L Cooper Wrote:
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> I am, of course, partial to my own squaring the
> circle theory on the subject - which makes use of
> the 11/14 and 8/9 ratios in creating the central
> diagrammatic construction for each pyramid. I
> believe that other considerations were then
> addressed within this over-arching scheme - such
> as the actual cubit length to be used, and the
> various numeric ratios one finds - many of which I
> believe stem from a recognition of Pythagorean
> Triples - both true triples and nearly true
> triples. I have discussed this important point
> before - the fact that theirs was an empirically
> based approach - and hence the importance of the
> 70-70-99 near triple that you cite, or similarly,
> the 99-99-140, the 68-72-99, the 49-50-70, etc.
> near, or 'empirical', triples. Since they had no
> way in which to determine any difference between
> these two categories ('true' and 'empirical'),
> both would have been seen as equivalent to them -
> hence the 'empirical' triples need to very much be
> in the discussion.
I have come across suggestions for the use of Pythagorean triangles by the AE, for example in the layout as put forward by the late Clive Ross -
By the way, where do the 68.72-99 and 49-50-70 triples occur?
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Returning to the question of Khufu 'design choices', as you put it, my take is as follows -
The key ratios are 14/11 and 140/99, shown here in the layout of the KC shafts -
The black rectangle measures 280 X 198.
In the diagram below, dissection of the root 2 figure (in red) locates a number of major features, notably the horizontal length of the Descending passage to north base as 205 (5 X 41) cubits -
The axis of the Descending passage is given by the intersection of square and circle A, or by projection of pyramid side B. The horizontal axis of the lower end intersects the square on the height at C (about 56 below base). This is the origin for the 'common' axis of the upper passages.
Regarding passage junctions, Legon (http://www.legon.demon.co.uk/greatpyr.htm) discovered the blueprint that controlled them, involving the ratios -
The scheme fits Petrie's survey with precision and is laid out using a cubit of 0.52375m -
The layout of the upper slopes is a refinement of the design. It has its origin in the intersection of the passage floorlines, using a module of 24 cubits -
However there is a discrepancy - pyramid base measures 439.81 cubits if measured by the passage system cubit (A 440 base requires a cubit of 0.52355m). Since it seems that the design imperatives, outlined above, were worked out before building it is difficult to account for this. Legon notes that the horizontal distance between the intersection and north base is the round number 141 cubits, but is this sufficient reason to abandon a precise 440 cubit base?
At any rate the integrated nature of Khufu passage design puts paid to the conceit that small differences can be ignored. The upper passage slopes are rigorously defined by horizontal and vertical dimensions which are fractional. The passages were not all intended to be laid out on a 1 to 2 rectangle.