8 - Squares Everywhere

Summary

We must say that there are as many squares as there are numbers.

Galileo Galilei

Squares have a special place in the world of quadrilaterals, just as equilateral triangles have a special place among all the triangles. We devote this chapter to theorems about squares, both in the geometric and number-theoretic sense. The two are closely related, as you read in Section 3.2 concerning the representation of an integer as the sum of two squares and will see again in Sections 8.2 and 8.3.

We present our theorems about squares according to the number of squares in the theorem. For example, the Pythagorean theorem can be thought of as a three-square theorem.

One-square theorems

The golden ratio ϕ appears in many constructions with regular polygons. In Section 2.3 we saw the close relationship between the golden ratio and the regular pentagon, and in Section 6.8 we discovered a relationship between the golden ratio and the equilateral triangle. The following theorem presents a similar result relating the golden ratio and the square.

Theorem 8.1.Inscribe a square in a semicircle as illustrated in Figure 8.1a. Then AB/BC = ϕ.

Proof. See Figure 8.1b. Choose the scale so that BC = 1 and let AB = x. The shaded triangles are similar, so x/1 = (x + 1)/x, and hence x² = x + 1.