Henry, I can't figure out what you mean by the angle of a pyramid, nor the aesthetic criteria motivating the numerology. FWIW, the pyramid whose apex is the center of the cube on its base (and thus apex solid angle 4π/6) has its four apex angles = ArcTan[√8] ~ 70.5º. Is this too steep? Too squat? --rwg On 2017-10-03 09:08, Henry Baker wrote:
The Great Pyramid angle is actually atan(14/11).
Why 14:11 ?
cf(14/11) = [1,3,1,2]
atan(14/11) = 51.843 degrees. = asin(14/sqrt(317)) = acos(11/sqrt(317))
["317" really weird coincidence: 22/7 = 3 1/7]
Apparently, 14/11 ~ 4/pi, if you believe pi=22/7.
cf(4/pi) = [1,3,1,1,1,15,...]
So, the angle the Egyptians really desired was
atan(4/pi) = 51.854 degrees.
Supposedly, the circumference of the base of the Great Pyramid ~ the circumference of a circle whose radius is the height of the Great Pyramid. (Note that the base corners would protrude outside the sphere of radius = height, so this sphere would NOT enclose the Pyramid; the Roman Pantheon is more mathematically interesting, as it perfectly illustrates Archimedes's "Hatbox Theorem".)
So if r is the height of the pyramid, the circle circumference is 2*pi*r ~ 2*(22/7)*r or (44/7)*r.
So each side of the Pyramid base is (11/7)*r, and the distance from the center of the base to the center of one of the base sides is (11/14)*r.
So we have a right triangle r, (11/14)*r, (sqrt(317)/14)*r through the center of the Pyramid for a face angle of atan(14/11).
David Ian Lightbody. "Biography of a Great Pyramid Casing Stone", The Journal of Ancient Egyptian Architecture vol. 1, 2016.
http://www.egyptian-architecture.com/JAEA1/article3/JAEA1_Lightbody.pdf