Lee, when we look at a pyramid we see merely a heap of blocks, yet when the Egyptians looked at a pyramid they felt in the presence of gods and goddesses. We have to consider this difference also when it comes to the mathematics of ancient Egypt, as I shall show in the case of the pyramid in RMP 57 and 58.
The base length measures 140 royal cubits, the sekad 5 palms 1 finger, the height 93 '3 royal cubits.
How long are the diagonals? Look out for a suiting number pair in the number column for the approximation of the square root of 2. The first lines are: 1 1 2, 2 3 4, 5 7 10, 12 17 24, 29 41 58, 70 99 140, ... If the side of a square measures 70a, the diagonals measure practically 99a. Side 140, diagonals 198 royal cubits.
Imagine a circle around the square. How long is the circumference? Look out for a suiting number pair in this sequence: 3/1 (plus 22/7) 25/8 47/15 69/22 91/29 113/36 135/43 157/50 179/57 201/64 223/71 245/78 267/85 289/92 311/99 333/106 355/113 377/120 ... If the diameter of a circle measures 99a, the circum- ference measures practically 311a. Diameter 198, circumference 622 royal cubits.
Now imagine a sphere and a hemisphere inside the pyramid. How long are their radii? Exactly 35 and 56 royal cubits respectively. An Egyptian pyramid symbolized the primeval hill, out of which rose the sun and the sky, personified in Ra and Nut, and if they can be symbolized in geometrical shapes, these are the sphere for the sun, and the hemisphere for the sky.