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Golden mean


The golden mean (proportio divina or sectio aurea), also called golden ratio, golden section, golden number or divine proportion, usually denoted by the Greek letter phi, is the number

the unique positive real number with

and the equally interesting property

Two quantities are said to be in the Golden ratio, if "the whole is to the larger as the larger is to the smaller", i.e. if
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:
After simple algebraic manipulations (multiply the first equation with a/b or the second equation with (a-b)/b), both of these equations are seen to be equivalent to

and hence

The fact that a length is divided into two parts of lengths a and b which stand in the golden ratio is also (in older texts) expressed as "the length is cut in extreme and mean ratio".

Uses and Aesthetics

ancient Egyptians and ancient Greeks already knew the number and, because they regarded it as an aesthetically pleasing ratio, often used it when building monuments (e.g., the Parthenon). The pentagram so popular among the Pythagoreans also contains the golden mean. It is also sometimes used in modern man-made constructions, such as stairs and buildings, woodwork, and in paper sizes, however it is a myth that the European formats (such as A4, which is actually cut to 4 decimal places of √2) are cut in the golden mean. Recent studies showed that the Golden ratio plays a role in human perception of beauty, as in body shapes and faces.

A possible reason for its supposed attractiveness is shown by the Golden rectangle, which is a rectangle whose sides a and b stand in the Golden ratio:

  |.......... a..........|

+-------------+--------+ - | | | . | | | . | B | A | b | | | . | | | . | | | . +-------------+--------+ -


If from this rectangle we remove square B with sides of length b, then the remaining rectangle A is again a Golden rectangle, since its side ratio is b/(a-b) = a/b = φ. By iterating this construction, one can produce a sequence of progressively smaller Golden rectangles; by drawing a quarter circle into each of the discarded squares, one obtains a figure which closely resembles the logarithmic spiral θ = (π/2log(φ)) * log r. (see polar coordinates)

The green spiral is made from quarter circle pieces as described above, the red spiral is a real logarithmic spiral. The similarity between the spirals should be noticeable. (If you instead only see a yellow spiral, look very carefully, there are actually two different spirals in the image.)

Since φ is defined to be the root of a polynomial equation, it is an algebraic number. It can be shown that φ is an irrational number. Because of 1+1/φ = φ, the continued fraction representation of φ is

The number φ turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles.

The ratios of justly tuned major and minor sixths, which equally tempered thirds approximate, considered with the thirdss the most dynamic and interesting consonances, are consecutive numbers of the fibonnaci sequence (13:8=1.625 and 8:5=1.6), making them the closet low integer ratios to the golden mean. James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

The explicit expression for the Fibonacci sequence involves the golden mean. Also, the limit of ratios of successive terms of the Fibonacci sequence equals the golden mean.

The golden mean has interesting properties when used as the base of a numeral system: see Golden mean base.

"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."
Johannes Kepler

The first few digits of the golden mean are:

1. 6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 1893911374 8475408807 5386891752 1266338622 2353693179 3180060766 7263544333 8908659593 9582905638 3226613199 2829026788 0675208766 8925017116 9620703222 1043216269 5486262963 1361443814 9758701220 3408058879 5445474924 6185695364 8644492410 4432077134 ...

See also:

The Doctrine of the Golden Mean (Zhong1 Yong2, 中庸), the name of a chapter in Li Ji (Li3 ji4, 禮記) is one of the "Four books" of classical Chinese writings.