View source for Golden ratio

{{About|the number|the Ace of Base album|The Golden Ratio (album)|the calendar dates|Golden number (time)}}
{{pp-semi-protected|small=yes}}
[[File:Golden ratio line.svg|thumb|Line segments in the golden ratio]]

[[File:SimilarGoldenRectangles.svg|right|thumb|A [[golden rectangle]] with longer side <span style="color:blue;">'''''a'''''</span> and shorter side <span style="color:red;">'''''b'''''</span>, when placed adjacent to a square with sides of length <span style="color:blue;">'''''a'''''</span>, will produce a [[Similarity (geometry)|similar]] golden rectangle with longer side <span style="color:green;">'''''a + b'''''</span> and shorter side <span style="color:blue;">'''''a'''''</span>. This illustrates the relationship <math> \frac{a+b}{a} = \frac{a}{b} \equiv \varphi</math>.]]

In [[mathematics]], two quantities are in the '''golden ratio''' if their [[ratio]] is the same as the ratio of their [[summation|sum]] to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities ''a'' and ''b'' with ''a''&nbsp;>&nbsp;''b''&nbsp;>&nbsp;0,

:<math> \frac{a+b}{a} = \frac{a}{b} \ \stackrel{\text{def}}{=}\ \varphi, </math>
where the Greek letter [[Phi (letter)|phi]] (<math>\varphi</math> or <math>\phi</math>) represents the golden ratio.{{efn|If the constraint on a and b each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. ϕ is defined as the positive solution. The negative solution can be written as <math>\frac{1-\sqrt{5}}{2}</math>. The sum of the two solutions is one, and the product of the two solutions is negative one.}} It is an [[irrational number]] that is a solution to the quadratic equation <math>x^2-x-1=0</math>, with a value of:

:<math>\varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots.</math><ref name="a001622">{{OEIS2C|id=A001622}}</ref><!-- PLEASE DO NOT add additional decimal places here. There is long-standing consensus that additional decimal places here do not contribute to understanding. Thank you.
-->

The golden ratio is also called the '''golden mean''' or '''golden section''' (Latin: ''sectio aurea'').{{sfn|Livio|2003|pp=3, 81}}<ref name="dunlap">Dunlap, Richard A., ''The Golden Ratio and Fibonacci Numbers'', World Scientific Publishing, 1997</ref> Other names include '''extreme and mean ratio''',<ref name="Elements 6.3">Euclid, ''[http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Elements]'', Book 6, Definition 3.</ref> '''medial section''', '''divine proportion''', '''divine section''' (Latin: ''sectio divina''), '''golden proportion''', '''golden cut''',<ref>Summerson John, ''Heavenly Mansions: And Other Essays on Architecture'' (New York: W.W. Norton, 1963) p.&nbsp;37. "And the same applies in architecture, to the [[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> and '''golden number'''.<ref>Jay Hambidge, ''Dynamic Symmetry: The Greek Vase'', New Haven CT: Yale University Press, 1920</ref><ref>William Lidwell, Kritina Holden, Jill Butler, ''Universal Principles of Design: A Cross-Disciplinary Reference'', Gloucester MA: Rockport Publishers, 2003</ref><ref name = "Pacioli">Pacioli, Luca. ''[[De divina proportione]]'', Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.</ref>

[[Mathematician]]s since [[Euclid]] have studied the properties of the golden ratio, including its appearance in the dimensions of a [[regular pentagon]] and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same [[aspect ratio]]. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as [[financial market]]s, in some cases based on dubious fits to data.<ref>{{Cite news|first=Steven|last=Strogatz|authorlink=Steven Strogatz|title=Me, Myself, and Math: Proportion Control|newspaper=[[The New York Times]]|date=September 24, 2012|url=http://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/}}</ref> The golden ratio appears in some [[patterns in nature]], including the [[phyllotaxis|spiral arrangement of leaves]] and other plant parts.

Some twentieth-century [[artist]]s and [[architect]]s, including [[Le Corbusier]] and [[Salvador Dalí]], have proportioned their works to approximate the golden ratio—especially in the form of the [[golden rectangle]], in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be [[aesthetics|aesthetically]] pleasing.

