Golden ratio

The golden ratio (also referred to as the golden number or golden section) or φ has been known for millennia. It is defined as "the ratio between two numbers such that the lesser is to the greater as the greater is to the sum"

The golden section: The golden ratio applied to the division of a line

A:B = B:(A+B)

B:A = φ = 1.61803398874989484820458683436563811772030917980576 (approx)

Its exact value is $\varphi=\frac{1+\sqrt5}{2}$.

It has generally been thought to be pleasing and harmonious to human perception and is the basis of much classical architecture. The usage of the Greek letter phi to represent the golden ratio was suggested by mathematician Mark Barr from the first letter of Phidias (ancient Greek, Φειδίας), the sculptor who was alleged to have used it in creating statues for the Parthenon.

The golden number (or an approximation) appears often in nature and is the convergent point of the ratio of successive terms of the Fibonacci sequence - 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,....

A rectangle with the ratio of adjacent sides equal to the golden ratio. It is supposedly particularly pleasant, visually.
 1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.66667 8/5 = 1.6 13/8 = 1.625 21/13 = 1.61539 34/21 = 1.61905 55/34 = 1.61765 89/55 = 1.61818 144/89 = 1.61798 233/144 = 1.61806 377/233 = 1.61803 610/377 = 1.61804 987/610 = 1.61803

It is interesting to note that: $\varphi - 1 = \frac{1}{\varphi}$ (by definition, of course).

Some people think that the golden ratio is an ideal way to proportion loudspeaker cabinets, but what do they know?

Footnotes

• The Golden Ratio: The Story of Phi, The World's Most Astonishing Number by Mario Livio, 2002
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