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{{Short description|Number, approximately 1.618}}
{{pp-semi-protected|small=yes}}
{{infobox non-integer number
| title = Golden ratio ({{mvar|φ}})
| image = Golden ratio line.svg
| image_alt = two line segments of lengths a and b in the golden ratio: a + b is to a as a is to b
| decimal = {{val|1.618033988749894}}...<ref name=a001622 />
| continued_fraction = <math>1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>
| algebraic = <math>\frac{1 + \sqrt5}{2}</math>
}}
[[File:SimilarGoldenRectangles.svg|right|thumb|A [[Golden rectangle|golden rectangle]] with long side {{math|1=<strong style="color:green;">''a'' + ''b''</strong>}} and short side {{mvar|1=<strong style="color:blue;">a</strong>}} can be divided into two pieces: a [[Similarity (geometry)|similar]] golden rectangle (shaded red, right) with long side {{mvar|1=<strong style="color:blue;">a</strong>}} and short side {{mvar|1=<strong style="color:red;">b</strong>}} and a [[Square|square]] (shaded blue, left) with sides of length {{mvar|1=<strong style="color:blue;">a</strong>}}.  This illustrates the relationship {{math|1={{sfrac|''a'' + ''b''|''a''}} = {{sfrac|''a''|''b''}} = ''φ''.}}]]

In [[Mathematics|mathematics]], two quantities are in the '''golden ratio''' if their [[Ratio|ratio]] is the same as the ratio of their [[Summation|sum]] to the larger of the two quantities. Expressed algebraically, for quantities <math>a</math> and <math>b</math> with <math>a > b > 0</math>,

{{bi|left=1.6|1=<math> \frac{a+b}{a} = \frac{a}{b} = \varphi</math>}}

where the Greek letter phi ({{nowrap|<math>\varphi</math>}} or <math>\phi</math>) denotes the golden ratio.{{efn|If the constraint on <math>a</math> and <math>b</math> each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. <math>\varphi</math> is defined as the positive solution. The negative solution is <math>-\varphi^{-1} = \tfrac12\bigl(1 - \sqrt5\bigr).</math> The sum of the two solutions is <math>1</math>, and the product of the two solutions is <math>-1</math>.}} The constant <math>\varphi</math> satisfies the [[Quadratic equation|quadratic equation]] <math>\varphi^2 = \varphi + 1</math> and is an [[Irrational number|irrational number]] with a value of<ref name=a001622 />

{{bi|left=1.6|1=<math>\varphi = \frac{1+\sqrt5}{2} = </math>{{math|{{val|1.618033988749}}....}}}}<!-- PLEASE DO NOT add additional digits to the value of φ in this equation; there is a long-standing consensus that additional digits do not add to understanding. Thank you.-->

The golden ratio was called the '''extreme and mean ratio''' by [[Biography:Euclid|Euclid]],<ref name="Elements 6.3" /> and the '''divine proportion''' by [[Biography:Luca Pacioli|Luca Pacioli]],<ref name=Pacioli /> and also goes by several other names.{{Efn|Other names include the ''golden mean'', ''golden section'',{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/3 3], [https://archive.org/details/goldenratiostory00livi/page/81 81]}} ''golden cut'',<ref name=goldencut /> ''golden proportion'', ''golden number'',{{sfn|Herz-Fischler|1998}} ''medial section'', and ''divine section''.}}

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the [[Dodecahedron|dodecahedron]] and [[Icosahedron|icosahedron]].{{sfn|Herz-Fischler|1998|pp=20–25}} A [[Golden rectangle|golden rectangle]]—that is, a rectangle with an aspect ratio of <math>\varphi</math>—may be cut into a square and a smaller rectangle with the same [[Aspect ratio|aspect ratio]]. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as [[Finance:Financial market|financial market]]s, in some cases based on dubious fits to data.<ref name="strogatz nytimes" /> The golden ratio appears in some [[Patterns in nature|patterns in nature]], including the [[Biology:Phyllotaxis|spiral arrangement of leaves]] and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be [[Social:Aesthetics|aesthetically]] pleasing. These uses often appear in the form of a golden rectangle.


==Calculation==
Two quantities <math>a</math> and <math>b</math> are in the ''golden ratio'' <math>\varphi</math> if<ref name=schielack />
{{bi|left=1.6|1=<math> \frac{a+b}{a} = \frac{a}{b} = \varphi.</math>}}

One method for finding a closed form for <math>\varphi</math> starts with the left fraction. Simplifying the fraction and substituting the reciprocal <math>b/a = 1/\varphi</math>,
{{bi|left=1.6|1=<math>\frac{a+b}{a} = \frac{a}{a}+\frac{b}{a} = 1 + \frac{b}{a} = 1 + \frac{1}{\varphi}.</math>}}

Therefore,
{{bi|left=1.6|1=<math> 1 + \frac{1}{\varphi} = \varphi. </math>}}

Multiplying by <math>\varphi</math> gives
{{bi|left=1.6|1=<math>\varphi + 1 = \varphi^2</math>}}
which can be rearranged to
{{bi|left=1.6|1=<math>{\varphi}^2 - \varphi - 1 = 0.</math>}}

The [[Quadratic formula|quadratic formula]] yields two solutions:
{{bi|left=1.6|1=<math>\frac{1 + \sqrt5}{2} = 1.618033\dots</math> and <math>\frac{1 - \sqrt5}{2} = -0.618033\dots.</math>}}

Because <math>\varphi</math> is a ratio between positive quantities, <math>\varphi</math> is necessarily the positive root.<ref name=peters /> The negative root is in fact the negative inverse <math>-\frac{1}{\varphi}</math>, which shares many properties with the golden ratio.

==History==

According to [[Biography:Mario Livio|Mario Livio]],

{{blockquote|Some of the greatest mathematical minds of all ages, from [[Biography:Pythagoras|Pythagoras]] and [[Biography:Euclid|Euclid]] in {{wipe|Ancient Greece}}, through the medieval Italian mathematician [[Biography:Fibonacci|Leonardo of Pisa]] and the Renaissance astronomer [[Biography:Johannes Kepler|Johannes Kepler]], to present-day scientific figures such as Oxford physicist [[Biography:Roger Penrose|Roger Penrose]], have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/6 6]}}|title=''The Golden Ratio: The Story of Phi, the World's Most Astonishing Number''}}

{{wipe|Ancient Greece}} mathematicians first studied the golden ratio because of its frequent appearance in [[Geometry|geometry]];{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/4 4]|ps=: "... line division, which [[Biography:Euclid|Euclid]] defined for ... purely geometrical purposes ..."}} the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular [[Pentagram#Geometry|pentagram]]s and [[Pentagon|pentagon]]s.{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/7 7–8]}} According to one story, 5th-century BC mathematician [[Hippasus]] discovered that the golden ratio was neither a whole number nor a fraction (it is [[Irrational number|irrational]]), surprising [[Philosophy:Pythagoreanism|Pythagoreans]].{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/4 4–5]}} [[Biography:Euclid|Euclid]]'s ''[[Euclid's Elements|Elements]]'' ({{nowrap|c. 300 BC}}) provides several [[Theorem|propositions]] and their proofs employing the golden ratio,{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/78 78]}}{{efn|Euclid, ''[http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Elements]'', Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.}} and contains its first known definition which proceeds as follows:<ref name=hemenway />

{{blockquote|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.{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/3 3]}}{{efn|"῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."<ref name="fitzpatrick elements" />}}}}

[[File:Michael Maestlin.jpg|thumb|upright|Michael Maestlin, the first to write a decimal approximation of the ratio]]
The golden ratio was studied peripherally over the next millennium. [[Biography:Abu Kamil|Abu Kamil]] (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of [[Biography:Fibonacci|Fibonacci]] (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the [[Fibonacci number]]s.{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/88 88–96]}}

[[Biography:Luca Pacioli|Luca Pacioli]] named his book ''[[Divina proportione]]'' (1509) after the ratio; the book, largely plagiarized from [[Biography:Piero della Francesca|Piero della Francesca]], explored its properties including its appearance in some of the Platonic solids.<ref name=mackinnon />{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/131 131–132]}} [[Biography:Leonardo da Vinci|Leonardo da Vinci]], who illustrated Pacioli's book, called the ratio the ''sectio aurea'' ('golden section').<ref name=baravalle /> Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/134 134–135]}} Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as [[Biography:Rafael Bombelli|Rafael Bombelli]] solved geometric problems using the ratio.{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/141 141]}}

German mathematician Simon Jacob (d.&nbsp;1564) noted that consecutive Fibonacci numbers converge to the golden ratio;<ref name=schreiber /> this was rediscovered by [[Biography:Johannes Kepler|Johannes Kepler]] in 1608.{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/151 151–152]}} The first known decimal approximation of the (inverse) golden ratio was stated as "about <math>0.6180340</math>" in 1597 by [[Biography:Michael Maestlin|Michael Maestlin]] of the [[Organization:University of Tübingen|University of Tübingen]] in a letter to Kepler, his former student.<ref name=mactutor /> The same year, Kepler wrote to Maestlin of the [[Kepler triangle]], which combines the golden ratio with the [[Pythagorean theorem]]. Kepler said of these:{{blockquote|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 mass of gold, the second we may call a precious jewel.<ref name=fink />}}

Eighteenth-century mathematicians [[Biography:Abraham de Moivre|Abraham de Moivre]], [[Biography:Nicolaus I Bernoulli|Nicolaus I Bernoulli]], and [[Biography:Leonhard Euler|Leonhard Euler]] used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by [[Biography:Jacques Philippe Marie Binet|Jacques Philippe Marie Binet]], for whom it was named "Binet's formula".<ref name="beutelspacher petri" /> Martin Ohm first used the German term ''goldener Schnitt'' ('golden section') to describe the ratio in 1835.{{sfn|Herz-Fischler|1998|pp=167–170}} James Sully used the equivalent English term in 1875.{{sfn|Posamentier|Lehmann|2011|p=8}}

By 1910, inventor Mark Barr began using the Greek letter phi {{nowrap|(<math>\varphi</math>)}} as a [[Symbol|symbol]] for the golden ratio.{{sfn|Posamentier|Lehmann|2011|p=285}}{{efn|After Classical Greek sculptor Phidias (c.&nbsp;490–430&nbsp;BC);<ref name=cook /> Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.<ref name=barr />}} It has also been represented by tau {{nowrap|(<math>\tau</math>),}} the first letter of the [[Social:Ancient Greek|ancient Greek]] τομή ('cut' or 'section').{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/5 5]}}

[[File:Dan Shechtman in 1985.jpg|thumb|right|Dan Shechtman demonstrates quasicrystals at the NIST in 1985 using a Zometoy model.]]
The [[Engineering:Zome|zome]] construction system, developed by Steve Baer in the late 1960s, is based on the [[Icosahedral symmetry|symmetry system]] of the [[Icosahedron|icosahedron]]/[[Dodecahedron|dodecahedron]], and uses the golden ratio ubiquitously. Between 1973 and 1974, [[Biography:Roger Penrose|Roger Penrose]] developed [[Penrose tiling]], a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.<ref name=gardner /> This gained in interest after [[Biography:Dan Shechtman|Dan Shechtman]]'s Nobel-winning 1982 discovery of [[Physics:Quasicrystal|quasicrystal]]s with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.<ref name=quasicrystals />

==Mathematics==

===Irrationality===
The golden ratio is an [[Irrational number|irrational number]]. Below are two short proofs of irrationality:

====Contradiction from an expression in lowest terms====
[[File:Whirling squares.svg|thumb|right|If {{mvar|φ}} were [[Rational number|rational]], then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the positive integers have a lower bound, so {{mvar|φ}} cannot be rational.]]

This is a [[Proof by infinite descent|proof by infinite descent]]. Recall that:
{{bi|left=1.6|1=the whole is the longer part plus the shorter part;<br />
the whole is to the longer part as the longer part is to the shorter part.}}

If we call the whole <math>n</math> and the longer part <math>m,</math> then the second statement above becomes

{{bi|left=1.6|1=<math>n</math> is to <math>m</math> as <math>m</math> is to <math>n-m.</math>}}

To say that the golden ratio <math>\varphi</math> is rational means that <math>\varphi</math> is a fraction <math>n/m</math> where <math>n</math> and <math>m</math> are integers. We may take <math>n/m</math> to be in lowest terms and <math>n</math> and <math>m</math> to be positive. But if <math>n/m</math> is in lowest terms, then the equally valued <math>m/(n-m)</math> is in still lower terms. That is a contradiction that follows from the assumption that <math>\varphi</math> is rational.

