1. The Golden Mean

The Golden Mean is the name for the ratio between the sides of a rectangle with the following property: If a square is cut off from one side of the rectangle ...
... the rectangle that remains ...
... has the same ratio of sides as the original rectangle.
If we call this ratio x, then a rectangle with sides 1 and x will have the correct ratio. The rectangle remaining after cutting away a square will have sides x-1 and 1.
Setting the two ratios equal leads to the equation x2-x-1 = 0.

It is just as natural to consider a rectangle with sides x and 1, letting x be the shorter side. Then the sides of the smaller rectangle are 1-x and x, and the equation becomes x²+x-1 = 0.

These two equations have opposite roots:

The two positive roots (which are reciprocals, as should be clear from the construction) have equal claim to be called the Golden Mean. In this column we will work with the larger one, but all our arguments could be recast in terms of the smaller.

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