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Golden ratio - Definition

Golden ratio - Definition: Encyclopedia II - Golden ratio - Definition

Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if where a is the larger part and b is the smaller part. Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are se ...

Golden ratio: Encyclopedia II - Golden ratio - Definition

Golden ratio - Definition

Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if

where a is the larger part and b is the smaller part.

Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if

After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are seen to be equivalent to

The Greek letter φ (phi) is conventionally used to denote the size of the larger part when the smaller part is 1, and this number φ is often called "the golden ratio". Thus we have

The equation labeled (*) above then becomes

or equivalently,

The solutions of this quadratic equation are

Since φ is a quantity it must be positive, hence we have