Phi, The Great Pyramid, and a statement attributed to Herodotus

         Suppose that h represents the height of the Pyramid, s represents the length of each side, and f represents the apothem (i.e. the distance from the midpoint of one side to the apex of the Pyramid). The area of a square with side h is just h2. The area of each face of the Pyramid is sf/2. In addition, one can apply the Pythagorean theorem to the right triangle formed by the following vertices: the center of the Pyramid's base, the apex, the midpoint of one side. This gives the relationship:

(s/2)2 + h2 = f2

Let us assume that h2=sf/2, as Herodotus supposedly learned from the priests. One then obtains the following equation:

(s/2)2 + (sf/2) = f2

Dividing both sides of this last equation by (s/2)2 leads to the simpler equation:

1 + x = x2

where we have written the ratio f/(s/2) as x. The quadratic equation x2=x+1 has two roots - one positive and one negative. The ratio f/(s/2) must be equal to the positive root, and that root is just the number φ=(1+sqrt(5))/2. This is the famous &phi relationship.

        Conversely, one can reverse the reasoning given above to show that the equality f/(s/2)=φ implies the equality h2=sf/2.