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 h^{2}. 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:

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

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

where we have written the ratio f/(s/2) as x. The quadratic equation x^{2}=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

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

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