Golden ratio

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This article is about the mathematical ratio and its applications. For the concept of "golden mean" in philosophy, see Golden mean (philosophy); for Aristotle's treatment of the latter, see Nicomachean Ethics.

The golden ratio is an irrational number, approximately 1.61803..., that possesses many interesting properties. Shapes defined by the golden ratio have long been considered aesthetically pleasing in Western cultures, reflecting nature's balance between symmetry and asymmetry and the ancient Pythagorean belief that reality is a numerical reality, except that numbers were not units as we define them today, but were expressions of ratios. The golden ratio is still used frequently in art and design. The golden ratio is also referred to as the golden proportion, golden mean, golden section, golden number, divine proportion or sectio divina.

Definition

Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a+b}{a} = \frac{a}{b}}

where a is the larger part and b is the smaller part.

File:Golden ratio line.png
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

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, i.e. if

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a}{b} = \frac{b}{a-b}.}

After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are seen to be equivalent to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{a}{b}\right)^2 = \frac{a}{b} + 1.\qquad\qquad(*)}

The Greek letter φ (phi) is conventionally used to denote the size of the larger part when the smaller part is 1, and this number φ is often called "the golden ratio". Thus we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a}{b} = \varphi.}

The equation labeled (*) above then becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^2=\varphi+1\,}

and the solutions of this quadratic equation are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1 \pm \sqrt{5} \over 2}.}

Since φ is positive, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi = {1 + \sqrt{5} \over 2}\approx\ 1.618033988\dots.}

History

The golden ratio was first studied by ancient mathematicians due to its frequent appearance in geometry and may have even been understood and used as far back in history as the Egyptians. More commonly, however, the discovery of the golden ratio is ascribed to the ancient Greeks, and is usually attributed to Pythagoras (or to the Pythagoreans, notably Theodorus) or to Hippasus of Metapontum. Euclid spoke of the "golden mean" this way, "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The golden ratio is symbolized by the Greek letter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} (phi) or less commonly by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} (tau).

A startlingly quick proof of irrationality

Recall that we denoted the "larger part" by a and the "smaller part" by b, and concluded that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a}{b} = \frac{b}{a-b}.}

This gives a startlingly quick proof that this number is irrational. An irrational number is one that cannot be written as a/b where a and b are integers. If a/b is such a fraction, in lowest terms, then b/(a − b) is in even lower terms — a contradiction. Thus this number cannot be so written; it is irrational.

Alternate forms

The formula Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi = 1 + 1/\varphi} can be expanded recursively to obtain a continued fraction for the golden ratio:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi = [1; 1, 1, 1, ...] = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}}

and its reciprocal:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^{-1} = [0; 1, 1, 1, ...] = 0 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}.}

Note that the successive convergents of these continued fractions are ratios of Fibonacci numbers.

The equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi^2 = 1 + \varphi} likewise produces the continued square root form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}.}

Also

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi=1+2\sin(\pi/10)=1+2\sin 18^\circ}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi=1/(2\sin(\pi/10))=1/(2\sin 18^\circ)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi=2\cos(\pi/5)=2\cos 36^\circ\,}

These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram.

Mathematical uses

File:FakeRealLogSpiral.png
Approximate and true Golden Spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smallest square is in the Golden Ratio.
File:Fibonacci Spiral.png
A Fibonacci spiral which also approximates the Golden Spiral.

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F\left(n\right) = {{\phi^n-(1-\phi)^n} \over {\sqrt 5}} = {{\phi^n-(-\phi)^{-n}} \over {\sqrt 5}}}

The limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the Fibonacci sequence is divided by its preceding number, it approximates φ. e.g., 987/610 ≈ 1.6180327868852. Alternatingly the approximation to φ is too small and too large, it gets better as the Fibonacci numbers get higher, and:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)| = \varphi.}

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

φ−2 = − φ + 2,
φ−1 = φ − 1,
φ0 = 1,
φ1 = φ,
φ2 = φ + 1,
φ3 = 2φ + 1,
φ4 = 3φ + 2,
φ5 = 5φ + 3,
φn = F(n)φ + F(n − 1),
...

Because φ is the only positive number that satisfies the identity φn = φn − 1 + φn − 2, any polynomial expression in φ may be decomposed into a linear expression. For example:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\varphi^3 - 5\varphi^2 + 4 = 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 = 3[(\varphi + 1) + \varphi] - 5(\varphi + 1) + 4 = \varphi + 2 \approx 3.618}

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 has been argued this is the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants). It is also the fundamental unit of the algebraic number field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

File:ParthenonGoldenRatio.png
The Parthenon showing various golden rectangles claimed to be used in its design.

It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positions or proportions of the Pyramids of Giza.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} (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] 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. For instance, the Acropolis, including the Parthenon, is often claimed to have been constructed using the golden ratio. This has encouraged modern artists, architects, photographers, and others, during the last 500 years, to incorporate the ratio in their work. As an example, a rule of thumb for composing a photograph is called the rule of thirds; it is said to be roughly based on the golden ratio.

It is also claimed that the human body has proportions close to the golden ratio.

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 Architect Le Corbusier used the golden ratio as the basis of his Modulor system of Architecture.

File:Golden section page.png
Golden ratio applied to page and margin dimensions in Book design

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}} and not on the golden ratio. The average ratio of the sides of great paintings, according to a recent analysis, is 1.34. [2]. Credit cards are generally 3 3/8 by 2 1/8 inches in size, which is less than 2 percent from the golden ratio.

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.

French composer Erik Satie used the Golden Section in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.

File:Divina proportione.png
Woodcut from the Divina Proportione by Luca Pacioli (1509) depicting the golden proportion as it applies to the human face.

The construction of a pentagram is based on the golden ratio. The pentagram can be seen as a geometric shape consisting of 5 straight lines arranged as a star with 5 points. The intersection of the lines naturally divides each length into 3 parts. The smaller part (which forms the pentagon inside the star) is proportional to the longer length (which form the points of the star) by a ratio of 1:1.618... It is thought by some that this fact may be a reason why the ancient philosopher Pythagoras chose the pentagram as the symbol of the secret fraternity of which he was both leader and founder.

Decimal expansion

(sequence A001622 in OEIS)

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...

This can also be found fairly easily on a calculator, using the formula

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {1+\sqrt{5} \over 2}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5^{0.5} \times 0.5 + 0.5}

See also

Other meanings

External links

Mathematics

Aesthetics

Tools

History

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