Thanks for that.
I have been thinking that I could do a paper for the Journal of Ancient Egyptian Architecture.
It would need about 5 or 6 illustrations to convey the picture in the context of the architecture of the King's Chamber and the Great Pyramid. Unfortunately, I don't think the journal is publishing any papers this year.
In the proposed model the internal space of the sarcophagus is equal to the volume of a sphere with a diameter of 2.5 cubits. The rectangular space for king’s body has an area corresponding to the area of a circle with a diameter of precisely 2.5 cubits, and a diagonal of 4 cubits (112 digits).
It is possible the volume of a sphere was determined as 2/3 times the volume of a cylinder by a practical method as this is the configuration which appears in the design of the sarcophagus which could be regarded as a rectilinear model of a practical determination.
We know the approximate area of a circle could have been determined from the formula in RMP if this formula was known as far back as the Pyramid Age.
The 22/7 formula is apparent as the ratio of circumference to diameter in the design of the King's Chamber as model of the size and shape of the pyramid on a scale of 1 digit to 1 cubit, but not necessarily in a formula to determine the area of a circle.
The internal space of the sarcophagus includes two significant ratios, D1/D2 (internal length / internal width) and D9/D3 (internal diagonal / internal depth). The advantage of evaluating ratios is that ratios are independent of conversion factor from cubits to millimetres, so the ratios remain the same if anyone were to evaluate the model for a cubit of 523 millimetres or 525 millimetres rather than 524 millimetres (20.63 inches).
In the RMP the area of a circle was regarded as a square with a side length equal to 8/9 times the diameter of the circle in a problem to determine the volume of a cylindrical granary. The measurements are closer to the 22/7 model than the 8/9 model for the ratio D1/D2; see below.
The D9/D3 ratio is independent of the method of determining the area of a circle with the depth D3 equal to 1.66 cubits as 2/3 times the diameter of the circle of 2.5 cubits. Ratios from the measurements are very close indeed to the theoretical D9/D3 ratio.
The Pyramid’s Equal Area Circle has a diameter of 280 cubits. Virtual equal area circles on the N/S and E/W axes of the pyramid may be envisaged as outlining a hypothetical sphere with a diameter of 280 cubits which is 112 times the diameter of the sphere latent in the sarcophagus. It follows that the open sarcophagus may be regarded as a 1/112 scale model of a virtual sphere with a diameter equal to the height of the pyramid.
Ratios are multiplied by 1000 to aid comparison:
D9/D3 = Internal diagonal / Internal depth
D9/D3 = 4 cubits / 1.66 cubits
D9/D3 x 1,000 = 2,400 (model), 2399.3 (Smyth), 2,397.9 (Petrie)
D1/D2 = Internal length / Internal width (irregular dimensions governed by D9)
D1/D2 x 1,000 = 2,916 (22/7 model), 2,916 (Smyth), 2,911 (Petrie)
D1/D2 x 1,000 = 2,886 (8/9 model), 2,916 (Smyth), 2,911 (Petrie)
Supporting data:
Internal diagonal D9 = 4 cubits = 112 digits (for 22/7 model and 8/9 model)
Internal depth D3 = 1.66 cubits (for 22/7 model and 8/9 model)
D1 x D2 = Area of a circle with a diameter of 2½ cubits (70 digits)
For 22/7 model: D1 (105.94 digits) x D2 (36.34 digits) = 3,850 square digits
where D9 = 112.0 digits
For 8/9 model: D1 (105.85 digits) x D2 (36.58 digits) = 3,871.6 square digits.
where D9 = 112.0 digits
The ratios infer a 22/7 determination of the area of a circle rather than 8/9 determination.
But does it really matter which was the case if the model of a sphere was in the mind of the designer?
Mark
Edited 2 time(s). Last edit at 11/09/2019 04:43AM by Mark Heaton.