{{TOC limit|3}}

==Calculation==
{| class="infobox bordered" cellpadding=5
| colspan="2" align="center" | {{Irrational numbers}}
|-
|[[Binary numeral system|Binary]]
| 1.1001111000110111011...
|-
| [[Decimal]]
| 1.6180339887498948482...<ref name="a001622"/>
|-
| [[Hexadecimal]]
| 1.9E3779B97F4A7C15F39...
|-
| [[Continued fraction]]
| <math>1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}}</math>
|-
| [[Algebraic number|Algebraic form]]
| <math>\frac{1 + \sqrt{5}}{2}</math>
|}
[[File:Phi uc lc.svg|thumb|upright|The Greek letter ''phi'' symbolizes the golden ratio. Usually, the lowercase form (φ or {{math|''φ''}}) is used. Sometimes the uppercase form (<math>\Phi</math>) is used for the [[Reciprocal (mathematics)|reciprocal]] of the golden ratio, 1/{{math|''φ''}}.<ref name="MathWorld GR Conjugate"/>]]

Two quantities ''a'' and ''b'' are said to be in the ''golden ratio'' {{math|''φ''}} if

:<math> \frac{a+b}{a} = \frac{a}{b} = \varphi.</math>

One method for finding the value of {{math|''φ''}} is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/{{math|''φ''}},

:<math>\frac{a+b}{a} = \frac{a}{a}+\frac{b}{a} = 1 + \frac{b}{a} = 1 + \frac{1}{\varphi}.</math>

Therefore,

:<math> 1 + \frac{1}{\varphi} = \varphi. </math>

Multiplying by {{math|''φ''}} gives

:<math>\varphi + 1 = \varphi^2</math>

which can be rearranged to

:<math>{\varphi}^2 - \varphi - 1 = 0.</math>

Using the [[quadratic formula]], two solutions are obtained:
:<math>\frac{1 + \sqrt{5}}{2} = 1.618\,033\,988\,7\dots</math> and <math>\frac{1 - \sqrt{5}}{2} = -0.618\,033\,988\,7\dots</math>

Because {{math|''φ''}} is the ratio between positive quantities, {{math|''φ''}} is necessarily positive:

:<math>\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots</math>

==History==
{{further|Mathematics and art}}
{{see also|Fibonacci number#History}}

According to [[Mario Livio]]:

{{quote|Some of the greatest mathematical minds of all ages, from [[Pythagoras]] and [[Euclid]] in [[ancient Greece]], through the medieval Italian mathematician [[Fibonacci|Leonardo of Pisa]] and the Renaissance astronomer [[Johannes Kepler]], to present-day scientific figures such as Oxford physicist [[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|2003|p=6}}}}

[[Ancient Greece|Ancient Greek]] mathematicians first studied what we now call the golden ratio because of its frequent appearance in [[geometry]];{{sfn|Livio|2003|p=4|ps=: "... line division, which [[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]]s.{{sfn|Livio|2003|pp=7–8}} According to one story, 5th-century BC mathematician [[Hippasus]] discovered that the golden ratio was neither a whole number nor a fraction (an [[irrational number]]), surprising [[Pythagoreanism|Pythagoreans]].{{sfn|Livio|2003|pp=4–5}} [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' ({{nowrap|c. 300 BC}}) provides several [[theorem|propositions]] and their proofs employing the golden ratio{{sfn|Livio|2003|p=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 the first known definition:<ref name=Hemenway>{{cite book|last=Hemenway|first=Priya|title=Divine Proportion: Phi In Art, Nature, and Science|year=2005|publisher=Sterling|location=New York|isbn=978-1-4027-3522-6|pages=20–21}}</ref>

{{quote|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|2003|p=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>{{cite book |title=Euclid's Elements of Geometry |author=Richard Fitzpatrick (translator) |year=2007 |isbn=978-0615179841|page=156}}</ref>}}}}

[[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. [[Abu Kamil]] (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of [[Fibonacci]] (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems, though never connected it to the [[Fibonacci number|series of numbers named after him]].{{sfn|Livio|2003|pp=88–96}} [[Luca Pacioli]] named his book ''[[Divina proportione]]'' ([[1509 in literature|1509]]) after the ratio and explored its properties including its appearance in some of the [[Platonic solids]].<ref name = "Pacioli"/>{{sfn|Livio|2003|pp=131–132}} [[Leonardo da Vinci]], who illustrated the aforementioned book, called the ratio the ''sectio aurea'' ('golden section').<ref>{{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}}</ref> 16th-century mathematicians such as [[Rafael Bombelli]] solved geometric problems using the ratio.{{sfn|Livio|2003|p=141}}

