Math in Engineering

Mathematics and Architecture

What is the golden ratio?

The golden ratio (also known as extreme and mean ratio, golden section, golden mean, or divine proportion) is a number that has many interesting properties; it is associated with the balance between symmetry and asymmetry used in art and design. Two quantities are said to be in the golden ratio if the “whole is to the larger as the larger is to the smaller.” Euclid expressed it as, “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.” This is seen in the accompanying illustration, in which for two segments “a” and “b,” the entire line is to the “a” segment as “a” is to the “b” segment.

The symbol for the golden ratio is φ (the Greek letter “phi,” or a circle with a vertical slash through it); it is equal to about 1.6103398 and is considered an irrational number. The calculation to reach the golden ratio is as follows:

This equals the quadratic equation:

which results in:


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