====By irrationality of {{math|{{sqrt|5}}}}====
Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the [[Closure (mathematics)|closure]] of rational numbers under addition and multiplication. If <math>\varphi = \tfrac12(1 + \sqrt5)</math> is rational, then <math>2\varphi - 1 = \sqrt5</math> is also rational, which is a contradiction if it is already known that the square root of all non-[[Square number|square]] [[Natural number|natural number]]s are irrational.

===Minimal polynomial===
[[File:Golden ratio parabolas.png|thumb|The golden ratio {{mvar|φ}} and its negative reciprocal {{math|−''φ''<sup>−1</sup>}} are the two roots of the [[Quadratic function|quadratic polynomial]] {{math|''x''<sup>2</sup> − ''x'' − 1}}. The golden ratio's negative {{math|−''φ''}} and reciprocal {{math|''φ''<sup>−1</sup>}} are the two roots of the quadratic polynomial {{math|''x''<sup>2</sup> + ''x'' − 1}}.]]
The golden ratio is also an [[Algebraic number|algebraic number]] and even an [[Algebraic integer|algebraic integer]]. It has [[Minimal polynomial (field theory)|minimal polynomial]]
{{bi|left=1.6|1=<math>x^2 - x - 1.</math>}}

This [[Quadratic function|quadratic polynomial]] has two [[Zero of a function|roots]], <math>\varphi</math> and <math>-\varphi^{-1}.</math>

The golden ratio is also closely related to the polynomial

{{bi|left=1.6|1=<math>x^2 + x - 1,</math>}}

which has roots <math>-\varphi</math> and <math>\varphi^{-1}.</math> As the root of a quadratic polynomial, the golden ratio is a [[Constructible number|constructible number]].<ref name=constructions />

===Golden ratio conjugate<!--'Golden ratio conjugate' redirects here--> and powers===
The [[Conjugate (square roots)|conjugate root]] to the minimal polynomial <math>x^2-x-1</math> is

{{bi|left=1.6|1=<math>-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt5}{2} = -0.618033\dots.</math>}}

The absolute value of this quantity {{nowrap|(<math>0.618\ldots</math>)}} corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, <math>b/a</math>).

This illustrates the unique property of the golden ratio among positive numbers, that
{{bi|left=1.6|1=<math>\frac1\varphi = \varphi - 1,</math>}}

or its inverse:
{{bi|left=1.6|1=<math>\frac1{1/\varphi} = \frac1\varphi + 1.</math>}}

The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with <math>\varphi</math>:

{{bi|left=1.6|1=<math>\begin{align}
\varphi^2 &= \varphi + 1 = 2.618033\dots, \\[5mu]
\frac1\varphi &= \varphi - 1 = 0.618033\dots.
\end{align}</math>}}

The sequence of powers of <math>\varphi</math> contains these values <math>0.618033\ldots,</math> <math>1.0,</math> <math>1.618033\ldots,</math> <math>2.618033\ldots;</math> more generally,
any power of <math>\varphi</math> is equal to the sum of the two immediately preceding powers:
{{bi|left=1.6|1=<math>\varphi^n = \varphi^{n-1} + \varphi^{n-2} = \varphi \cdot \operatorname{F}_n + \operatorname{F}_{n-1}.</math>}}

As a result, one can easily decompose any power of <math>\varphi</math> into a multiple of <math>\varphi</math> and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of <math>\varphi</math>:

If <math> \lfloor n/2 - 1 \rfloor = m,</math> then:
{{bi|left=1.6|1=<math>\begin{align}
\varphi^n &= \varphi^{n-1} + \varphi^{n-3} + \cdots + \varphi^{n-1-2m} + \varphi^{n-2-2m} \\[5mu]
\varphi^n - \varphi^{n-1} &= \varphi^{n-2}.
\end{align}</math>}}

===Continued fraction and square root===
[[File:Golden mean.png|right|thumb|Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers]]

The formula <math>\varphi = 1 + 1/\varphi</math> can be expanded recursively to obtain a [[Continued fraction|continued fraction]] for the golden ratio:<ref name="Concrete Abstractions" />
{{bi|left=1.6|1=<math>\varphi = [1; 1, 1, 1, \dots] = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>}}

It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
{{bi|left=1.6|1=<math>\varphi^{-1} = [0; 1, 1, 1, \dots] = 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>}}

The convergents of these continued fractions {{nowrap|(<math>1/1,</math>}} <math>2/1,</math> <math>2/1,</math> <math>3/2,</math> <math>5/3,</math> <math>8/5,</math> <math>13/8,</math> ... or <math>1/1,</math> <math>1/2,</math> <math>2/3,</math> <math>3/5,</math> <math>5/8,</math> {{nowrap|<math>8/13,</math> ...)}} are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the [[Hurwitz's theorem (number theory)|Hurwitz inequality]] for [[Diophantine approximation]]s, which states that for every irrational <math>\xi</math>, there are infinitely many distinct fractions <math>p/q</math> such that,
<blockquote><math>\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}.</math></blockquote>
This means that the constant <math>\sqrt{5}</math> cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.<ref name=hardy />

A [[Nested radical|continued square root]] form for <math>\varphi</math> can be obtained from <math>\varphi^2 = 1 + \varphi</math>, yielding:<ref name=sizer/>
{{bi|left=1.6|1=<math>\varphi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}.</math>}}

===Relationship to Fibonacci and Lucas numbers===
{{details|Fibonacci number#Relation to the golden ratio}}

{{multiple image|align=right|direction=vertical|image1=Fibonacci Spiral.svg|image2=Lucas number spiral.svg|width=250|footer=A [[Fibonacci sequence|Fibonacci spiral]] (top) which approximates the [[Golden spiral|golden spiral]], using [[Fibonacci number|Fibonacci sequence]] square sizes up to {{math|21}}. A different approximation to the golden spiral is generated (bottom) from stacking squares whose lengths of sides are numbers belonging to the sequence of Lucas numbers, here up to {{math|76}}.}}

[[Fibonacci number]]s and [[Lucas number]]s have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence <math>0,1</math>:

{{bi|left=1.6|1=<math>0,</math> <math>1,</math> <math>1,</math> <math>2,</math> <math>3,</math> <math>5,</math> <math>8,</math> <math>13,</math> <math>21,</math> <math>34,</math> <math>55,</math> <math>89,</math> <math>\ldots</math>({{OEIS2C|id=A000045}}).}}

The sequence of Lucas numbers (not to be confused with the generalized [[Lucas sequence]]s, of which this is part) is like the Fibonacci sequence, in which each term is the sum of the previous two, however instead starts with <math>2,1</math>:

{{bi|left=1.6|1=<math>2,</math> <math>1,</math> <math>3,</math> <math>4,</math> <math>7,</math> <math>11,</math> <math>18,</math> <math>29,</math> <math>47,</math> <math>76,</math> <math>123,</math> <math>199,</math> <math>\ldots</math>({{OEIS2C|id=A000032}}).}}

Exceptionally, the golden ratio is equal to the [[Limit of a sequence|limit]] of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:<ref name=tattersall />
{{bi|left=1.6|1=<math>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\lim_{n\to\infty}\frac{L_{n+1}}{L_n}=\varphi.</math>}}

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates <math>\varphi</math>.

For example, <math>\frac{F_{16}}{F_{15}} = \frac{987}{610} = 1.6180327\ldots, </math> and <math>\frac{L_{16}}{L_{15}} = \frac{2207}{1364} = 1.6180351\ldots.</math>

These approximations are alternately lower and higher than <math>\varphi,</math> and converge to <math>\varphi</math> as the Fibonacci and Lucas numbers increase.

[[Closed-form expression]]s for the Fibonacci and Lucas sequences that involve the golden ratio are:

{{bi|left=1.6|1=<math>F\left(n\right)
 = {{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}},</math>}}
{{bi|left=1.6|1=<math>L\left(n\right)
= \varphi^n + (- \varphi)^{-n}= \left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n\, .</math>}}

Combining both formulas above, one obtains a formula for <math>\varphi^n</math> that involves both Fibonacci and Lucas numbers:
{{bi|left=1.6|1=<math>\varphi^n = {{L_n + F_n \sqrt{5}} \over 2}\, .</math>}}

Between Fibonacci and Lucas numbers one can deduce <math>L_{2n} = 5 F_n^2 + 2(-1)^n = L_n^2 - 2(-1)^n,</math> which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five:
{{bi|left=1.6|1=<math>\lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}.</math>}}
Indeed, much stronger statements are true:
:<math> \vert L_n - \sqrt{5} F_n \vert = \frac{2}{\varphi^n} \to 0, </math>
:<math> (L_{3n}/2)^2 = 5 (F_{3n}/2)^2 + (-1)^n. </math>

These values describe <math>\varphi</math> as a [[Fundamental unit (number theory)|fundamental unit]] of the [[Algebraic number field|algebraic number field]] <math>\mathbb{Q}(\sqrt5)</math>.

Successive powers of the golden ratio obey the Fibonacci [[Recurrence relation|recurrence]], i.e. <math>\varphi^{n+1} = \varphi^n + \varphi^{n-1}.</math>

The reduction to a linear expression can be accomplished in one step by using:
{{bi|left=1.6|1=<math>\varphi^n = F_n \varphi + F_{n-1}.</math>}}

This identity allows any polynomial in <math>\varphi</math> to be reduced to a linear expression, as in:

{{bi|left=1.6|1=<math>\begin{align}
3\varphi^3 - 5\varphi^2 + 4
&= 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\[5mu]
&= 3[(\varphi + 1) + \varphi] - 5(\varphi + 1) + 4 \\[5mu]
&= \varphi + 2 \approx 3.618033.
\end{align}</math>}}

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by [[Series (mathematics)|infinite summation]]:
{{bi|left=1.6|1=<math>\sum_{n=1}^{\infty}|F_n\varphi-F_{n+1}| = \varphi.</math>}}

In particular, the powers of <math>\varphi</math> themselves round to Lucas numbers (in order, except for the first two powers, <math>\varphi^0</math> and <math>\varphi</math>, are in reverse order):

{{bi|left=1.6|1=<math>\begin{align}
\varphi^0 &= 1, \\[5mu]
\varphi^1 &= 1.618033989\ldots \approx 2, \\[5mu]
\varphi^2 &= 2.618033989\ldots \approx 3, \\[5mu]
\varphi^3 &= 4.236067978\ldots \approx 4, \\[5mu]
\varphi^4 &= 6.854101967\ldots \approx 7,
\end{align}</math>}}

and so forth.<ref name=parker4d /> The Lucas numbers also directly generate powers of the golden ratio; for <math>n \ge 2</math>:
{{bi|left=1.6|1=<math> \varphi^n = L_n - (- \varphi)^{-n}.</math>}}

Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of ''third'' consecutive Fibonacci numbers equals a Lucas number, that is <math>L_n = F_{n-1}+F_{n+1}</math>; and, importantly, that <math> L_n = \frac{F_{2n}}{F_n}</math>.

Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the [[Golden spiral|golden spiral]] (which is a special form of a [[Logarithmic spiral|logarithmic spiral]]) using quarter-circles with radii from these sequences, differing only slightly from the ''true'' golden logarithmic spiral. ''Fibonacci spiral'' is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.

===Geometry===
The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the [[Pentagon|pentagon]], and extends to form part of the coordinates of the vertices of a [[Regular dodecahedron#Cartesian coordinates|regular dodecahedron]], as well as those of a [[5-cell#Coordinates|5-cell]]. It features in the [[Kepler triangle]] and Penrose tilings too, as well as in various other polytopes.

====Construction====
{{multiple image|align=right|direction=vertical|image1=Goldener Schnitt Konstr beliebt.svg|image2=Goldener Schnitt (Äußere Teilung).svg|width=175|footer=Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.}}

'''Dividing by interior division'''
# Having a line segment <math>AB,</math> construct a perpendicular <math>BC</math> at point <math>B,</math> with <math>BC</math> half the length of <math>AB.</math> Draw the [[Hypotenuse|hypotenuse]] <math>AC.</math>
# Draw an arc with center <math>C</math> and radius <math>BC.</math> This arc intersects the hypotenuse <math>AC</math> at point <math>D.</math>
# Draw an arc with center <math>A</math> and radius <math>AD.</math> This arc intersects the original line segment <math>AB</math> at point <math>S.</math> Point <math>S</math> divides the original line segment <math>AB</math> into line segments <math>AS</math> and <math>SB</math> with lengths in the golden ratio.

