Golden ratio
The golden ratio (proportio divina or sectio aurea), also called the golden mean, golden section, golden number or divine proportion, usually denoted by the Greek letter phi, is the number
| Table of contents |
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2 Mathematical uses 3 Aesthetic uses 4 See also 5 Other meanings 6 Deriving value from continued fraction 7 External links and references |
It is the unique positive real number with
Properties
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:
The golden rectangle, whose sides a and b stand in the golden ratio, is illustrated below:
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)
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
Mathematical uses
"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."
|.......... a..........|
+-------------+--------+ -
| | | .
| | | .
| B | A | b
| | | .
| | | .
| | | .
+-------------+--------+ -
|......b......|..a-b...|
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.)
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 explicit expression for the Fibonacci sequence involves the golden ratio. Also, the limit of ratios of successive terms of the Fibonacci sequence equals the golden ratio. From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem. It is also the fundamental unit of the algebraic number field and is a Pisot-Vijayaraghavan number.
The golden ratio has interesting properties when used as the base of a numeral system: see Golden mean base.
Aesthetic uses
The ancient Egyptians and ancient Greeks already knew the number and had identified its nature. They found this mathematical proportion, which they called The Golden Mean throughout nature, and it impacted their art, architecture, paideia, and philosophy. The golden mean was found throughout nature, in structure nautilus shells, the size of leaves, the branching patterns of trees, and in the human body. The Greeks thought that the golden mean described the dimensions of average, and by inference "ideal", body features such as the face and the torso, and the proportions of arms and legs to the size of the body.
The golden ratio was used as a guide for accurately creating human likenesses in painting and sculpture. Because the ratio was so common in nature, it was considered auspicious and aesthetically pleasing and was used for other creations, even if not dictated by nature. Many paintings are arranged according to golden ratios; buildings and courtyards were designed with golden rectangles, as were great monuments (e.g., the Parthenon and the Great Pyramid at Giza). The pentagram so popular among the Pythagoreans also contains the golden ratio.
The golden mean continuees to be used for design. It is sometimes used in modern man-made constructions, such as stairs and buildings, woodwork, and in paper sizes; however, the series of standard sizes that includes A4 is based on a ratio of and not on the golden ratio. Recent studies show that the golden ratio continues to plays a role in human perception of beauty, have confirmed its presence in body shapes and faces.
The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the Fibonacci sequence, making them the closest low integer ratios to the golden ratio. 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 first few digits of the golden ratio 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
Other meanings
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.
Deriving value from continued fraction
Deriving the value from nested radicals
External links and references