German mathematician Simon Jacob (d.&nbsp;1564) noted that [[#Relationship to Fibonacci sequence|consecutive Fibonacci numbers converge to the golden ratio]];<ref>{{cite journal|doi= 10.1006/hmat.1995.1033|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|year=1995|last1=Schreiber|first1=Peter|bibcode=}}</ref> this was rediscovered by [[Johannes Kepler]] in 1608.{{sfn|Livio|2003|pp=151–152}} The first known [[Decimal fractions|decimal]] approximation of the (inverse) golden ratio was stated as "about 0.6180340" in 1597 by [[Michael Maestlin]] of the [[University of Tübingen]] in a letter to Kepler, his former student.<ref>{{cite web| url = http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html | title = The Golden Ratio | work = The MacTutor History of Mathematics archive | accessdate = 2007-09-18}}</ref> The same year, Kepler wrote to Maestlin of the [[Kepler triangle]], which combines the golden ratio with the [[Pythagorean theorem]]. Kepler said of these:{{quote|Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel.}}

18th-century mathematicians [[Abraham de Moivre]], [[Daniel Bernoulli]], and [[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 [[Jacques Philippe Marie Binet]], for whom it was named "Binet's formula".<ref>{{MathWorld|title=Binet's Fibonacci Number Formula|urlname=BinetsFibonacciNumberFormula}}</ref> [[Martin Ohm]] first used the German term ''goldener Schnitt'' ('golden section') to describe the ratio in 1835.<ref>{{Cite book| title = A Mathematical History of Division in Extreme and Mean Ratio | last=Herz-Fischler |first=Roger | publisher = Wilfrid Laurier University Press | year = 1987 | isbn = 978-0889201521 | page = }}</ref> [[James Sully]] used the equivalent English term in 1875.<ref>{{Cite book|last1=Posamentier|first1=Alfred S.|authorlink1=Alfred S. Posamentier|last2=Lehmann |first2=Ingmar|title=The Glorious Golden Ratio|url=https://books.google.com/books?id=Gw-lqvE6fNgC&pg=PA8&pg=PT8|year=2011|publisher=[[Prometheus Books]]|isbn=9-781-61614-424-1|page=8}}</ref>

By 1910, mathematician [[Mark Barr]] began using the [[Greek alphabet|Greek letter]] [[Phi]] ('''φ''') as a [[symbol]] for the golden ratio.<ref>{{Cite book|last1=Posamentier|first1=Alfred S.|authorlink1=Alfred S. Posamentier|last2=Lehmann |first2=Ingmar|title=The Glorious Golden Ratio|url=https://books.google.com/books?id=Gw-lqvE6fNgC&pg=PT285|year=2011|publisher=[[Prometheus Books]]|isbn=9-781-61614-424-1|page=285}}</ref>{{efn|After Classical Greek sculptor [[Phidias]] (c.&nbsp;490–430&nbsp;BC);<ref>{{Cite book|last=Cook|first=Theodore Andrea|authorlink=Theodore Andrea Cook|title=The Curves of Life|year=1914|page=420|url=https://archive.org/details/cu31924028937179/page/n455|publisher=Constable and Company Ltd|location=London}}</ref> Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.<ref>{{cite magazine|first=Mark|last=Barr|title=Parameters of beauty|magazine=[[Architecture (magazine, 1900–1936)|Architecture]] (NY)|volume=60|page=325|year=1929}} Reprinted: {{cite magazine|title=Parameters of beauty|magazine=Think|volume= 10&ndash;11|publisher=International Business Machines Corporation|year=1944}}</ref>}} It has also been represented by [[tau]] ('''τ'''), the first letter of the [[ancient Greek]] τομή  ('cut' or 'section').{{sfn|Livio|2003|p=5}}<ref>{{Mathworld|title=Golden Ratio|urlname=GoldenRatio}}</ref>

