Hi Hermione,



".....generally speaking the knowledge of some triplets in AE is not incompatible with the surviving mathematical sources. It is not necessary to suggest that the AE were acquainted with more or less complicated versions of the Theorem of Pythagorus as early as the OK. The peculiarity of the Pythagorean triplets - that is, the fact that they correspond to right-angled triangles in which three sides are equal to whole numbers - is not necessarily related to their squares." (Rossi 2004: 218)



"Some historians of mathematics, wishing to decry the mathematical ability of the Egyptians as compared to the Babylonians and the Greeks, have doubted whether the Egyptians even knew of the 3:4:5 right-angled triangle. We do not concur with this view, since we think that the best explanation for the appearance of a 5 1/4 seked with the pyramid of Khephren in the 4th dynasty and its seemingly universal adoption in the 6th dynasty is that it gives 3:4:5 proportion to the half-base width, height, and apothem of the face. This, pace Legon, would have facilitated the cutting of casing stones..............the 3,4,5 result could have been got very easily from inspection of a table of squares, and splitting 25 into 16 and 9. We do not claim that the Egyptians necessarily did use the method of calculation outlined above, although they could have, but we give it because it can readily yield any Pythagorean triple, and because of the slur, which we think to be unjustified, that the Egyptians were mathematically inept, and hampered by being harnessed to unit fractions. In fact, some of the calculations in the mathematical papyri are quite complex. Moreover, it would be wrong to suppose that what survives in such student manuals represents the sum total of ancient Egyptian mathematical knowledge." (Robins & Shute, DE 18, 1990: 43-53)



The 4:3 ratio often occurs in early Old Kingdom funerary architecture. A pyramid with sqd 5 1/4 defined slope (a ratio of 4:3), was a popular model - for example, the slope of Khafra's pyramid had a 4:3 ratio.


The ground plan of the funerary complex of the first king of Dyn 5, Userkaf, had a length / width ratio of 4:3 (noted by Verner, 2001: 274-5). This means the length, width, diagonal of his complex were 3:4:5. The slope of his pyramid was defined by sqd 5 1/4, again 4:3.


The slopes of three Dyn 6 kings were defined by sqd 5 1/4. The sarcophagus chamber in Khufu's pyramid had a 10:20 (1:2) cubit proportion for the 2- dimensional floor area and a 15:20:25 (3:4:5) cubit proportion for the 3-dimensional space.


The 1:2 and 3:4:5 proportions are found in AE architectural design, the question is, are they coincidental or intentional.


I once asked AS if the 1:2 proportion was intentional, and he replied yes. When I asked him how did he know, he replied "they achieved it".


So how did they "achieve" a 1:2 proportion but not a 4:3 proportion? - the diagonal of 5 is important because it means the corners of a funerary complex or the 3-dimensional space of Khufu's sarcophagus chamber, can be checked with a whole cubit number length to make sure the corners are square - hence 3:4:5.

FBI maxim: Never believe in coincidence, but never dismiss them.

CT