'''Dividing by exterior division'''
# Draw a line segment <math>AS</math> and construct off the point <math>S</math> a segment <math>SC</math> perpendicular to <math>AS</math> and with the same length as <math>AS.</math>
# Do bisect the line segment <math>AS</math> with <math>M.</math>
# A circular arc around <math>M</math> with radius <math>MC</math> intersects in point <math>B</math> the straight line through points <math>A</math> and <math>S</math> (also known as the extension of <math>AS</math>). The ratio of <math>AS</math> to the constructed segment <math>SB</math> is the golden ratio.
Application examples you can see in the articles [[Pentagon#Side length is given|Pentagon with a given side length]], [[Decagon#Construction|Decagon with given circumcircle and Decagon with a given side length]].

Both of the above displayed different [[Algorithm|algorithm]]s produce geometric constructions that determine two aligned [[Line segment|line segment]]s where the ratio of the longer one to the shorter one is the golden ratio.

====Golden angle====
{{main|Golden angle}}
[[File:Golden Angle.svg|175px|thumb|{{center|{{math|''g'' ≈ 137.508°}}}}]]
When two angles that make a full circle have measures in the golden ratio, the smaller is called the ''golden angle'', with measure <math display=inline>g\colon</math>

{{bi|left=1.6|1=<math>\begin{align}
\frac{2\pi - g}{g} &= \frac{2\pi}{2\pi - g} = \varphi,  \\[8mu]
2\pi - g &= \frac{2\pi}{\varphi} \approx 222.5^\circ, \\[8mu]
g &= \frac{2\pi}{\varphi^2} \approx 137.5^\circ.
\end{align}</math>}}

This angle occurs in [[Biology:Phyllotaxis|patterns of plant growth]] as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.<ref name=phyllotaxis />

====Pentagonal symmetry system====

=====Pentagon and pentagram=====
[[File:Pentagram-phi.svg|right|thumb|A pentagram colored to distinguish its line segments of different lengths. The four
lengths are in golden ratio to one another.]]

In a [[Pentagon#Regular pentagons|regular pentagon]] the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying [[Ptolemy's theorem]] to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are <math>a,</math> and short edges are <math>b,</math> then Ptolemy's theorem gives <math>a^2 = b^2 + ab.</math> Dividing both sides by <math>ab</math> yields (see {{slink|#Calculation}} above),
{{bi|left=1.6|1=<math>\frac ab = \frac{a + b}{a} = \varphi.</math>}}

The diagonal segments of a pentagon form a [[Pentagram|pentagram]], or five-pointed [[Star polygon|star polygon]], whose geometry is quintessentially described by <math>\varphi</math>. Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is <math>\varphi,</math> as the four-color illustration shows.

Pentagonal and pentagrammic geometry permits us to calculate the following values for <math>\varphi</math>:
{{bi|left=1.6|1=<math>\begin{align}
\varphi &= 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ, \\[5mu]
\varphi &= \tfrac12\csc(\pi/10) = \tfrac12\csc 18^\circ, \\[5mu]
\varphi &= 2\cos(\pi/5)=2\cos 36^\circ, \\[5mu]
\varphi &= 2\sin(3\pi/10)=2\sin 54^\circ.
\end{align}</math>}}

=====Golden triangle and golden gnomon=====
{{main|Golden triangle (mathematics)}}
[[File:Golden triangle (math).svg|235px|right|thumb|A [[Golden triangle (mathematics)|golden triangle]] {{mvar|ABC}} can be subdivided by an angle bisector into a smaller golden triangle {{mvar|CXB}} and a golden gnomon {{mvar|XAC}}.]]
The triangle formed by two diagonals and a side of a regular pentagon is called a ''golden triangle'' or ''sublime triangle''. It is an acute [[Isosceles triangle|isosceles triangle]] with apex angle 36° and base angles 72°.<ref name=fletcher /> Its two equal sides are in the golden ratio to its base.<ref name=loeb /> The triangle formed by two sides and a diagonal of a regular pentagon is called a ''golden gnomon''. It is an obtuse isosceles triangle with apex angle 108° and base angle 36°. Its base is in the golden ratio to its two equal sides.<ref name=loeb /> The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a [[Pentagram|regular pentagram]] are golden triangles,<ref name=loeb /> as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.<ref name=miller />

Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.<ref name=loeb />

If the apex angle of the golden gnomon is [[Angle trisection|trisected]], the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.<ref name=loeb />

=====Penrose tilings=====
{{main|Penrose tiling}}
[[File:Kite Dart.svg|thumb|The kite and dart tiles of the Penrose tiling. The colored arcs divide each edge in the golden ratio; when two tiles share an edge, their arcs must match.]]
The golden ratio appears prominently in the ''Penrose tiling'', a family of [[Aperiodic tiling|aperiodic tiling]]s of the plane developed by [[Biography:Roger Penrose|Roger Penrose]], inspired by [[Biography:Johannes Kepler|Johannes Kepler]]'s remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.<ref name="Tilings and Patterns" /> Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
*Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.<ref name=pentaplexity />
*The kite and dart Penrose tiling uses [[Kite (geometry)|kites]] with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.<ref name="Tilings and Patterns" />
*The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context ''Robinson triangles'', can be used as the prototiles for a form of the Penrose tiling.<ref name="Tilings and Patterns" /><ref name=robinson />
*The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals <math>1:\varphi</math>, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.<ref name="Tilings and Patterns" />

{{multiple image
|align=left
|image1=Penrose Tiling (P1).svg|caption1=Original four-tile Penrose tiling
|image2=PenroseTilingFilled4.svg|caption2=Rhombic Penrose tiling
|total_width=540}}

{{clear|left}}

====In triangles and quadrilaterals====

=====Odom's construction=====
[[File:Odom.svg|thumb|upright|Odom's construction: {{math|1=AB&thinsp;:&thinsp;BC = AC&thinsp;:&thinsp;AB = ''φ''&thinsp;:&thinsp;1}}]]

George Odom found a construction for <math>\varphi</math> involving an [[Equilateral triangle|equilateral triangle]]: if the line segment joining the midpoints of two sides is extended to intersect the circumcircle, then the two midpoints and the point of intersection with the circle are in golden proportion.<ref name=triangleconstruction />

=====Kepler triangle=====
{{main|Kepler triangle}}
{{multiple image
|image1=Kepler triangle.svg|caption1=Geometric progression of areas of squares on the sides of a Kepler triangle
|image2=Kepler and the Deathly Hallows.svg|caption2=An isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length
|total_width=480}}
The ''Kepler triangle'', named after [[Biography:Johannes Kepler|Johannes Kepler]], is the unique [[Right triangle|right triangle]] with sides in [[Geometric progression|geometric progression]]:
{{bi|left=1.6|1=<math>1\mathbin{:}\sqrt\varphi\mathbin{:}\varphi</math>.}}
These side lengths are the three Pythagorean means of the two numbers <math>\varphi \pm 1</math>. The three squares on its sides have areas in the golden geometric progression <math>1\mathbin{:}\varphi\mathbin{:}\varphi^2</math>.

Among isosceles triangles, the ratio of inradius to side length is maximized for the triangle formed by two [[Reflection (mathematics)|reflected copies]] of the Kepler triangle, sharing the longer of their two legs.<ref name="Liber mensurationum" /> The same isosceles triangle maximizes the ratio of the radius of a [[Semicircle|semicircle]] on its base to its [[Perimeter|perimeter]].<ref name=bruce />

For a Kepler triangle with smallest side length <math>s</math>, the [[Area|area]] and acute internal angles are:
{{bi|left=1.6|1=<math>\begin{align}
A &= \tfrac{s^2}{2}\sqrt\varphi, \\[5mu]
\theta &= \sin^{-1}\frac{1}{\varphi}\approx 38.1727^\circ, \\[5mu]
\theta &= \cos^{-1}\frac{1}{\varphi}\approx 51.8273^\circ.
\end{align}</math>}}

=====Golden rectangle=====
{{main|Golden rectangle}}
[[File:Golden Rectangle Construction.svg|175px|thumb|To construct a golden rectangle [[Straightedge and compass construction|with only a straightedge and compass]] in four simple steps:
{|
|-
|Draw a square.
|-
|Draw a line from the midpoint of one side of the square to an opposite corner.
|-
|Use that line as the radius to draw an arc that defines the height of the rectangle.
|-
|Complete the golden rectangle.
|-
|}
]]

The golden ratio proportions the adjacent side lengths of a ''golden rectangle'' in <math>1:\varphi</math> ratio.{{sfn|Posamentier|Lehmann|2011|p=11}} Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in <math>\varphi</math> ratio. They can be generated by ''golden spirals'', through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the [[Regular icosahedron|icosahedron]] as well as in the [[Regular dodecahedron|dodecahedron]] (see section below for more detail).<ref name=BurgerStarbird />

=====Golden rhombus=====
{{main|Golden rhombus}}
A ''golden rhombus'' is a [[Rhombus|rhombus]] whose diagonals are in proportion to the golden ratio, most commonly <math>1:\varphi</math>.<ref name=hexecontahedron /> For a rhombus of such proportions, its acute angle and obtuse angles are:

{{bi|left=1.6|1=<math>\begin{align}
\alpha &= 2\arctan{1\over\varphi}\approx63.43495^\circ, \\[5mu]
\beta  &= 2\arctan\varphi=\pi-\arctan2 = \arctan1+\arctan3 \approx116.56505^\circ.
\end{align}</math>}}

The lengths of its short and long diagonals <math>d</math> and <math>D</math>, in terms of side length <math>a</math> are:

{{bi|left=1.6|1=<math>\begin{align}
d &= {2a\over\sqrt{2+\varphi}}=2\sqrt{{3-\varphi}\over5}a\approx1.05146a, \\[5mu]
D &= 2\sqrt{{2+\varphi}\over5}a\approx1.70130a.
\end{align}</math>}}

Its area, in terms of <math>a</math>,and <math>d</math>:

{{bi|left=1.6|1=<math>\begin{align}
A &= (\sin(\arctan2))~a^2 = {2\over\sqrt5}~a^2 \approx 0.89443a^2, \\[5mu]
A &= {{\varphi}\over2}d^2\approx 0.80902d^2.
\end{align}</math>}}

Its inradius, in terms of side <math>a</math>:

{{bi|left=1.6|1=<math>r=\frac{a}{\sqrt{5}}.</math>}}

Golden rhombi form the faces of the [[Rhombic triacontahedron|rhombic triacontahedron]], the two golden rhombohedra, the [[Bilinski dodecahedron]],<ref name="golden rhombohedra" /> and the [[Rhombic hexecontahedron|rhombic hexecontahedron]].<ref name=hexecontahedron />

====Golden spiral====
{{main|Golden spiral}}
[[File:FakeRealLogSpiral.svg|thumb|The [[Golden spiral|golden spiral]] (red) and its approximation by quarter-circles (green), with overlaps shown in yellow]]
[[File:Golden triangle and Fibonacci spiral.svg|175px|thumb|A [[Logarithmic spiral|logarithmic spiral]] whose radius grows by the golden ratio per 108° of turn, surrounding nested golden isosceles triangles. This is a different spiral from the [[Golden spiral|golden spiral]], which grows by the golden ratio per 90° of turn.<ref name=loeb-varney />]]
Logarithmic spirals are self-similar spirals where distances covered per turn are in [[Geometric progression|geometric progression]]. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the [[Golden spiral|golden spiral]]. These spirals can be approximated by quarter-circles that grow by the golden ratio,<ref name=quarter-circles /> or their approximations generated from Fibonacci numbers,<ref name=diedrichs /> often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the [[Polar coordinate system|polar equation]] with <math>(r,\theta)</math>:
{{bi|left=1.6|1=<math>r = \varphi^{2\theta/\pi}.</math>}}

Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each 108° that it turns, instead of the 90° turning angle of the golden spiral.<ref name=loeb-varney /> Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.<ref name=quarter-circles />

====In the dodecahedron and icosahedron====
[[File:Dodecahedron vertices.png|240px|right|thumb|
{|
|- valign=top
| colspan=2 |
|- valign=top
|Cartesian coordinates of the [[Dodecahedron|dodecahedron]] :
<br />
|- valign=top
|<span style="color:#dd4400">{{math|(±1, ±1, ±1)}}</span>
|- valign=top
|<span style="color:#007722">{{math|(0, ±'''φ''', ±{{sfrac|1|'''φ'''}})}}</span>
|- valign=top
|<span style="color:#0011bb">{{math|(±{{sfrac|1|'''φ'''}}, 0, ±'''φ''')}}</span>
|- valign=top
| <span style="color:#cc0055">{{math|(±'''φ''', ±{{sfrac|1|'''φ'''}}, 0)}}</span>
|- valign=top
|A nested cube inside the dodecahedron is represented with <span style="color:#dd4400">dotted</span> lines.
| colspan=2 |
|}
]]