Between 1973 and 1974, [[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>{{cite book |last=Gardner |first=Martin |authorlink=Martin Gardner |title=The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems|publisher=W. W. Norton & Company|year=2001|isbn=9780393020236|pp=77, 88|url=https://books.google.com/books?id=orz0SDEakpYC&pg=PA88}}.</ref> This led to [[Dan Shechtman]]'s early 1980s discovery of [[quasicrystal]]s,<ref>{{cite web|url=http://www.bloomberg.com/news/2011-10-05/technion-s-shechtman-wins-chemistry-nobel-for-discovery-of-quasicrystals.html|title=Tecnion's Shechtman Wins Nobel in Chemistry for Quasicrystals Discovery|last=Gerlin|first=Andrea|date=October 5, 2011|website=[[Bloomberg News|Bloomberg]]|archiveurl=https://web.archive.org/web/20141205014945/http://www.bloomberg.com/news/2011-10-05/technion-s-shechtman-wins-chemistry-nobel-for-discovery-of-quasicrystals.html|archivedate=December 5, 2014|accessdate=January 4, 2019}}</ref><ref>{{citation|title=Introduction to the Mathematics of Quasicrystals|first=Marko V.|last=Jaric|publisher=Elsevier|year=2012|isbn=9780323159470|page=x|url=https://books.google.com/books?id=OToVjZW9CKMC&pg=PR10|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.}}</ref> some of which exhibit [[icosahedral symmetry]].{{sfn|Livio|2003|pp=203–209}}<ref>{{cite journal |doi=|title=Quasicrystalline Materials|journal=[[American Scientist]]|volume=84|issue=3|pages=230–241|year=1996|last1=Goldman|first1=Alan I.|display-authors= et al.|bibcode=}}</ref>

==Applications and observations==

===Architecture===
{{further|Mathematics and architecture}}
[[File:Grande Mosquée de Kairouan, vue du minaret depuis la cour.jpg|thumb|upright|Minaret as seen from the courtyard of the [[Great Mosque of Kairouan]]]]
A 2004 geometrical analysis of earlier research into the [[Great Mosque of Kairouan]] (670) reveals an application of the golden ratio in much of the design.<ref name=Kairouan>Boussora, Kenza and Mazouz, Said, ''[http://www.emis.de/journals/NNJ/BouMaz.html The Use of the Golden Section in the Great Mosque of Kairouan], Nexus Network Journal'', vol. 6 no. 1 (Spring 2004).</ref> They found ratios close to the golden ratio in the overall layout and in the dimensions of the prayer space, the court, and the [[minaret]]. However, the areas with ratios close to the golden ratio were not part of the original plan, and were likely added in a reconstruction.<ref name=Kairouan/>

It has been speculated that the golden ratio was used by the designers of the [[Naqsh-e Jahan Square]] (1629) and the adjacent [[Lotfollah Mosque]].<ref>{{Cite book|last=Elliot | first = Jason | title = Mirrors of the Unseen: Journeys in Iran | year = 2006 | pages = 277, 284 | publisher = Macmillan | isbn = 978-0-312-30191-0 | url = https://books.google.com/?id=Gcs4IjUx3-4C&pg=PA284&dq=intitle:%22Mirrors+of+the+Unseen%22+golden-ratio+maidan}}</ref>

The Swiss [[architect]] [[Le Corbusier]], famous for his contributions to the [[modernism|modern]] [[International Style (architecture)|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>Le Corbusier, ''The Modulor'' p.&nbsp;25, as cited in Padovan, Richard, ''Proportion: Science, Philosophy, Architecture'' (1999), p.&nbsp;316, Taylor and Francis, {{isbn|0-419-22780-6}}</ref><ref>Frings, Marcus, ''[http://belveduto.de/text-nnj.htm The Golden Section in Architectural Theory], Nexus Network Journal'' vol. 4 no. 1 (Winter 2002).</ref>

Le Corbusier explicitly used the golden ratio in his [[Modulor]] system for the [[scale (ratio)|scale]] of [[Proportion (architecture)|architectural proportion]]. He saw this system as a continuation of the long tradition of [[Vitruvius]], Leonardo da Vinci's "[[Vitruvian Man]]", the work of [[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 [[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>Le Corbusier, ''The Modulor'', p.&nbsp;35, as cited in Padovan, Richard, ''Proportion: Science, Philosophy, Architecture'' (1999), p.&nbsp;320. Taylor & Francis. {{isbn|0-419-22780-6}}: "Both the paintings and the architectural designs make use of the golden section".</ref>

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>Urwin, Simon. ''Analysing Architecture'' (2003) pp.&nbsp;154–155, {{isbn|0-415-30685-X}}</ref>

