# Golden ratio

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

The golden ratio, golden mean, golden number, or golden section is the mathematical constant

$\varphi=\frac{1+\sqrt5}{2}\approx1.6180339\ldots$

More importantly, it is the ratio of two quantities A and B such that the ratio from A to B (where A is the smaller one) is the same as the ratio from B to A + B; this comes from the fact that it's the positive real root of $x^2-x-1$ .

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 ratio is closely associated with the Fibonacci sequence $1,1,2,3,5,8,13,21,34,\ldots$ Among many other things, the ratios of successive Fibonacci numbers converge to phi:

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

##  Phi woo

Phi and the Fibonacci numbers lend to a lot of very fascinating mathematical properties, but some cranks are willing to push it further with a good dose of pareidolia.[1][2] A classic example is nautilus shells: it is often said that they're golden spirals, when in fact they're just logarithmic spirals with ratios usually around 1.3 or so. Others claim to have found the golden ratio or Fibonacci numbers in human facial beauty, historical architecture (sometimes legitimate), Apple products, planets, musical instruments, ideal loudspeaker cabinets, ... the list goes on and on. In many cases, they've actually found the golden ratio in object X, but it's not a particularly special result; with enough perseverance, you can find the golden ratio in just about goddamn anything.

The distinction between an actual case of the golden ratio's magic and a mere crank sighting is when the presence of the golden ratio can be actually explained. That is, the golden ratio should appear in both theoretical models describing X and measured outcomes in X. Through the study of phyllotaxis, botanists have not only observed sunflower seeds growing in Fibonacci-numbered spirals, but they've provided a scientific explanation for why that happens: an example of the golden ratio legitimately appearing in nature.

The numerological woo surrounding phi and Fibonacci is an example of the strong law of small numbers; that is, there are not very many small numbers (or visually distinguishable ratios between 1 and 2) and they'll show up in many unrelated places. See also Ramsey theory.

## References

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