The [[Regular dodecahedron|regular dodecahedron]] and its [[Dual polyhedron|dual polyhedron]] the [[Regular icosahedron|icosahedron]] are [[Platonic solid]]s whose dimensions are related to the golden ratio. An icosahedron is made of <math>12</math> regular pentagonal faces, whereas the icosahedron is made of <math>20</math> [[Equilateral triangle|equilateral triangle]]s; both with <math>30</math> [[Edge (geometry)|edges]].<ref name ="Regular dodecahedron">{{harvtxt|Livio|2002|pp=70–72}}</ref>

For a dodecahedron of side <math>a</math>, the [[Radius|radius]] of a circumscribed and inscribed sphere, and [[Midsphere|midradius]] are (<math>r_u, </math> <math>r_i,</math> and <math>r_m,</math> respectively):

{{bi|left=1.6|1=<math>r_u = a\, \frac{\sqrt{3}\varphi}{2},</math> <math>r_i = a\, \frac{\varphi^2}{2 \sqrt{3-\varphi}},</math> and <math>r_m = a\, \frac{\varphi^2}{2}.</math>}}

While for an icosahedron of side <math>a</math>, the radius of a circumscribed and inscribed sphere, and [[Midsphere|midradius]] are:

{{bi|left=1.6|1=<math>r_u = a\frac{\sqrt{\varphi \sqrt{5}}}{2},</math> <math>r_i = a\frac{\varphi^2}{2 \sqrt{3}},</math> and <math>r_m = a\frac{\varphi}{2}.</math>}}

The volume and surface area of the dodecahedron can be expressed in terms of <math>\varphi</math>:
{{bi|left=1.6|1=<math>A_d = \frac{15\varphi}{\sqrt{3-\varphi}}</math> and <math>V_d = \frac{5\varphi^3}{6-2\varphi}</math>.}}

As well as for the icosahedron:

{{bi|left=1.6|1=<math>A_i = 20\frac{\varphi^{2}}{2}</math> and <math>V_i = \frac{5}{6}(1 + \varphi).</math>}}

[[File:Icosahedron-golden-rectangles.svg|175px|right|thumb|Three golden rectangles touch all of the {{math|12}} vertices of a [[Regular icosahedron|regular icosahedron]].]]

These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving <math>\varphi</math>. The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the [[Cyclic permutation|cyclic permutation]]s of:

{{bi|left=1.6|1=<math>(0,\pm1,\pm\varphi)</math>, <math>(\pm1,\pm\varphi,0)</math>, <math>(\pm\varphi,0,\pm1).</math>}}

Sets of three golden rectangles intersect [[Perpendicular|perpendicular]]ly inside dodecahedra and icosahedra, forming [[Borromean rings]].<ref name=borromean /><ref name=BurgerStarbird /> In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain <math>12</math> vertices of the icosahedron, or equivalently, intersect the centers of <math>12</math> of the dodecahedron's faces.<ref name="Regular dodecahedron" />

A [[Cube|cube]] can be [[Inscribed figure|inscribed]] in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is <math>\tfrac{2}{2+\varphi}</math> times that of the dodecahedron's.<ref name=hume /> In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in <math>\varphi : \varphi^{2}</math> ratio. On the other hand, the [[Octahedron|octahedron]], which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's <math>12</math> vertices touch the <math>12</math> edges of an octahedron at points that divide its edges in golden ratio.<ref name="59 Icosahedra" />

Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the [[Compound of five cubes|compound of five cubes]], [[Compound of five octahedra|compound of five octahedra]], [[Compound of five tetrahedra|compound of five tetrahedra]], the [[Compound of ten tetrahedra|compound of ten tetrahedra]], [[Rhombic triacontahedron|rhombic triacontahedron]], [[Icosidodecahedron|icosidodecahedron]], [[Truncated icosahedron|truncated icosahedron]], [[Truncated dodecahedron|truncated dodecahedron]], and [[Rhombicosidodecahedron|rhombicosidodecahedron]], [[Rhombic enneacontahedron|rhombic enneacontahedron]], and Kepler-Poinsot polyhedra, and [[Rhombic hexecontahedron|rhombic hexecontahedron]]. In four dimensions, the dodecahedron and icosahedron appear as faces of the [[120-cell|120-cell]] and [[600-cell|600-cell]], which again have dimensions related to the golden ratio.

===Other properties===

The golden ratio's ''decimal expansion'' can be calculated via root-finding methods, such as [[Newton's method]] or [[Halley's method]], on the equation <math>x^2-x-1=0</math> or on <math>x^2-5=0</math> (to compute <math>\sqrt{5}</math> first). The time needed to compute <math>n</math> digits of the golden ratio using Newton's method is essentially <math>O(M(n))</math>, where <math>M(n)</math> is [[Multiplication algorithm#Computational complexity|the time complexity of multiplying]] two <math>n</math>-digit numbers.<ref name=muller /> This is considerably faster than known algorithms for [[Pi|<math>\pi</math>]] and [[E (mathematical constant)|<math>e</math>]]. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers <math>F_{25001}</math> and <math>F_{25000},</math> each over <math>5000</math> digits, yields over <math>10{,}000</math> significant digits of the golden ratio. The decimal expansion of the golden ratio <math>\varphi</math><ref name=a001622 /> has been calculated to an accuracy of ten trillion {{nowrap|1=(<math>1 \times 10^{13} = 10{,}000{,}000{,}000{,}000</math>)}} digits.<ref name=ycruncher />

In the [[Complex plane|complex plane]], the fifth [[Root of unity|roots of unity]] <math>z = e^{2\pi k i/5}</math> (for an integer <math display=inline>k</math>) satisfying <math>z^5 = 1</math> are the vertices of a pentagon. They do not form a [[Ring (mathematics)|ring]] of [[Quadratic integer|quadratic integer]]s, however the sum of any fifth root of unity and its [[Complex conjugate|complex conjugate]], <math>z + \bar z,</math> ''is'' a quadratic integer, an element of <math display=inline>\mathbb{Z}[\varphi].</math> Specifically,

{{bi|left=1.6|1=<math>\begin{align}
e^{0} + e^{-0} &= 2, \\[5mu]
e^{2\pi i / 5} + e^{-2\pi i / 5} &= \varphi^{-1} = -1 + \varphi, \\[5mu]
e^{4\pi i / 5} + e^{-4\pi i / 5} &= -\varphi.
\end{align}</math>}}

This also holds for the remaining tenth roots of unity satisfying <math>z^{10} = 1,</math>

{{bi|left=1.6|1=<math>\begin{align}
e^{\pi i} + e^{-\pi i} &= -2, \\[5mu]
e^{\pi i / 5} + e^{-\pi i / 5} &= \varphi, \\[5mu]
e^{3\pi i / 5} + e^{-3\pi i / 5} &= -\varphi^{-1} = 1 - \varphi.
\end{align}</math>}}

For the [[Gamma function|gamma function]] <math>\Gamma</math>, the only solutions to the equation <math>\Gamma(z-1) = \Gamma(z+1)</math> are <math>z = \varphi</math> and <math>z = -\varphi^{-1}</math>.

When the golden ratio is used as the base of a [[Numeral system|numeral system]] (see [[Golden ratio base|golden ratio base]], sometimes dubbed ''phinary'' or <math>\varphi</math>''-nary''), [[Quadratic integer|quadratic integer]]s in the ring <math>\mathbb{Z}[\varphi]</math> – that is, numbers of the form <math>a + b\varphi</math> for <math>a, b \in \mathbb{Z}</math> – have [[Repeating decimal|terminating]] representations, but rational fractions have non-terminating representations.

The golden ratio also appears in [[Hyperbolic geometry|hyperbolic geometry]], as the maximum distance from a point on one side of an [[Ideal triangle|ideal triangle]] to the closer of the other two sides: this distance, the side length of the [[Equilateral triangle|equilateral triangle]] formed by the points of tangency of a circle inscribed within the ideal triangle, is <math>4\log(\varphi).</math><ref name=horocycle />

The golden ratio appears in the theory of [[Modular form|modular functions]] as well. For <math>\left|q\right|<1</math>, let
{{bi|left=1.6|1=<math>R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\ddots}}}}.</math>}}
Then
{{bi|left=1.6|1=<math>R(e^{-2\pi})=\sqrt{\varphi\sqrt5}-\varphi ,\quad R(-e^{-\pi})=\varphi^{-1}-\sqrt{2-\varphi^{-1}}</math>}}
and
{{bi|left=1.6|1=<math>R(e^{-2\pi i/\tau})=\frac{1-\varphi R(e^{2\pi i\tau})}{\varphi+R(e^{2\pi i\tau})}</math>}}
where <math>\operatorname{Im}\tau>0</math> and <math>(e^z)^{1/5}</math> in the continued fraction should be evaluated as <math>e^{z/5}</math>. The function <math>\tau\mapsto R(e^{2\pi i\tau})</math> is invariant under <math>\Gamma (5)</math>, a [[Modular group#Congruence subgroups|congruence subgroup of the modular group]]. Also for [[Positive real numbers|positive real numbers]] <math>a, b \in \mathbb{R}^+</math> and <math>ab = \pi^2,</math> then<ref name=rrcf />

{{bi|left=1.6|1=<math>\begin{align}
\Bigl(\varphi+R{\bigl(e^{-2a}\bigr)}\Bigr)\Bigl(\varphi+R{\bigl(e^{-2b}\bigr)}\Bigr)&=\varphi\sqrt5, \\[5mu]
\Bigl(\varphi^{-1}-R{\bigl({-e^{-a}}\bigr)}\Bigr)\Bigl(\varphi^{-1}-R{\bigl({-e^{-b}}\bigr)}\Bigr)&=\varphi^{-1}\sqrt5.
\end{align}</math>}}

<math>\varphi</math> is a [[Pisot–Vijayaraghavan number]].<ref name=duffin />

==Applications and observations==
[[File:GoldenSquare_6.png|thumb|Rhythms apparent to the eye: rectangles in aspect ratios {{math|''φ''}} (left, middle) and {{math|''φ''<sup>2</sup>}} (right side) tile the square.]]

===Architecture===
The Swiss architect Le Corbusier, famous for his contributions to the [[Philosophy:Modernism|modern]] international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."<ref name=modulor /><ref name=Frings />

Le Corbusier explicitly used the golden ratio in his [[Modulor]] system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "[[Vitruvian Man]]", the work of [[Biography:Leon Battista Alberti|Leon Battista Alberti]], and others who used the proportions of the human body to improve the appearance and function of architecture.

In addition to the golden ratio, Le Corbusier based the system on [[Biology:Anthropometry|human measurements]], Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the [[Modulor]] system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.<ref name=modulor2 />

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.<ref name=urwin />

===Art===
[[File:Divina proportione - Illustration 13, crop & monochrome.jpg|thumb|right|[[Biography:Leonardo da Vinci|Da Vinci]]'s illustration of a dodecahedron from Pacioli's ''[[Divina proportione|Divina proportione]]'' (1509)]]
[[Biography:Leonardo da Vinci|Leonardo da Vinci]]'s illustrations of polyhedra in Pacioli's ''Divina proportione'' have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his ''Mona Lisa'', for example, employs golden ratio proportions, is not supported by Leonardo's own writings.<ref name="livio plus"/> Similarly, although Leonardo's ''[[Vitruvian Man]]'' is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.<ref name=devlin /><ref name=simanek />

Salvador Dalí, influenced by the works of Matila Ghyka,<ref name=dalidimension /> explicitly used the golden ratio in his masterpiece, ''The Sacrament of the Last Supper''. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind [[Biography:Jesus|Jesus]] and dominates the composition.<ref name="livio plus" /><ref name="hunt gilkey" />

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is <math>1.34,</math> with averages for individual artists ranging from <math>1.04</math> (Goya) to <math>1.46</math> (Bellini).<ref name=olariu /> On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and <math>\sqrt5</math> proportions, and others with proportions like <math>\sqrt2,</math> <math>3,</math> <math>4,</math> and <math>6.</math><ref name=tosto />

[[File:Medieval manuscript framework.svg|thumb|Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."<ref name=tschichold />]]

===Books and design===

According to Jan Tschichold,

<blockquote>There was a time when deviations from the truly beautiful page proportions <math>2\mathbin{:}3,</math> <math>1\mathbin{:}\sqrt3,</math> and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.<ref name=tschichold2 /></blockquote>

According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.<ref name=miscellany />

