square.
If the Egyptians had intended to make ACGE also a double square, they would also, without knowing it, have made triangle ABG a 3:4:5 triangle.
Part a) I will deal with below.
The definition of the double remen (= √2 cubits) seems to suggest Pythagoras (cf. extension of this idea in "City of Revelation" p. 106), but here again this is a modern interpretation of a practical result. The double remen appears to have come into use for “doubling up” areas of land. A field of side x by y double remens has exactly double the area of a field x by y cubits. “Doubling up” was very much a feature of cumbersome Egyptian arithmetic, and the derivation of a double remen by a practical exercise involving areas, and with no concept of Pythagoras at all, is featured in fig. 2.
Fig. 2. The area of the 1 double-remen square is double the area of a 1 cubit square.
(In this section RMP56 means problem study 56 in the Rhind Mathematical Papyrus. See Gillings for details.)
In accordance with the criteria of simplicity & consistency with known Egyptian Mathematics, the following present themselves:
Fig. 3. Definition of the seked
RMP 56–60 inclusive concern pyramids and their sekeds. In particular RMP 57–59 incl. concern pyramids with sekeds 5 palms 1 finger, or, as we would measure it, a slope of arc cot 5¼/7 = arc cot 3/4, which is of course the slope of Chephren’s Pyramid but without recourse to the 3:4:5 triangle, though of course giving the same ‘end product’ as the use of that triangle.
The Great Pyramid, on the other hand, is accurately described by a seked of 5 palms 2 fingers, or, as we would say, a slope of arc cot 5½/7 = arc cot 11/14, and from this it easy to see how the pi illusion (and hence the phi delusion) came to arise.
And Mycerinus? Petrie gives 51° 0′ ± 10′ as its slope. A seked of 5 palms 3 fingers would imply an angle of slope of 50° 36′.
And the bent pyramid? The lower portion conforms to a seked of 5 palms, and the upper to a seked of 7 palms 2 fingers (though the latter is 0.3 degrees out, so not satisfactory; the actual slope falls between a seked of 7½ palms & 7¼ palms.)
In conclusion, therefore, using only a practical approach involving sekeds of palms and fingers, and using no mysticism whatsover, the Giza Group as a whole, plus at least half of the Bent Pyramid (!?!) follow the same angle-of-slope design principle.
To finish with a passing fancy: the Great Pyramid, with its seked of 5 palms 2 fingers: in view of the volume of bizarre literature centred about the Great Pyramid, those 2 fingers (like the indication of Merton Sewage Works – FT 18, p. 15), carry a wickedly symbolic messsge for posterity …
Robert Forrest. March 77.
“The Egyptians were a practical people, and they reveal through the products of their arts and crafts their particular genius. In classical times these early Egyptians were also credited by the Greeks with great knowledge and wisdom; but the evidence provided by Egyptian writings does not support this Greek opinion. It is probable that Greek travellers in Egypt, impressed by the grandeur and antiquity of the monuments of the land and misled by the accounts of past ages given to them by their priestly guides, grossly misinterpreted the evidence and jumped to unwarranted conclusions. Unlike the Greeks, the Egyptians were not philosophically inclined, intellectually inquisitive, or prone to theorising. They tended to accept the world as they saw it and to make use of its advantages without looking too deeply into the properties of its parts. They were good engineers and builders, but not good mathematicians, limiting their interest in calculation to the solution of practical problems.”