===Art===
{{See also|Mathematics and art|History of aesthetics before the 20th century}}
[[File:Divina proportione - Illustration 13, crop & monochrome.jpg|thumb|left|upright|[[Leonardo da Vinci|Leonardo]]'s illustration of a dodecahedron from [[Pacioli]]'s ''[[Divina proportione]]'' (1509)]]
''[[Divina proportione]]'' (''Divine proportion''), a three-volume work by [[Luca Pacioli]], was published in 1509. Pacioli, a [[Franciscan]] [[friar]], was known mostly as a mathematician, but he was also trained and keenly interested in art. ''Divina proportione'' explored the mathematics of the golden ratio. 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 [[Vitruvius|Vitruvian]] system of rational proportions.{{sfn|Livio|2003|pp=134–135}} Pacioli also saw Catholic religious significance in the ratio, which led to his work's title.

[[Leonardo da Vinci]]'s illustrations of [[polyhedra]] in ''Divina proportione''<ref>{{cite web|url=http://www.georgehart.com/virtual-polyhedra/leonardo.html|title=Leonardo da Vinci's Polyhedra|last=Hart|first=George W.|authorlink=George W. Hart|year=1999|website=George W. Hart|accessdate=March 10, 2019}}</ref> 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">{{cite web|url=https://plus.maths.org/content/os/issue22/features/golden/index|title=The golden ratio and aesthetics|last=Livio|first=Mario|date=November 1, 2002|website=Plus Magazine|accessdate=November 26, 2018}}</ref> Similarly, although the ''[[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>{{cite web|author=Keith Devlin|url=http://www.maa.org/external_archive/devlin/devlin_05_07.html|title=The Myth That Will Not Go Away|accessdate=September 26, 2013|date=May 2007|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>{{cite web|author=Donald E. Simanek|url=http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm|title=Fibonacci Flim-Flam|accessdate=April 9, 2013|url-status=dead|archiveurl=https://web.archive.org/web/20100109045556/http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm|archivedate=January 9, 2010|df=}}</ref>

[[Salvador Dalí]], influenced by the works of [[Matila Ghyka]],<ref>{{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> 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 [[Jesus]] and dominates the composition.<ref name="livio plus"/><ref>Hunt, Carla Herndon and Gilkey, Susan Nicodemus. ''Teaching Mathematics in the Block'' pp. 44, 47, {{isbn|1-883001-51-X}}</ref>

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 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).<ref>Olariu, Agata, ''Golden Section and the Art of Painting'' [https://arxiv.org/abs/physics/9908036/ Available online]</ref> 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 root-5 proportions, and others with proportions like root-2, 3, 4, and 6.<ref>Tosto, Pablo, ''La composición áurea en las artes plásticas – El número de oro'', Librería Hachette, 1969, pp.&nbsp;134–144</ref>

[[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>[[Jan Tschichold]].  ''The Form of the Book'', p. 43 Fig 4. "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>]]

===Books and design===
{{Main|Canons of page construction}}

According to [[Jan Tschichold]],

<blockquote>There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.<ref>{{cite book|last=Tschichold|first=Jan|title=The Form of the Book|publisher=Hartley & Marks (1991)|isbn=0-88179-116-4|pp=27–28}}</ref></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>
{{Cite journal
 |title=The golden section: A most remarkable measure
 |first=Ronald|last=Jones
 |journal=The Structurist
 |volume=11|year=1971
 |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?
}}</ref><ref>
{{cite book
 | title = Famous problems and their mathematicians
 | author = Johnson, Art
 | publisher = Libraries Unlimited
 | year = 1999
 | isbn = 978-1-56308-446-1
 | page = 45
 | url = https://books.google.com/books?id=STKX4qadFTkC&pg=PA45&dq=light+switch
 | quote = The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
 }}</ref>{{sfn|Stakhov|Olsen|2009|p=21|ps=. "A credit card has a form of the golden rectangle."}}<ref>
{{cite book
 | title = Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown's bestselling novel
 | author = Cox, Simon
 | publisher = Barnes & Noble Books
 | year = 2004
 | isbn = 978-0-7607-5931-8
 | url = https://books.google.com/?id=TbjwhwLCEeAC&q=postcard&dq=postcard
 | 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.
 | page = 62
 }}</ref>