===Flags===
[[File:Flag of Togo.svg|thumb|right|The flag of Togo, whose [[Aspect ratio|aspect ratio]] uses the golden ratio]]

The [[Aspect ratio|aspect ratio]] (width to height ratio) of the flag of Togo was intended to be the golden ratio, according to its designer.<ref>{{harvnb|Posamentier|Lehmann|2011}}, chapter 4, footnote 12: "The Togo flag was designed by the artist Paul Ahyi (1930–2010), who claims to have attempted to have the flag constructed in the shape of a golden rectangle".</ref>

===Music===
[[Biography:Ernő Lendvai|Ernő Lendvai]] analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,<ref name=lendvai /> though other music scholars reject that analysis.{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/190 190]}} French composer Erik Satie used the golden ratio in several of his pieces, including ''Sonneries de la Rose+Croix''. The golden ratio is also apparent in the organization of the sections in the music of Debussy's ''Reflets dans l'eau (Reflections in Water)'', from ''Images'' (1st series, 1905), in which "the sequence of keys is marked out by the intervals {{math|34,}} {{math|21,}} {{math|13}} and {{math|8,}} and the main climax sits at the phi position".<ref name=Smith />

The musicologist Roy Howat has observed that the formal boundaries of Debussy's ''La Mer'' correspond exactly to the golden section.<ref name=howat /> Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.<ref name=trezise />

Music theorists including Hans Zender and Heinz Bohlen have experimented with the [[833 cents scale|833 cents scale]], a musical scale based on using the golden ratio as its fundamental musical interval. When measured in [[Cent (music)|cents]], a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.<ref name=833cents />

===Nature===
[[File:Aeonium tabuliforme.jpg|thumb|upright|Detail of the saucer plant, ''[[Biology:Aeonium tabuliforme|Aeonium tabuliforme]]'', showing the multiple spiral arrangement (parastichy)]]
{{main|Patterns in nature}}

Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/154 154]}}

The psychologist Adolf Zeising noted that the golden ratio appeared in [[Biology:Phyllotaxis|phyllotaxis]] and argued from these [[Patterns in nature|patterns in nature]] that the golden ratio was a universal law.<ref name=padovan /> Zeising wrote in 1854 of a universal [[Philosophy:Orthogenesis|orthogenetic]] law of "striving for beauty and completeness in the realms of both nature and art".<ref name=zeising />

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.<ref name=pommersheim />

===Physics===
The quasi-one-dimensional [[Physics:Ising model|Ising]] [[Physics:Ferromagnet|ferromagnet]] <chem display="inline">CoNb2O6</chem> (cobalt niobate) has 8 predicted excitation states (with [[E8 (mathematics)|E<sub>8</sub> symmetry]]), that when probed with neutron scattering, showed its lowest two were in golden ratio. Specifically, these quantum phase transitions during spin excitation, which occur at near absolute zero temperature, showed pairs of kinks in its ordered-phase to spin-flips in its [[Physics:Paramagnetic|paramagnetic]] phase; revealing, just below its [[Physics:Critical field|critical field]], a spin dynamics with sharp modes at low energies approaching the golden mean.<ref name=ising />

===Optimization===
There is no known general [[Algorithm|algorithm]] to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, ''[[Thomson problem]]'' or ''[[Tammes problem]]''). However, a useful approximation results from dividing the sphere into parallel bands of equal [[Surface area|surface area]] and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. <math>360^\circ/\varphi \approx 222.5^\circ.</math> This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.<ref name=disco />
{{Clear}}

The golden ratio is a critical element to [[Golden-section search|golden-section search]] as well.

==Disputed observations==
Examples of disputed observations of the golden ratio include the following:
[[File:NautilusCutawayLogarithmicSpiral.jpg|thumb|Nautilus shells are often erroneously claimed to be golden-proportioned.]]
* Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive [[Biology:Phalanx bone|phalangeal]] and [[Biology:Metacarpal bones|metacarpal bones]] (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.<ref name=pheasant /><ref name=vanLaack />
* The shells of mollusks such as the nautilus are often claimed to be in the golden ratio.<ref name=dunlap /> The growth of nautilus shells follows a [[Logarithmic spiral|logarithmic spiral]], and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio,<ref name=falbo /> or sometimes claimed that each new chamber is golden-proportioned relative to the previous one.<ref name=moscovich /> However, measurements of nautilus shells do not support this claim.<ref name=shellspirals />
* Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is <math>1.45.</math><ref name=gutenberg />
* Studies by psychologists, starting with [[Biography:Gustav Fechner|Gustav Fechner]] {{Circa|1876}},<ref name=Fechner /> have been devised to test the idea that the golden ratio plays a role in human perception of [[Philosophy:Beauty|beauty]]. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/7 7]}}<ref name="livio plus" />
* In investing, some practitioners of [[Finance:Technical analysis|technical analysis]] use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.<ref name=osler/> The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. [[Finance:Elliott wave principle|Elliott wave principle]] and [[Fibonacci retracement]]). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.<ref name=magicdow />

===Egyptian pyramids===
[[File:Egypt. Pyramids. Pyramids and palm grove reflections LOC matpc.23063.jpg|thumb|The Great Pyramid of Giza]]
The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by [[Unsolved:Pyramidology|pyramidologists]] as having a doubled [[Kepler triangle]] as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on [[Pi|pi]] or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.<ref name=greatpyramid />

===The Parthenon===
[[File:The_Parthenon_in_Athens.jpg|thumb|280px|Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/74 74–75]}}]]

The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles.<ref name=Polemic /> Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, [[Biography:Keith Devlin|Keith Devlin]] says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation."<ref name=mathinstinct /> Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."<ref name=gazalé />

From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries.<ref name=foutakis />
Later sources like Vitruvius (first century&nbsp;BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

===Modern art===
[[File:Albert Gleizes, 1912, Les Baigneuses, oil on canvas, 105 x 171 cm, Paris, Musée d'Art Moderne de la Ville de Paris.jpg|thumb|Albert Gleizes, ''Les Baigneuses'' (1912)]]
The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with [[Cubism]] and Orphism.<ref name=centrepompidou1 /> Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat.<ref name=centrepompidou2 /> (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat’s writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.)<ref name=seuratclaims /> The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception".<ref name=herbert /> However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 ''Salon de la Section d'Or'' exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not,{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/169 169]}} and Marcel Duchamp said as much in an interview.<ref name=camfield /> On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition.<ref name=camfield /><ref name=juangris /> Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition's name also refers to the earlier ''Bandeaux d'Or'' group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved.<ref name=allard />

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,<ref name=bouleau /> though other experts (including critic Yve-Alain Bois) have discredited these claims.<ref name="livio plus" />{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/177 177–178]}}

==See also==
* List of works designed with the golden ratio
* Metallic mean
* [[Plastic ratio]]
* [[Sacred geometry]]
* [[Supergolden ratio]]
* [[Silver ratio]]

==References==
=== Explanatory footnotes ===
{{notelist}}

=== Citations ===
{{reflist|30em|refs=

<ref name="a001622">{{cite OEIS
|A001622 |2=Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2}}
</ref>

<ref name=Pacioli>
{{cite book
| last = Pacioli
| first = Luca
| title = [[De divina proportione]]
| publisher = Luca Paganinem de Paganinus de Brescia (Antonio Capella)
| year = 1509
| publication-place = Venice
}}
</ref>

<ref name="Elements 6.3">
{{cite book
| last = Euclid
| title = [[Euclid's Elements|Elements]]
| orig-year = c. 300 BCE
| chapter = Book 6, Definition 3
}}
</ref>

<ref name=goldencut>
{{cite book
| last = Summerson
| first = John
| year = 1963
| title = Heavenly Mansions and Other Essays on Architecture
| publication-place = New York
| publisher = W.W. Norton
| page = 37
| url = https://archive.org/details/heavenlymansions0000summ/page/37/
| quote = And the same applies in architecture, to the [[Rectangle|rectangle]]s representing these and other ratios (e.g., the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.
}}
</ref>

<ref name="strogatz nytimes">
{{Cite news
| first = Steven
| last = Strogatz
| title = Me, Myself, and Math: Proportion Control
| newspaper = {{wipe|The New York Times}}
| date = 2012-09-24
| url = http://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/
}}
</ref>

<ref name=schielack>
{{Cite journal
| last = Schielack
| first = Vincent P.
| year = 1987
| title = The Fibonacci Sequence and the Golden Ratio
| journal = The Mathematics Teacher
| volume = 80 | issue = 5 | pages = 357–358 | doi = 10.5951/MT.80.5.0357
| jstor = 27965402
}} This source contains an elementary derivation of the golden ratio's value.
</ref>

<ref name=peters>
{{Cite journal
| last = Peters
| first = J. M. H.
| year = 1978
| title = An Approximate Relation between π and the Golden Ratio
| journal = The Mathematical Gazette
| volume = 62
| issue = 421
| pages = 197–198
| doi = 10.2307/3616690
| jstor = 3616690
| s2cid = 125919525
}}</ref>

<ref name="fitzpatrick elements">
{{cite book
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| translator-first = Richard
| year = 2007
| title = Euclid's Elements of Geometry
| isbn = 978-0615179841
| page = 156
| publisher = Lulu.com
}}
</ref>

<ref name=hemenway>
{{cite book
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| first = Priya
| title = Divine Proportion: Phi In Art, Nature, and Science
| url = https://archive.org/details/divineproportion0000heme/page/20/
| url-access = limited
| year = 2005
| publisher = Sterling
| location = New York
| pages = 20–21
| isbn = 9781402735226
}}
</ref>

<ref name=mackinnon>
{{cite journal
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| first = Nick
| year = 1993
| title = The Portrait of Fra Luca Pacioli
| volume = 77
| issue = 479
| journal = The Mathematical Gazette
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| doi = 10.2307/3619717
| jstor = 3619717
| s2cid = 195006163
}}
</ref>

<ref name=baravalle>
{{cite journal
| last = Baravalle
| first = H. V.
| title = The geometry of the pentagon and the golden section
| journal = Mathematics Teacher
| volume = 41
| year = 1948
| pages = 22–31
| doi = 10.5951/MT.41.1.0022
}}
</ref>

<ref name=schreiber>
{{cite journal
| last = Schreiber
| first = Peter
| year = 1995
| title = A Supplement to J. Shallit's Paper 'Origins of the Analysis of the Euclidean Algorithm'
| journal = Historia Mathematica
| volume = 22
| issue = 4
| pages = 422–424
| doi = 10.1006/hmat.1995.1033
| doi-access=free
}}
</ref>

<ref name=mactutor>
{{cite web
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| first1 = John J.
| last2 = Robertson
| first2 = Edmund F.
| year = 2001
| title = The Golden Ratio
| work = [[MacTutor History of Mathematics archive]]
| url = http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html
| access-date = 2007-09-18
}}
</ref>

<ref name="beutelspacher petri">
{{cite book
| last1 = Beutelspacher
| first1 = Albrecht
| last2 = Petri
| first2 = Bernhard
| year = 1996
| contribution = Fibonacci-Zahlen
| title = Der Goldene Schnitt
| publisher = Vieweg+Teubner Verlag
| doi = 10.1007/978-3-322-85165-9_6
| language = de
| pages = 87–98
}}
</ref>

<ref name=fink>
{{cite book
| title = A Brief History of Mathematics
| last1 = Fink
| first1 = Karl
| translator-last1 = Beman
| translator-first1 = Wooster Woodruff
| translator-last2 = Smith
| translator-first2 = David Eugene
| translator-link2 = David Eugene Smith
| year = 1903
| publisher = Open Court
| location = Chicago
| edition = 2nd
| page = 223
| url = https://archive.org/details/bub_gb_3hkPAAAAIAAJ/page/n238
}} (Originally published as ''Geschichte der Elementar-Mathematik''.)
</ref>

<ref name=cook>
{{Cite book
| last = Cook
| first = Theodore Andrea
| title = The Curves of Life
| year = 1914
| page = 420
| publisher = Constable
| location = London
| url = https://archive.org/details/cu31924028937179/page/n455
}}
</ref>

<ref name=barr>
{{cite magazine
|last=Barr
|first=Mark
|title=Parameters of beauty
|magazine=Architecture (NY)
|volume=60
|page=325
|year=1929
}} Reprinted: {{cite magazine
| title = Parameters of beauty
| magazine = Think
| volume = 10–11
| publisher = [[Company:IBM|IBM]]
| year = 1944
}}
</ref>

