To continue the recent discussion and recap a few points.
In a series of papers, Miatello presented data from architectural dimensions, texts, and iconography to suggest, among other things, that certain numerical sets formed the basis of pyramid design -
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- These numbers will be familiar to members of this forum - eg. 7/11 - Khufu 280/440, Meidum 175/275. He cites multiples built on other prime factors. Particularly interesting is his presentation of the dimensions of successive enlargments of Zoser's mastabas -
120 X 120. 136 X 136. 136 X 152.
- and of successive steps of the pyramid -
209 X 231. 198 X 220. 165 X 187. 132 X 154.
If this data is accurate does it mean that Imhotep codified a system of theological number, and that later architects drew on this 'canon'? Perhaps this was the case at Meidum and Khufu, but things are not so simple at Dashur. Using Dorner's figure for the Red base of 418 Rc, Miatello accepts the slope at 45 degrees. Yet at the same time he promotes the familiar design for the Bent pyramid based on old data - lower height 90, upper height 110 - requiring a slope lower than that Dorner now appears to promote.
So what happened? The common view is that emerging weaknesses in the structure caused the builders to lower the intended slope - the lower part of the bent contains about 8 million cu.Rc and 3 million more would be required to project the sides to an apex. The present upper part adds 0.5 million cu.Rc.
The builders also laid stones horizontally in the upper part as they did in the Red, indicating a relation between the two.
So we might expect the slopes to be the same, and if the slopes of the Red and Bent satellite are now estimated at 45 degrees, this is the most probable slope for the upper Bent - and this kills the 90/110 height division. At the same time the pleasing ratio between the Bent and Red - 360 : 6/7 : 420 disappears. Not that 'canonical' ratios are absent at Dashur - the repetition of factors 7 and 11 in the design and spacing of inner chambers and numbers of corbels appears to bear this out. Miatello states that Red height equals Zoser base (209 Rc. = 11 X 19). At Giza factor 19 is seen in inter-pyramid spacing.
An important element of design was root two - expressed as 99/70 (derived from Theon's series) and developed in harmony with the 14/11 ratio in Khufu to account for the positions of interior features. Number thus appears to have trumped the geometry upon which it was based - kind of 'proto-Pythagorean' approach.
Further, the lower slope of the Bent clearly is based on root two (but which approximation was used? 10/7? 10/7-7/5? 17/12? And what then the measures of the upper part?) whereas the upper slope is the diagonal of a square, as is the arris slope of the lower part. As earlier noted, this interplay between the square and its diagonal could well have represented a kind of religious metaphor (death and rebirth?) and it is interesting that for Khufu the diagonal at King's Chamber floor level (220 Rc) equals the half base, a possible development of the concept.
The interior arrangements of the Bent approach Khufu in complexity and a proper analysis might give clues to the overall design of the pyramid, as is the case with Khufu.