===Music===
[[Ernő Lendvai]] analyzes [[Béla Bartók]]'s works as being based on two opposing systems, that of the golden ratio and the [[acoustic scale]],<ref>Lendvai, Ernő (1971). ''Béla Bartók: An Analysis of His Music''. London: Kahn and Averill.</ref> though other music scholars reject that analysis.{{sfn|Livio|2003|p=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 34, 21, 13 and 8, and the main climax sits at the phi position".<ref name=Smith>Smith, Peter F. ''[https://books.google.com/books?id=ZgftUKoMnpkC&pg=PA83&dq=bartok+intitle:The+intitle:Dynamics+intitle:of+intitle:Delight+intitle:Architecture+intitle:and+intitle:Aesthetics&as_brr=0&ei=WkkSR5L6OI--ogLpmoyzBg&sig=Ijw4YifrLhkcdQSMVAjSL5g4zVk The Dynamics of Delight: Architecture and Aesthetics]'' (New York: Routledge, 2003) pp 83, {{isbn|0-415-30010-X}}</ref>

The musicologist [[Roy Howat]] has observed that the formal boundaries of Debussy's ''[[La Mer (Debussy)|La Mer]]'' correspond exactly to the golden section.<ref>{{Cite book| title = Debussy in Proportion: A Musical Analysis | author = Roy Howat | url = https://books.google.com/?id=4bwKykNp24wC&pg=PA169&dq=intitle:Debussy+intitle:in+intitle:Proportion+golden+la-mer | publisher = Cambridge University Press | year = 1983 | isbn = 978-0-521-31145-8 }}</ref> Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.<ref>{{Cite book| title = Debussy: La Mer | author = Simon Trezise | publisher = Cambridge University Press | year = 1994 | isbn = 978-0-521-44656-3 | page = 53 | url = https://books.google.com/?id=THD1nge_UzcC&pg=PA53&dq=inauthor:Trezise+golden+evidence }}</ref>

[[Pearl Drums]] positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a [[patent]] on this innovation.<ref>
{{cite web
 |url=http://www.pearldrum.com/premium-birch.asp 
 |title=Pearl Masters Premium 
 |accessdate=December 2, 2007 
 |publisher=Pearl Corporation 
 |url-status=dead 
 |archiveurl=https://web.archive.org/web/20071219191703/http://www.pearldrum.com/premium-birch.asp 
 |archivedate=December 19, 2007 
 |df= 
}}</ref>

Though [[Heinz Bohlen]] proposed the non-octave-repeating [[833 cents scale]] based on [[combination tone]]s, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618... is 833.090... cents ({{audio|Golden ratio on C.mid|Play}}).<ref>"[http://www.huygens-fokker.org/bpsite/833cent.html An 833 Cents Scale: An experiment on harmony]", ''Huygens-Fokker.org''. Accessed December 1, 2012.</ref>

===Nature===
[[File:Aeonium tabuliforme.jpg|thumb|upright|Detail of the saucer plant, ''[[Aeonium tabuliforme]]'', showing the multiple spiral arrangement ([[parastichy]])]]
{{main|Patterns in nature}}
{{see also|Fibonacci number#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|2003|p=154}}

The psychologist [[Adolf Zeising]] noted that the golden ratio appeared in [[phyllotaxis]] and argued from these [[patterns in nature]] that the golden ratio was a universal law.<ref>{{Cite book | title = Proportion | author = Richard Padovan | publisher = Taylor & Francis | year = 1999 | isbn = 978-0-419-22780-9 | pages = 305–306 | url = https://books.google.com/?id=Vk_CQULdAssC&pg=PA306&dq=%22contained+the+ground-principle+of+all+formative+striving%22 }}</ref><ref>{{cite journal|journal=Nexus Network Journal|first=Richard|last=Padovan|title=Proportion: Science, Philosophy, Architecture |volume=4 |pages=113–122 |doi=10.1007/s00004-001-0008-7 |year=2002 |issue=1}}</ref> Zeising wrote in 1854 of a universal [[orthogenesis|orthogenetic]] law of "striving for beauty and completeness in the realms of both nature and art".<ref>{{cite book |first=Adolf |last=Zeising |title=Neue Lehre van den Proportionen des meschlischen Körpers |year=1854 |page=preface |nopp=true}}</ref>

In 2010, the journal ''Science'' reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.<ref>{{cite web|url=http://www.eurekalert.org/pub_releases/2010-01/haog-grd010510.php |title=Golden ratio discovered in a quantum world |publisher=Eurekalert.org |date=2010-01-07 |accessdate=2011-10-31}}</ref>

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.<ref>Pommersheim, James E., Tim K. Marks, and [[Erica Flapan|Erica L. Flapan]], eds. 2010. "Number Theory: A Lively Introduction with Proofs, Applications, and Stories". John Wiley and Sons: 82.</ref>

===Optimization===
The golden ratio is key to the [[golden-section search]].

