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December 15, 2019, 8:44 am UTC    
November 25, 2019 02:42PM
The design of the Grand Gallery provides an explanation of how to square the circle for the pi approximation 22/7.

Marking off twice the diameter of a circle on the floor of the Grand Gallery of the Great Pyramid would result in a vertical rise corresponding to the precise side length of the circle’s equal area square. It is, of course, impossible to square the circle exactly, but this proposition requires a theoretical slope of precisely 26.30 degrees, or 26.31 degrees for the Pi approximation 22/7.

Smyth determined the angle of the slope as a weighted average of approximately 26.29 degrees (26 degrees 17 minutes 37 seconds). A build angle would be within 0.05 degrees of the intended angle for a positive error of 1 part in 1000 on the rise and a negative error of 1 part in 1000 on the run, or vice versa, and the proposed model is approximately 0.02 less than the theoretical angle.

Petrie measured offsets from a mean axis of 26.275 degrees (26 degrees 16 minutes 30 seconds), or approximately 0.035 degrees less than the theoretical slope. A simple 1 in 2 slope would require an angle of precisely 26.565 degrees which is 0.255 degrees steeper than the theoretical slope, so this may have been the intended slope but only if the build standard was low.

The diameter of the Pyramid’s Equal Area Circle is 280 cubits. The theoretical slope to square the circle for the Pi approximation 22/7 is defined exactly by a right angled triangle with a long side of 15 cubits and a hypotenuse equal to the square root of 280 multiplied a cubit (approx 16.73 cubits). Remarkably, the gallery’s vertical height above its entrance may be regarded as the diameter of a 1/sr280 scale model of the Pyramid’s Equal Area Circle.

Smyth’s first three measurements of the vertical height near the entrance, as taken on the inclined floor rather than the broken section, were 344.2 inches, 343.7 inches and 346.0 inches. The mean of 344.6 inches (approx 8.753 metres) is 0.6 inches (approx 15 mm) less than the theoretical vertical height of sr280 cubits. Applying Smyth’s determination of the average slope of the gallery to the vertical height yields a perpendicular height of 308.9 inches (approx 7.846 metres) which is 0.55 inches (approx 14 mm) less than the theoretical perpendicular height of 15 cubits, as calculated for a cubit of 20.63 inches (524 mm). The Grand Gallery was built perpendicular to the slope.

The model circle with a diameter of sr280 cubits is equal to a square with a side length of 2 x sr55 cubits and an area of 220 square cubits. This means the Pyramid’s Equal Area Circle with a diameter of 280 cubits has an area 280 times greater than the model circle:

Area of triangular cross-section of pyramid = 1/2 x 440 x 280 = 61,600 square cubits
Pyramid’s Equal Area Circle = 280 x 220 square cubits = 61,600 square cubits
22/7 x 140 cubits (radius) x 140 cubits (radius) = 61,600 square cubits

This model provides an explanation for the unusual soaring height of the Grand Gallery, as assessed at the entrance near the north end wall, so squaring the area of a circle for the Pi approximation 22/7 is naturally applicable to the Pyramid’s Equal Area Circle:
Side length of equal square = [sr55/sr280] x (2 x diameter of 280) = sr55 x sr280 x 2 cubits
Area of pyramid’s equal area square = 55 x 280 x 4 square cubits = 61,600 square cubits

Applying this model to a circle with a diameter of 70 digits yields an area of 3850 square digits. The volume of the sphere latent in the dimensions of the sarcophagus would then be 2/3 x 70 digits x 3850 cubic digits which is exactly equal to the volume of a sphere with a diameter of 70 digits as calculated for the Pi approximation 22/7.

Mark



Edited 5 time(s). Last edit at 11/25/2019 02:53PM by Mark Heaton.
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