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Golden ratio

The golden ratio is a number, approximately 1.618, that possesses many interesting properties. It was studied by ancient mathematicians due to its frequent appearance in geometry. Shapes defined by the golden ratio have long been considered aesthetically pleasing in western cultures, reflecting nature's balance between symmetry and asymmetry. The ratio is still used frequently in art and design. The golden ratio is also known as the golden mean, golden section, golden number or divine proportion.

It is usually denoted by the Greek letter φ (phi).

$\phi = \frac{\sqrt{5} + 1}{2} \approx 1.618 033 988 749 894 848 204 586 834 366 \$

Contents

1 Origin of name

2 Definition of the Golden Ratio

3 Properties

4 Alternate forms

5 Mathematical uses

6 Aesthetic uses

7 In nature

8 The golden ratio to 1024 decimal places

9 See also

10 Other meanings

11 External links

11.1 Mathematics
11.2 History
11.3 Other

Origin of name

The name "golden ratio" first seemed to have been used in the form sectio aurea, "golden section", by Leonardo da Vinci. The use of the symbol φ to represent the golden ratio was invented by the American mathematician Mark Barr and taken from the first Greek letter in the name of the Greek sculptor Phidias, who was long believed to have used the golden ratio in his designs.

Definition of the Golden Ratio

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

$\frac{a+b}{a} = \frac{a}{b}.$

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:

$\frac{a}{b} = \frac{b}{a-b}.$

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

$\left(\frac{a}{b}\right)^2 = \frac{a}{b} + 1$

and hence

$\frac{a}{b} = \phi.$

A line is divided into two segments a and b. The entire line is to the a segment as a is to the b segment

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". This can be easily visualized using a line that is divided into two segments, as in the diagram.

Properties

φ is an irrational number, and the unique positive real number with

$\phi^2 = \phi + 1 \$

$\phi^3 = \frac{\phi + 1}{\phi - 1}$

$\phi^{-1} = \phi - 1 \$

The last quantity is called the golden ratio conjugate (erroneously called the silver ratio or silver mean) and is also represented by $\hat\phi$ .

Alternate forms

The formula $φ = 1 + 1 / φ$ can be expanded recursively to obtain a continued fraction for the golden ratio:

$\phi = [1; 1, 1, 1, ...] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}$

and its conjugate:

$\hat\phi = [0; 1, 1, 1, ...] = 0 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}$

The equation $φ 2 = 1 + φ$ likewise produces the continued square root form:

$\phi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}}}$

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."

—Johannes Kepler

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)

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.

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 and its conjugate. Also, the limit of ratios of successive terms of the Fibonacci sequence equals the golden ratio. The successive powers of φ obey the Fibonacci recurrence. 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 $\mathbb{Q}(\sqrt{5})$ 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

summary of classical construction
upon a unit square

The ancient Greeks already knew the golden ratio from their investigations into geometry, but there is no evidence they thought the number warranted special attention above that for numbers like $π$ (Pi), for example. Studies by psychologists have been devised to test the idea that the golden ratio plays a role in human perception of beauty. They are, at best, inconclusive. [1] http://plus.maths.org/issue22/features/golden/ Despite this, a large corpus of beliefs about the aesthetics of the golden ratio has developed. These beliefs include the mistaken idea that the purported aesthetic properties of the ratio was known in antiquity. This has encouraged modern artists, architects, and others, during the last 500 years, to incorporate the ratio in their work.

In 1509 Luca Pacioli published the Divina Proportione, which explored not only the mathematics of the golden ratio, but also its use in architectural design. This was a major influence on subsequent generations of artists and architects. Leonardo Da Vinci drew the illustrations, leading many to speculate that he himself incorporated the golden ratio into his work, although there is no evidence supporting this.

The ratio 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 $\sqrt{2}$ and not on the golden ratio. It's also interesting to note that the average ratio of the sides of great paintings, according to a recent analysis, is 1.34. [2] http://arxiv.org/abs/physics/9908036/

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.

In nature

The golden ratio turns up in nature as a result of the dynamics of some systems - for instance, in the angular spacing of tree limbs around a trunk, or sunflower seeds. In both cases, the problem is "wedge this next one into the biggest available space".

You can draw a nice sunflower by plotting the points $(\theta = {{2 \pi} \over {\phi}} i, r = \sqrt i), i = 1 .. N$

The shape of the shell of the chambered nautilus (Nautilus pompilius) is often claimed to be related to the golden ratio. (Please see discussion about the "Golden ratio" page!)

The golden ratio to 1024 decimal places

 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 4947049565 8467885098 7433944221
   2544877066 4780915884 6074998871 2400765217 0575179788
   3416625624 9407589069 7040002812 1042762177 1117778053
   1531714101 1704666599 1466979873 1761356006 7087480710
   1317952368 9427521948 4353056783 0022878569 9782977834
   7845878228 9110976250 0302696156 1700250464 3382437764
   8610283831 2683303724 2926752631 1653392473 1671112115
   8818638513 3162038400 5222165791 2866752946 5490681131
   7159934323 5973494985 0904094762 1322298101 7261070596
   1164562990 9816290555 2085247903 5240602017 2799747175
   3427775927 7862561943 2082750513 1218156285 5122248093
   9471234145 1702237358 0577278616 0086883829 5230459264
   7878017889 9219902707 7690389532 1968198615 1437803149
   9741106926 0886742962 2675756052 3172777520 3536139362
   1076738937 6455606060 5922...

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.

External links

Mathematics

The Golden Section: Phi http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html
PHI, the golden ratio http://astronomy.swin.edu.au/~pbourke/analysis/phi/

History

Golden Ratio - University of St. Andrews site http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html
Amazon.com link to "The Golden Ratio", by Mario Livio http://www.amazon.com/exec/obidos/tg/detail/-/0767908163/ which discusses the occurrences of the golden ratio in nature, art and geometry.

Other

The Golden Number http://www.goldennumber.net/