<ref name=gardner>
{{cite book
| last = Gardner
| first = Martin
| title = The Colossal Book of Mathematics
| publisher = Norton
| year = 2001
| chapter = 7. Penrose Tiles
| pages=73–93
| chapter-url = https://archive.org/details/martingardnerthecolossalbookofmathematics/page/n89
}}
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<ref name=quasicrystals>
{{harvnb|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/203 203–209]}}
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| url = https://hal.archives-ouvertes.fr/hal-02379494
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| url-access = limited
| quote = Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
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| pages = 230–241
| title = Quasicrystalline materials
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<ref name=constructions>
{{ cite book
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| first = George E.
| year = 1998
| title = Geometric Constructions
| title-link = Geometric Constructions
| series = Undergraduate Texts in Mathematics
| publisher = Springer
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| doi = 10.1007/978-1-4612-0629-3
| isbn = 978-1-4612-6845-1
}}
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<ref name=sizer>{{cite journal
 | last = Sizer | first = Walter S.
 | doi = 10.1080/0025570X.1986.11977215
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 | title = Continued roots
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<ref name="Concrete Abstractions">
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| last3 = Knight
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| year = 1999
| title = Concrete Abstractions: An Introduction to Computer Science Using Scheme
| publisher = Brooks/Cole
| page = 63
| url = https://archive.org/details/ost-computer-science-concreteabstractions/page/n78
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<ref name=hardy>
{{cite book
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| title = An Introduction to the Theory of Numbers
| edition = 4th
| publisher = Oxford University Press
| chapter-url = https://archive.org/details/introductiontoth0000hard/page/163
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<ref name=tattersall>
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| title = Elementary number theory in nine chapters
| publisher = Cambridge University Press
| page = 28
| url = https://archive.org/details/elementarynumber0000tatt/page/28/
| url-access = limited
}}
</ref>

<ref name=parker4d>
{{cite book
| last1 = Parker
| first1 = Matt
| date = 2014
| title = Things to Make and Do in the Fourth Dimension
| publisher = Farrar, Straus and Giroux
| page = 284
| isbn = 9780374275655
| url = https://archive.org/details/thingstomakedoin0000park_r9y6/page/284/
| url-access = limited
}}
</ref>

<ref name=phyllotaxis>{{cite journal
| last1 = King
| first1 = S.
| last2 = Beck
| first2 = F.
| last3 = Lüttge
| first3 = U.
| year = 2004
| title = On the mystery of the golden angle in phyllotaxis
| journal = Plant, Cell and Environment
| volume = 27
| issue = 6
| pages = 685–695
| doi = 10.1111/j.1365-3040.2004.01185.x
}}
</ref>

<ref name=triangleconstruction>
{{cite journal
| last1 = Odom
| first1 = George
| last2 = van de Craats
| first2 = Jan
| date = 1986
| title = E3007: The golden ratio from an equilateral triangle and its circumcircle
| department = Problems and solutions
| journal = The American Mathematical Monthly
| volume = 93
| issue = 7
| page = 572
| doi = 10.2307/2323047
| jstor = 2323047
}}
</ref>

<ref name="Liber mensurationum">
{{cite journal
| last = Busard
| first = Hubert L. L.
| date = 1968
| title = L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr
| journal = Journal des Savants
| volume = 1968
| issue = 2
| language = fr,la
| pages = 65–124
| url = https://www.persee.fr/doc/jds_0021-8103_1968_num_2_1_1175
| doi = 10.3406/jds.1968.1175
}} See problem 51, reproduced on p. 98
</ref>

<ref name=bruce>
{{cite journal
| last = Bruce
| first = Ian
| year = 1994
| title = Another instance of the golden right triangle
| journal = [[Fibonacci Quarterly]]
| volume = 32
| issue = 3
| pages = 232–233
| url = https://www.mathstat.dal.ca/FQ/Scanned/32-3/bruce.pdf
}}
</ref>

<ref name=fletcher>
{{cite journal
| last = Fletcher
| first = Rachel
| year = 2006
| title = The golden section
| journal = Nexus Network Journal
| volume = 8
| issue = 1
| pages = 67–89
| doi = 10.1007/s00004-006-0004-z
| s2cid = 120991151
| doi-access = free
}}
</ref>

<ref name=loeb>
{{cite book
| last = Loeb
| first = Arthur
| year = 1992
| chapter = The Golden Triangle
| title = Concepts & Images: Visual Mathematics
| publisher = Birkhäuser
| pages = 179–192
| doi = 10.1007/978-1-4612-0343-8_20
| isbn = 978-1-4612-6716-4
}}
</ref>

<ref name=hexecontahedron>
{{cite journal
| last = Grünbaum
| first = Branko
| year = 1996
| journal = Geombinatorics
| volume = 6
| issue = 1
| pages = 15–18
| title = A new rhombic hexecontahedron
| url = https://faculty.washington.edu/moishe/branko/BG207.New%20rhombic%20hexecontah.pdf
}}
</ref>

<ref name=quarter-circles>{{cite conference
 | last1 = Reitebuch | first1 = Ulrich
 | last2 = Skrodzki | first2 = Martin
 | last3 = Polthier | first3 = Konrad
 | editor1-last = Swart | editor1-first = David
 | editor2-last = Farris | editor2-first = Frank
 | editor3-last = Torrence | editor3-first = Eve
 | contribution = Approximating logarithmic spirals by quarter circles
 | contribution-url = https://archive.bridgesmathart.org/2021/bridges2021-95.html
 | isbn = 978-1-938664-39-7
 | location = Phoenix, Arizona
 | pages = 95–102
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture
 | year = 2021}}</ref>

<ref name=diedrichs>{{cite journal
 | last = Diedrichs | first = Danilo R.
 | date = February 2019
 | doi = 10.1017/mag.2019.7
 | issue = 556
 | journal = The Mathematical Gazette
 | pages = 52–64
 | title = Archimedean, Logarithmic and Euler spirals – intriguing and ubiquitous patterns in nature
 | volume = 103| s2cid = 127189159
 }}</ref>

<ref name=loeb-varney>{{cite book
 | last1 = Loeb | first1 = Arthur L.
 | last2 = Varney | first2 = William
 | editor1-last = Hargittai | editor1-first = István
 | editor2-last = Pickover | editor2-first = Clifford A.
 | contribution = Does the golden spiral exist, and if not, where is its center?
 | contribution-url = https://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA47
 | date = March 1992
 | doi = 10.1142/9789814343084_0002
 | pages = 47–61
 | publisher = World Scientific
 | title = Spiral Symmetry}}</ref>

<ref name=BurgerStarbird>
{{cite book
| first1 = Edward B.
| last1 = Burger
| first2 = Michael P.
| last2 = Starbird
| year = 2005
| orig-year = 2000
| edition = 2nd
| title = The Heart of Mathematics: An Invitation to Effective Thinking
| publisher = Springer
| page = 382
| isbn = 9781931914413
| url = https://archive.org/details/heartofmathemati0000burg_z4x5/page/382/
}}
</ref>

<ref name="golden rhombohedra">
{{cite book
| last = Senechal
| first = Marjorie
| year = 2006
| contribution = Donald and the golden rhombohedra
| editor1-last = Davis
| editor1-first = Chandler
| editor2-last = Ellers
| editor2-first = Erich W.
| title = The Coxeter Legacy
| isbn = 0-8218-3722-2
| pages = 159–177
| publisher = American Mathematical Society
}}
</ref>

<ref name="Tilings and Patterns">
{{cite book
| last1 = Grünbaum
| first1 = Branko
| last2 = Shephard
| first2 = G. C.
| title = Tilings and Patterns
| location = New York
| publisher = W. H. Freeman
| year = 1987
| pages = 537–547
| isbn = 9780716711933
| url = https://archive.org/details/isbn_0716711931/page/537/
}}
</ref>

<ref name=pentaplexity>
{{cite magazine
| first = Roger
| last = Penrose
| year = 1978
| title = Pentaplexity
| magazine = [[Organization:Eureka (University of Cambridge magazine)|Eureka]]
| volume =  39
| pages = 32
| url = https://archive.org/details/eureka-39/page/16/
}} ([https://www.archim.org.uk/eureka/archive/Eureka-39.pdf#page=16 original PDF])
</ref>

<ref name=borromean>
{{cite conference
| last1 = Gunn
| first1 = Charles
| last2 = Sullivan
| first2 = John M.
| year = 2008
| title = The Borromean Rings: A video about the New IMU logo
| url = https://archive.bridgesmathart.org/2008/bridges2008-63.html
| editor1-last = Sarhangi
| editor1-first = Reza
| editor2-last = Séquin
| editor2-first = Carlo H.
| book-title = Proceedings of Bridges 2008
| conference = Leeuwarden, the Netherlands
| pages = 63–70
| publisher = Tarquin Publications
}}; Video at {{cite web
| title = The Borromean Rings: A new logo for the IMU
| website = International Mathematical Union
| url = http://torus.math.uiuc.edu/jms/Videos/imu/
| archive-url = https://web.archive.org/web/20210308065039/http://torus.math.uiuc.edu/jms/Videos/imu/
| archive-date = 2021-03-08
}}
</ref>

<ref name=hume>
{{cite journal
| last = Hume
| first = Alfred
| year = 1900
| title = Some propositions on the regular dodecahedron
| journal = [[The American Mathematical Monthly]]
| volume = 7
| issue = 12
| pages = 293–295
| doi = 10.2307/2969130
| jstor = 2969130
}}
</ref>

<ref name="59 Icosahedra">
{{Cite book
| last1 = Coxeter
| first1 = H.S.M.
| last2 = du Val
| first2 = Patrick
| last3 = Flather
| first3 = H.T.
| last4 = Petrie
| first4 = J.F.
| year = 1938
| title = The Fifty-Nine Icosahedra
| title-link = The Fifty-Nine Icosahedra
| publisher = University of Toronto Studies
| volume = 6
| page = 4
| quote = Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section.
}}
</ref>

<ref name=muller>
{{cite book
| last1 = Muller
| first1 = J. M.
| date = 2006
| title = Elementary functions : algorithms and implementation
| edition = 2nd
| publisher = Birkhäuser
| location = Boston
| isbn = 978-0817643720
| page = 93
}}
</ref>

<ref name=ycruncher>
{{Cite web
| first =  Alexander J.
| last = Yee
| date = 2021-03-13
| title = Records Set by y-cruncher
| website = numberword.org
| url = http://www.numberworld.org/y-cruncher/records.html
}} Two independent computations done by Clifford Spielman.
</ref>

<ref name=horocycle>
[http://www.cabri.net/abracadabri/GeoNonE/GeoHyper/KBModele/Biss3KB.html Horocycles exinscrits : une propriété hyperbolique remarquable], cabri.net, retrieved 2009-07-21.
</ref>

<ref name=rrcf>
{{cite journal
| last1 = Berndt
| first1 = Bruce C.
| last2 = Chan
| first2 = Heng Huat
| last3 = Huang
| first3 = Sen-Shan
| last4 = Kang
| first4 = Soon-Yi
| last5 = Sohn
| first5 = Jaebum
| last6 = Son
| first6 = Seung Hwan
| year = 1999
| title = The Rogers–Ramanujan Continued Fraction
| journal = Journal of Computational and Applied Mathematics
| volume = 105
| number = 1–2
| pages = 9–24
| url = https://faculty.math.illinois.edu/~berndt/articles/rrcf.pdf
| doi = 10.1016/S0377-0427(99)00033-3
}}
</ref>

<ref name=duffin>
{{cite journal
| last = Duffin
| first = Richard J.
| year = 1978
| title = Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers
| journal = [[Pacific Journal of Mathematics]]
| volume = 74
| issue = 1
| pages = 47–56
| url = https://projecteuclid.org/euclid.pjm/1102810434
| doi = 10.2140/pjm.1978.74.47
| doi-access = free
}}
</ref>

<ref name=modulor>Le Corbusier, ''The Modulor'', {{nobr|p. 25}}, as cited in
{{cite book
| last = Padovan
| first = Richard
| year = 1999
| title = Proportion: Science, Philosophy, Architecture
| page = 316
| publisher = Taylor & Francis
| doi = 10.4324/9780203477465
| isbn = 9781135811112
}}</ref>

<ref name=modulor2>Le Corbusier, ''The Modulor'', {{nobr|p. 35}}, as cited in
{{cite book
| last = Padovan
| first = Richard
| year = 1999
| title = Proportion: Science, Philosophy, Architecture
| page = 320
| publisher = Taylor & Francis
| doi = 10.4324/9780203477465
| isbn = 9781135811112
| quote = Both the paintings and the architectural designs make use of the golden section
}}</ref>

