I need to study your post to respond, but I agree that the volume of a sphere was important to the architect of the Great Pyramid.

The design of the sarcophagus in the King's Chamber is a clear indication that the architect of the Great Pyramid knew the formula for the volume of a sphere.

The internal volume of the sarcophagus equates to the volume of a sphere with a diameter of 2.5 royal cubits (70 digits), and a radius of 35 digits.

The modern formula of 4/3 x pi x r x r x r = (4/3 x radius) x (area of a circle)

The internal area of the sarcophagus is equal to the area of a circle with a radius of 35 digits.

The internal depth is equal to 4/3 x radius = 4/3 x 35 digits.

The internal volume is equal to a sphere with a radius of 35 digits.

The internal diagonal is precisely 4 royal cubits.

The internal diagonal constrains both the internal width and the internal length in order to equate to the area of a circle with a radius of 35 digits.

22/7 x 35 x 35 = Area of Circle

Theoretical dimensions by modern analysis:
Length of 105.94 digits x width of 36.34 digits = Area of Circle

105.94 digits = 77.98 inches (Piazzi Smyth 77.93 inches)
36.34 digits = 26.75 inches (Piazzi Smyth 26.73 inches)

The internal depth of 4/3 x 35 digits = 34.35 inches (Piazzi Smyth 34.34 inches)

Piazzi's Smyth's model indicates that the length of the royal cubit converts to 20.61 inches, and this is the same as the best estimate from the base square (J.H. Cole 1926).

Piazzi Smyth had great skill in deciding where to measure from, with an eye for picking out the intended plane of the surface.

Unfortunately, Piazzi Smyth failed to convert his measurements to royal cubits.

Petrie determined the volume absolutely, taking into account depressions in surfaces.

Petrie:
''The volume has also been attributed to a sphere of 2 and* 1/2 cubits (or 1/4 width of the chamber) in diameter; by the true contents this would need a cubit of 20.644, which is very close to the best determinations.''
*'and' inserted by me for clarity

Petrie seems to have overlooked the formula (4/3 x radius) x (area of a circle) for the volume of a sphere, because it is not just the the total volume that fits the model, but also the rational components in line with a simple formula that preserves the cross-sectional area of the sphere.

All this and more was published in Appendix C 'King's Coffer' of my monograph on the Grand Gallery, along with illustrations.

Mark




Edited 7 time(s). Last edit at 12/04/2014 02:58AM by Mark Heaton.