At-home lessons: the golden ratio

By Michael Avidon, math editor

The ancient Greeks thought that the “most aesthetically pleasing” (nicest looking) rectangle was one whose sides conform to the following proportion:

We wish to determine the numerical value of the ratio of the sides.

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You can choose any unit of measurement without changing the proportion. For convenience, we’ll choose one such that the smaller side measures 2 units. The larger side is greater than 2 units. To make a certain part of the algebra work out simply, we are going to label this side “1 + x,” where x represents a number greater than 1.

We can solve this equation for x:

(1 + x)(1 + x) = 2(3 + x)  Cross multiply
1 + x + x + x² = 6 + 2x. distribute
1 + 2x + x² = 6 + 2x. Combine like terms
x² = 5. subtract 1 + 2x from both sides
x = √5. take the square root of both sides

With our choice of units, the larger side measures 1 + √5 units.

What we are really interested in is the ratio given on either side of the proportion.

This is called the golden ratio, and is denoted by the Greek letter 𝜙.

We have shown that

𝜙 = ^{1 + √5} ⁄ ₂.

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