<ref name=Frings>
{{cite journal
| last = Frings
| first = Marcus
| year = 2002
| title = The Golden Section in Architectural Theory
| journal = Nexus Network Journal
| volume = 4
| number = 1
| pages =  9–32
| doi = 10.1007/s00004-001-0002-0
| s2cid = 123500957
| url = http://belveduto.de/text-nnj.htm
| doi-access = free
}}
</ref>

<ref name=urwin>
{{cite book
| last = Urwin
| first = Simon
| year = 2003
| title = Analysing Architecture
| edition = 2nd
| publisher = Routledge
| pages = 154–155
| url = https://archive.org/details/analysingarchite0000unwi/page/154/
| url-access = limited
}}
</ref>

<ref name=devlin>
{{cite web
| first = Keith
| last = Devlin
| year = 2007
| title = The Myth That Will Not Go Away
| access-date = September 26, 2013
| url = http://www.maa.org/external_archive/devlin/devlin_05_07.html
| quote = Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.
}}
</ref>

<ref name="livio plus">
{{cite web
| last = Livio
| first = Mario
| year = 2002
| title = The golden ratio and aesthetics
| url = https://plus.maths.org/content/os/issue22/features/golden/index
| ref = none
| website = Plus Magazine
| access-date = November 26, 2018
}}
</ref>

<ref name=simanek>
{{cite web
| first = Donald E.
| last =  Simanek
| title = Fibonacci Flim-Flam
| url = http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
| access-date = April 9, 2013
| url-status = dead
| archive-url = https://web.archive.org/web/20100109045556/http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
| archive-date = January 9, 2010
}}
</ref>

<ref name=dalidimension>
{{cite video
| people = Salvador Dalí
| date = 2008
| title = The Dali Dimension: Decoding the Mind of a Genius
| format = DVD
| publisher = Media 3.14-TVC-FGSD-IRL-AVRO
| url = http://www.dalidimension.com/eng/index.html
}}
</ref>

<ref name="hunt gilkey">
{{cite book
| last1 = Hunt
| first1 = Carla Herndon
| last2 = Gilkey
| first2 = Susan Nicodemus
| year = 1998
| title = Teaching Mathematics in the Block
| pages = 44, 47
| publisher = Eye On Education
| isbn = 1-883001-51-X
}}
</ref>

<ref name=olariu>
{{cite arXiv
| last = Olariu
| first = Agata
| year = 1999
| title = Golden Section and the Art of Painting
| eprint = physics/9908036
}}
</ref>

<ref name=tosto>
{{cite book
| last = Tosto
| first = Pablo
| year = 1969
| title = La composición áurea en las artes plásticas
| lang = es
| trans-title = The golden composition in the plastic arts
| publisher = Hachette
| pages = 134–144
}}
</ref>

<ref name=tschichold>
{{cite book
| last = Tschichold
| first = Jan
| year = 1991
| title = The Form of the Book
| publisher = Hartley & Marks
| isbn = 0-88179-116-4
| page = 43 Fig 4
| url = https://archive.org/details/the-form-of-the-book-jan-tschichold/page/n59/
| quote = Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well.
}}
</ref>

<ref name=tschichold2>
{{cite book
| last = Tschichold
| first = Jan
| year = 1991
| title = The Form of the Book
| publisher = Hartley & Marks
| isbn = 0-88179-116-4
| pages = 27–28
| url = https://archive.org/details/the-form-of-the-book-jan-tschichold/page/n43/
}}
</ref>

<ref name=miscellany>
{{Cite journal
| last = Jones
| first = Ronald
| title = The golden section: A most remarkable measure
| year = 1971
| journal = The Structurist
| volume = 11
| pages = 44–52
| quote = Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
}}
<p>
{{cite book
| last = Johnson
| first = Art
| year = 1999
| title = Famous problems and their mathematicians
| publisher = Teacher Ideas Press
| page = 45
| isbn = 9781563084461
| url = https://archive.org/details/famousproblemsth0000john/page/45
| url-access = limited
| quote = The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
}}
<p>
{{cite book
| last1 = Stakhov
| first1 = Alexey P.
| last2 = Olsen
| first2 = Scott
| year = 2009
| title = The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science
| chapter = §1.4.1 A Golden Rectangle with a Side Ratio of ''τ''
| publisher = World Scientific
| pages = 20–21
| quote = A credit card has a form of the golden rectangle
| chapter-url = https://archive.org/details/alexy-stakhov-the-mathematics-of-harmony-from-euclid-to-contemporary-mathematics/page/21/
}}
<p>
{{cite book
| last = Cox
| first = Simon
| year = 2004
| title = Cracking the Da Vinci Code
| publisher = Barnes & Noble
| page = 62
| isbn = 978-1-84317-103-4
| url = https://archive.org/details/crackingdavincic00coxs/page/62/
| quote = The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.
}}
</ref>

<ref name=lendvai>
{{cite book
| last = Lendvai
| first = Ernő
| year = 1971
| title = Béla Bartók: An Analysis of His Music
| publication-place = London
| publisher = Kahn and Averill
}}
</ref>

<ref name=Smith>
{{cite book
| last = Smith
| first = Peter F.
| year = 2003
| title = The Dynamics of Delight: Architecture and Aesthetics
| url = https://books.google.com/books?id=ZgftUKoMnpkC&pg=PA83
| publisher = Routledge
| page = 83
| isbn = 9780415300100
}}
</ref>

<ref name=howat>
{{cite book
| last= Howat
| first = Roy
| year = 1983
| title = Debussy in Proportion: A Musical Analysis
| chapter = 1. Proportional structure and the Golden Section
| chapter-url = https://archive.org/details/debussyinproport0000howa/page/1
| publisher = Cambridge University Press
| pages = 1–10
}}
</ref>

<ref name=trezise>
{{cite book
| last = Trezise
| first = Simon
| year = 1994
| title = Debussy: La Mer
| publisher = Cambridge University Press
| page = 53
| isbn = 9780521446563
| url = https://books.google.com/books?id=THD1nge_UzcC&pg=PA53
}}
</ref>

<ref name=833cents>
{{cite journal
| last = Mongoven
| first = Casey
| year = 2010
| title = A style of music characterized by Fibonacci and the golden ratio
| url = https://www.mat.ucsb.edu/Publications/mongoven_CongressusNumerantium2010.pdf
| journal = Congressus Numerantium
| volume = 201
| pages = 127–138
}}
<p>
{{cite journal
| last = Hasegawa
| first = Robert
| year = 2011
| title = ''Gegenstrebige Harmonik'' in the Music of Hans Zender
| journal = Perspectives of New Music
| volume = 49
| issue = 1
| doi = 10.1353/pnm.2011.0000
| publisher = Project Muse
| jstor = 10.7757/persnewmusi.49.1.0207
| pages = 207–234
}}
<p>
{{cite conference
| last = Smethurst
| first = Reilly
| year = 2016
| title = Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal
| url = https://archive.bridgesmathart.org/2016/bridges2016-519.html
| editor1-last = Torrence
| editor1-first = Eve
| editor2-last = Torrence
| editor2-first = Bruce
| editor3-last = Séquin
| editor3-first = Carlo
| editor4-last = McKenna
| editor4-first = Douglas
| editor5-last = Fenyvesi
| editor5-first = Kristóf
| editor6-last = Sarhangi
| editor6-first = Reza
| display-editors = 1
| book-title = Proceedings of Bridges 2016
| conference = Jyväskylä, Finland
| pages = 519–522
| publisher = Tessellations Publishing

}}
</ref>

<ref name=padovan>
{{Cite book
| last = Padovan
| first = Richard
| year = 1999
| title = Proportion: Science, Philosophy, Architecture
| publisher = Taylor & Francis
| pages = 305–306
| doi = 10.4324/9780203477465
| isbn = 9781135811112
}}
<p>
{{cite journal
| last = Padovan
| first = Richard
| year=2002
| title = Proportion: Science, Philosophy, Architecture
| journal = Nexus Network Journal
| volume = 4
| issue=1
| pages = 113–122
| doi=10.1007/s00004-001-0008-7
| doi-access=free
}}
</ref>

<ref name=zeising>
{{cite book
| first = Adolf
| last = Zeising
| title = Neue Lehre von den Proportionen des menschlichen Körpers
| trans-title = New doctrine of the proportions of the human body
| lang = de
| year = 1854
| publisher = Weigel
| chapter = Einleitung [preface]
| pages = 1–10
| chapter-url = https://archive.org/details/neuelehrevondenp00zeis/page/n25/
}}
</ref>

<ref name=pommersheim>
{{cite book
| editor1-last = Pommersheim
| editor1-first = James E.
| editor2-last = Marks
| editor2-first = Tim K.
| editor3-last = Flapan
| editor3-first = Erica L.
| year = 2010
| title = Number Theory: A Lively Introduction with Proofs, Applications, and Stories
| publisher = Wiley
| page = 82
}}
</ref>

<ref name=ising>
{{cite journal
| last1 = Coldea
| first1 = R.
| last2 = Tennant
| first2 = D.A.
| last3 = Wheeler
| first3 = E.M.
| last4 = Wawrzynksa
| first4 = E.
| last5 = Prabhakaran
| first5 = D.
| last6 = Telling
| first6 = M.
| last7 = Habicht
| first7 = K.
| last8 = Smeibidl
| first8 = P.
| last9 = Keifer
| first9 = K.
| year = 2010
| title = Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry
| journal = Science
| volume = 327
| issue = 5962
| pages = 177–180
| doi = 10.1126/science.1180085
| pmid = 20056884
| arxiv = 1103.3694
| bibcode = 2010Sci...327..177C
| s2cid = 206522808
}}
</ref>

<ref name=disco>
{{cite web
| url = https://science.nasa.gov/science-news/science-at-nasa/2001/ast09oct_1/
| title = A Disco Ball in Space
| publisher = NASA
| date = 2001-10-09
| access-date = 2007-04-16
}}
</ref>

<ref name=pheasant>
{{cite book
| last = Pheasant
| first = Stephen
| year = 1986
| title = Bodyspace
| publisher = Taylor & Francis
| isbn = 9780850663402
| url = https://archive.org/details/bodyspaceanthrop0000phea/
| url-access = limited
}}
</ref>

<ref name=vanLaack>
{{cite book
| last = van Laack
| first = Walter
| title = A Better History Of Our World: Volume 1 The Universe
| location = Aachen
| publisher = van Laach
| year = 2001
}}
</ref>

<ref name=dunlap>
{{cite book
| last = Dunlap
| first = Richard A.
| year = 1997
| title = The Golden Ratio and Fibonacci Numbers
| publisher = World Scientific
| page = [https://books.google.com/books?id=qBftCgAAQBAJ&pg=PA130 130]
}}
</ref>

<ref name=falbo>
{{cite journal
| last = Falbo
| first = Clement
| date = March 2005
| title = The golden ratio—a contrary viewpoint
| journal = The College Mathematics Journal
| volume = 36
| issue = 2
| pages = 123–134
| doi = 10.1080/07468342.2005.11922119
| s2cid = 14816926
}}
</ref>

<ref name=miller>
{{cite journal
| last = Miller
| first = William
| year = 1996
| title = Pentagons and Golden Triangles
| journal = Mathematics in School
| volume = 25
| issue = 4
| pages = 2–4
| jstor = 30216571
}}
</ref>

<ref name=robinson>
{{cite web
| last1 = Frettlöh
| first1 = D.
| last2 = Harriss
| first2 = E.
| last3 = Gähler
| first3 = F.
| title = Robinson Triangle
| website = Tilings Encyclopedia
| url = https://tilings.math.uni-bielefeld.de/substitution/robinson-triangle/
}}
<p>
{{cite journal
| last = Clason
| first = Robert G
| year = 1994
| title = A family of golden triangle tile patterns.
| journal = The Mathematical Gazette
| volume = 78
| number = 482
| pages = 130–148
| doi = 10.2307/3618569
| jstor = 3618569
| s2cid = 126206189
}}
</ref>

<ref name=moscovich>
{{cite book
| last = Moscovich
| first = Ivan
| year = 2004
| title = The Hinged Square & Other Puzzles
| location = New York
| publisher = Sterling
| url = https://books.google.com/books?id=4n7PvKC1dfwC&pg=PA122
| page = 122
| isbn = 9781402716669
}}
</ref>

<ref name=shellspirals>
{{cite journal
| last = Peterson
| first = Ivars
| date = 1 April 2005
| title = Sea shell spirals
| journal = Science News
| url = http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
| access-date = 10 November 2008
| archive-date = 3 October 2012
| archive-url = https://web.archive.org/web/20121003045834/http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
| url-status = dead
}}
</ref>

