Hi Mark,
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What you now propose is just the scheme that has already mentioned on this thread for which you raised the objection that it did not explain the upper slope of the Bent Pyramid, so you now want to focus on just the lower section to avoid your own question mark which is reasonable enough.
I do not propose to avoid the question of the reasons for the observed shape of the upper section of the Bent pyramid, but only try to consider a simpler situation, then to move on to a more complex one.
Since the reasons for the existence of the double-slope design are not completely clear (there are arguments both in favor of the assumption that: a) it was originally planned; and in favor of the assumption that: b) it arose as a result of structural instability), it makes sense to consider the lower section separately, in order to try to understand what shape of the pyramid the builders planed to get in case b) and in what way this shape could be outstanding to choose it from other possible shapes. If no acceptable explanation will be found, this will be an additional argument in favor of the case a) - the shape of the lower section was chosen such to implement some relationships with the upper section.
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Just look at your table again and imagine that the state of construction was with the north side finished with casing stones to 90 cubits, the south side not started with respect to casing stones, and the east and west sides finished with casing stones to a level of 45 cubits. Then the plan is changed to a truncated pyramid. The only opportunity to get an entirely new slope below the level of the bend was on the south side in this hypothesis, but a good attempt was made on the east and west sides by overcompensating, aiming for the seked of 5 palms.
Why do you think the supposed decision to change the plan (uniform slope -> dual slope) should have caused a slight change in the slope of the lower section (55° -> 54.5°)?
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The height of the Bent Pyramid would be 198 cubits assuming 360 base for seked of 5 palms to 90 cubits then seked of 7.5 palms to peak.
The assumption about the length of the base side of 360c and the height of the lower section of 90c requires two different lengths of cubit simultaneously.
Petrie: Base = 189.46m (189.46m / 360c = 0.5263m); The height of the lower section = 47.17 m (47.17m / 90c = 0.5241m).
Dorner: Base = 189.61m (189.61m / 360c = 0.5266m); The height of the lower section = 47.04 m (47.04m / 90c = 0.5227m).
Thus, either the side length is greater than 360c or the height of the lower section is less than 90c.
In my opinion, the most probable values are 362c of 0.5238m/c and 89.86c = ~ 89c 6p of 0.5238m/c.
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The crucial symbolism at the Bent Pyramid became the south side facing the Satellite Pyramid when a decision was made to change the shape to a truncated pyramid, and the south side faces the Satellite Pyramid with a slope corresponding to the same cotangent 100/99.
Mark, imagine that you are the chief architect of the first Egyptian true pyramid. You can choose the shape of the pyramid at your discretion (if the shape of the pyramids is due to the geometry, and not due to a stellar event). Which shape will you choose from all the possible shapes?
Logic suggests that the shape must be unique, must be outstanding in some aspect in order to be chosen from many other variants (of course, it cannot be said for certain that the Egyptians' choice was based on such logic, but this is more likely than an unjustified choice).
In this regard, the question arises - what is the uniqueness of profiles 10 : 7 or 100 : 99 so that they were chosen?
Although I am a sympathizer of the hypotheses that the shape of the Bent pyramid is caused by the stallar event, Legon’s suggestion of a rt2 : 1 ratio (height : half of the base) seems reasonable, and the resulting pyramid is unique in that its faces are equilateral triangles.
Thus, three consecutive Snofru designs are possibly purely geometric, unique and interconnected:
1) Inner Bent: equilateral triangle in vertical section;
2) lower Bent: faces are equilateral triangles (vertical section of previous design applied to faces)
3) the Red pyramid: the height is equal to half the base (and also again the vertical section of the previous design applied to the faces).
The problematic points here are:
- some facts testify against the existence of an internal 60° pyramid
- the angle of slope of the Red Pyramid is slightly less than 45° (although Dorner believes that the difference of 16' is negligible).
Alex.