<ref name=gutenberg>
{{cite book
| last = Man
| first = John
| year = 2002
| title = Gutenberg: How One Man Remade the World with Word
| pages = 166–167
| publisher = Wiley
| isbn = 9780471218234
| url = https://archive.org/details/gutenberghowonem00john/page/166
| quote = The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.
}}
</ref>

<ref name=Fechner>
{{cite book
| title = Vorschule der Ästhetik
| lang = de
| trans-title = Preschool of Aesthetics
| last = Fechner
| first = Gustav
| year = 1876
| publisher = Breitkopf & Härtel
| location = Leipzig
| pages = 190–202
| url = https://archive.org/details/vorschulederaest12fechuoft/page/n203/
}}
</ref>

<ref name=osler>
{{cite journal
| last = Osler
| first = Carol
| title = Support for Resistance: Technical Analysis and Intraday Exchange Rates
| journal = Federal Reserve Bank of New York Economic Policy Review
| year = 2000
| volume = 6
| issue = 2
| pages= 53–68
| quote = 38.2 percent and 61.8 percent retracements of recent rises or declines are common,
| url= http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf
| archive-url = https://web.archive.org/web/20070512155447/http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf
| archive-date = 2007-05-12
| url-status = live
}}
</ref>

<ref name=magicdow>
{{cite report
| last1 = Batchelor
| first1 = Roy
| last2 = Ramyar
| first2 = Richard
| year = 2005
| title = Magic numbers in the Dow
| publisher = Cass Business School
| url = http://openaccess.city.ac.uk/16276/
| pages = 13, 31
}}
Popular press summaries can be found in:
{{cite news
| last = Stevenson
| first = Tom
| date = 2006-04-10
| title = Not since the 'big is beautiful' days have giants looked better
| newspaper = The Daily Telegraph
| url = https://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html
}}
{{cite news
| date = 2006-09-23
| title = Technical failure
| magazine = The Economist
| url = https://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html
}}
</ref>

<ref name=greatpyramid>
{{cite book
| last = Herz-Fischler
| first = Roger
| year = 2000
| isbn = 0-88920-324-5
| publisher = Wilfrid Laurier University Press
| title = The Shape of the Great Pyramid
}} The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
<p>
{{cite book
| last = Rossi
| first = Corinna
| title = Architecture and Mathematics in Ancient Egypt
| year = 2004
| publisher = Cambridge University Press
| pages = 67–68
| url = https://archive.org/details/architechture-and-mathematics-in-ancient-egypt-corianna-rossi-2003/page/67/
| quote = there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to <math>\varphi</math>, and <math>\varphi</math> itself as a number, do not fit with the extant Middle Kingdom mathematical sources
}}; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
<p>
{{cite journal
| last1 = Rossi
| first1 = Corinna
| last2 = Tout
| first2 = Christopher A.
| year = 2002
| title = Were the Fibonacci series and the Golden Section known in ancient Egypt?
| journal = Historia Mathematica
| volume = 29
| issue = 2
| doi = 10.1006/hmat.2001.2334
| pages = 101–113
| hdl = 11311/997099
| hdl-access = free
}}
<p>
{{cite journal
| last = Markowsky
| first = George
| year = 1992
| title = Misconceptions about the Golden Ratio
| journal = [[Organization:The College Mathematics Journal|The College Mathematics Journal]]
| volume = 23
| issue = 1
| doi = 10.2307/2686193
| url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
| publisher = Mathematical Association of America
| pages = 2–19
| jstor = 2686193
| quote = It does not appear that the Egyptians even knew of the existence of <math>\varphi</math> much less incorporated it in their buildings
| access-date = 2012-06-29
}}
</ref>

<ref name=Polemic>
{{cite journal
| last = Van Mersbergen
| first = Audrey M.
| year = 1998
| title = Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic
| journal = Communication Quarterly
| volume = 46
| number = 2
| pages = 194–213
| doi = 10.1080/01463379809370095
}}
</ref>

<ref name=mathinstinct>
{{cite book
| last = Devlin
| first = Keith J.
| year = 2005
| title = The Math Instinct
| url = https://archive.org/details/isbn_9781560258391/page/108
| location = New York
| publisher = Thunder's Mouth Press
| page = 108
}}
</ref>

<ref name=gazalé>
{{cite book
| last = Gazalé
| first = Midhat J.
| year = 1999
| title = Gnomon: From Pharaohs to Fractals
| publisher = Princeton
| page = 125
| isbn = 9780691005140
| url = https://archive.org/details/gnomonfrompharao0000gaza/page/125/
| url-access = limited
}}
</ref>

<ref name=foutakis>
{{cite journal
| last = Foutakis
| first = Patrice
| year = 2014
| title = Did the Greeks Build According to the Golden Ratio?
| journal = Cambridge Archaeological Journal
| volume = 24
| number = 1
| pages = 71–86
| doi = 10.1017/S0959774314000201
| s2cid = 162767334
}}
</ref>

<ref name=centrepompidou1>
[http://mediation.centrepompidou.fr/education/ressources/ENS-futurisme2008/ENS-futurisme2008-07-section-or.html ''Le Salon de la Section d'Or''], October 1912, Mediation Centre Pompidou
</ref>

<ref name=centrepompidou2>
[http://bibliothequekandinsky.centrepompidou.fr/imagesbk/RP225%5C001/M5050_X0031_PER_P02251912001.pdf ''Jeunes Peintres ne vous frappez pas !'', La Section d'Or: Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, n° 1, 9 octobre 1912, pp.&nbsp;1–7] {{Webarchive
| url = https://web.archive.org/web/20201030080014/http://bibliothequekandinsky.centrepompidou.fr/imagesbk/RP225%5C001/M5050_X0031_PER_P02251912001.pdf
| date = 2020-10-30
}}, Bibliothèque Kandinsky
</ref>

<ref name=seuratclaims>
{{cite journal
| last = Herz-Fischler
| first = Roger
| year = 1983
| title = An Examination of Claims Concerning Seurat and the Golden Number
| journal = Gazette des Beaux-Arts
| volume = 101
| pages = 109–112
| url = https://people.math.carleton.ca/~rhfischl/PUBLICATIONS/seurat_GN.pdf
}}</ref>

<ref name=herbert>
{{cite book
| last = Herbert
| first = Robert
| year = 1968
| title = Neo-Impressionism
| publisher = Guggenheim Foundation
| url = https://archive.org/details/neoimpressionism0000herb
| url-access = limited
</ref>

<ref name=camfield>
{{cite journal
| last = Camfield
| first = William A.
| date = March 1965
| doi = 10.1080/00043079.1965.10788819
| issue = 1
| journal = The Art Bulletin
| pages = 128–134
| title = Juan Gris and the golden section
| volume = 47
}}
</ref>

<ref name=juangris>
{{cite book
| last = Green
| first = Christopher
| year = 1992
| title = Juan Gris
| publisher = Yale
| pages = 37–38
| isbn = 9780300053746
| url = https://archive.org/details/juangris0000gree/page/37/
| url-access = limited
}}
<p>
{{cite book
| last = Cottington
| first = David
| year = 2004
| title = Cubism and Its Histories
| publisher = Manchester University Press
| page = 112, 142
| url = https://books.google.com/books?isbn=0719050049
}}
</ref>

<ref name=allard>
{{cite journal
| last = Allard
| first = Roger
| date = June 1911
| title = Sur quelques peintres
| journal = Les Marches du Sud-Ouest
| pages = 57–64
}}
Reprinted in
{{cite book
| editor1-last = Antliff
| editor1-first = Mark
| editor2-last = Leighten
| editor2-first = Patricia
| title = A Cubism Reader, Documents and Criticism, 1906–1914
| year = 2008
| publisher = The University of Chicago Press
| pages = 178–191
| url = https://archive.org/details/cubismreaderdocu0008unse/page/178/
}}
</ref>

<ref name=bouleau>
{{cite book
| last = Bouleau
| first = Charles
| year = 1963
| title = The Painter's Secret Geometry: A Study of Composition in Art
| publisher = Harcourt, Brace & World
| pages = 247–248
| url = https://archive.org/details/painterssecretge0000char/page/247/
}}
</ref>

<!-- END REFLIST --> }}

===Works cited===
{{refbegin|30em}}
* {{Cite book
| last = Herz-Fischler
| first = Roger
| year = 1998
| orig-year = 1987
| title = A Mathematical History of the Golden Number
| publisher = Dover
| isbn = 9780486400075
| url-access = registration
| url = https://archive.org/details/mathematicalhist0000herz
 }} (Originally titled ''A Mathematical History of Division in Extreme and Mean Ratio''.)
* {{cite book
| last = Livio
| first = Mario
| year = 2002
| title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
| url = https://archive.org/details/goldenratiostory00livi/
| url-access = registration
| publisher = Broadway Books
| location = New York
| isbn = 9780767908153
}}
* {{cite book
| last1 = Posamentier
| first1 = Alfred S.
| last2 = Lehmann
| first2 = Ingmar
| year = 2011
| title = The Glorious Golden Ratio
| url = https://books.google.com/books?id=Gw-lqvE6fNgC
| publisher = Prometheus Books
| isbn = 9-781-61614-424-1
}}
{{refend}}

==Further reading==
{{refbegin|30em}}
* {{Cite book
| last = Doczi
| first = György
| title = The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture
| year = 1981
| publisher = Shambhala
| location = Boston
| url = https://archive.org/details/poweroflimits00gyeo/
| url-access = registration
}}
* {{cite book
| editor-last = Hargittai
| editor-first = István
| title = Fivefold Symmetry
| publisher = World Scientific
| year = 1992
| isbn = 9789810206000
| url = https://archive.org/details/fivefoldsymmetry0000unse/
| url-access = registration
| ref = CITEREFHargittai2002
}}
* {{Cite book
| last = Huntley
| first = H. E.
| title = The Divine Proportion: A Study in Mathematical Beauty
| year = 1970
| publisher = Dover
| location = New York
| isbn = 978-0-486-22254-7
}}
* {{cite book
| editor-last=Schaaf
| editor-first=William L.
| year=1967
| title=The Golden Measure
| publisher=Stanford University
| series=California School Mathematics Study Group Reprint Series
| url=https://files.eric.ed.gov/fulltext/ED175696.pdf |archive-url=https://web.archive.org/web/20150425083400/http://files.eric.ed.gov/fulltext/ED175696.pdf |archive-date=2015-04-25 |url-status=live
}}
* {{Cite book
| last = Scimone
| first = Aldo
| title = La Sezione Aurea. Storia culturale di un leitmotiv della Matematica
| year = 1997
| publisher = Sigma Edizioni
| location = Palermo
| isbn = 978-88-7231-025-0
}}
* {{Cite book
| last = Walser
| first = Hans
| others = Peter Hilton trans.
| title = The Golden Section
| orig-year = ''Der Goldene Schnitt'' 1993
| year = 2001
| publisher = The Mathematical Association of America
| location = Washington, DC
| isbn = 978-0-88385-534-8
}}
{{refend}}

==External links==
<!--======================== {{No more links}} ============================
    | PLEASE BE CAUTIOUS IN ADDING MORE LINKS TO THIS ARTICLE. Wikipedia  |
    | is not a collection of links nor should it be used for advertising. |
    =======================================================================-->

* {{MathWorld|title=Golden Ratio|urlname=GoldenRatio}}
* {{cite web
| last = Bogomolny
| first = Alexander
| year = 2018
| title = Golden Ratio in Geometry
| website = Cut-the-Knot
| url = http://www.cut-the-knot.org/do_you_know/GoldenRatio.shtml
}}
* {{cite web | url = http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html| title = The Golden section ratio: Phi| last = Knott| first = Ron}} Information and activities by a mathematics professor.
* [http://www.maa.org/external_archive/devlin/devlin_05_07.html The Myth That Will Not Go Away], by [[Biography:Keith Devlin|Keith Devlin]], addressing multiple allegations about the use of the golden ratio in culture.
* [https://xkcd.com/spiral/ Spurious golden spirals] collected by [[Biography:Randall Munroe|Randall Munroe]]
* [https://www.youtube.com/watch?v=NdTVvWrD6r0 YouTube lecture on Zeno's mice problem and logarithmic spirals]

{{Algebraic numbers}}
{{Irrational number}}
{{Metallic ratios}}
{{Ancient Greek mathematics}}
{{Mathematical art}}


{{DEFAULTSORT:Golden Ratio}}
[[Category:Golden ratio| ]]
[[Category:Euclidean plane geometry]]
[[Category:Irrational numbers]]
[[Category:Quadratic irrational numbers]]
[[Category:Mathematical constants]]

{{Sourceattribution|Golden ratio}}