==Mathematics==

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

====Contradiction from an expression in lowest terms====
[[File:Whirling squares.svg|thumb|right|If &phi; 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 integers have a lower bound, so &phi; cannot be rational.]]

Recall that:

: the whole is the longer part plus the shorter part;
: the whole is to the longer part as the longer part is to the shorter part.

If we call the whole ''n'' and the longer part ''m'', then the second statement above becomes

: ''n'' is to ''m'' as ''m'' is to ''n''&nbsp;−&nbsp;''m'',

or, algebraically

: <math> \frac nm = \frac{m}{n-m}.\qquad (*) </math>

To say that the golden ratio {{math|''φ''}} is rational means that {{math|''φ''}} is a fraction ''n''/''m'' where ''n'' and ''m'' are integers.  We may take ''n''/''m'' to be in [[lowest terms]] and ''n'' and ''m'' to be positive. But if ''n''/''m'' is in lowest terms, then the identity labeled (*) above says ''m''/(''n''&nbsp;−&nbsp;''m'') is in still lower terms. That is a contradiction that follows from the assumption that {{math|''φ''}} is rational.

====By irrationality of {{radic|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>\textstyle\frac{1 + \sqrt{5}}{2}</math> is rational, then <math>\textstyle2\left(\frac{1 + \sqrt{5}}{2}\right) - 1= \sqrt{5}</math> is also rational, which is a contradiction if it is already known that the square root of a non-[[square number|square]] [[natural number]] is irrational.

===Minimal polynomial===
The golden ratio is also an [[algebraic number]] and even an [[algebraic integer]]. It has [[minimal polynomial (field theory)|minimal polynomial]]

:<math>x^2 - x - 1.</math>

Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.

===Golden ratio conjugate===
The conjugate root to the minimal polynomial x<sup>2</sup> − x − 1 is

:<math>-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt{5}}{2} = -0.61803\,39887\dots.</math>

The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ''b/a''), and is sometimes referred to as the ''golden ratio conjugate''.<ref name="MathWorld GR Conjugate">{{MathWorld|title=Golden Ratio Conjugate|urlname=GoldenRatioConjugate}}</ref> It is denoted here by the capital Phi ('''<math>\Phi</math>'''):

:<math>\Phi = {1 \over \varphi} = \varphi^{-1} = 0.61803\,39887\ldots.</math>

Alternatively, '''<math>\Phi</math>''' can be expressed as

:<math>\Phi = \varphi -1 = 1.61803\,39887\ldots -1 = 0.61803\,39887\ldots.</math>

This illustrates the unique property of the golden ratio among positive numbers, that

:<math>{1 \over \varphi} = \varphi - 1,</math>

or its inverse:

:<math>{1 \over \Phi} = \Phi + 1.</math>

This means 0.61803...:1 = 1:1.61803....

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

The formula {{math|''φ''}} = 1 + 1/{{math|''φ''}} can be expanded recursively to obtain a [[continued fraction]] for the golden ratio:<ref>{{Cite book| title = Concrete Abstractions: An Introduction to Computer Science Using Scheme
|author1=Max. Hailperin |author2=Barbara K. Kaiser |author3=Karl W. Knight | publisher = Brooks/Cole Pub. Co | year = 1998 | isbn = 978-0-534-95211-2 | url = https://books.google.com/?id=yYyVRueWlZ8C&pg=PA63&dq=continued-fraction+substitute+golden-ratio }}</ref>

:<math>\varphi = [1; 1, 1, 1, \dots] = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>

and its reciprocal:

:<math>\varphi^{-1} = [0; 1, 1, 1, \dots] = 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>

The [[Convergent (continued fraction)|convergent]]s of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive [[Fibonacci numbers]].

The equation {{math|''φ''}}<sup>2</sup> = 1 + {{math|''φ''}} likewise produces the [[Nested radical|continued square root]], or infinite surd, form:

:<math>\varphi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}.</math>

An infinite series can be derived to express ''φ'':<ref>Brian Roselle, [http://sites.google.com/site/goldenmeanseries/ "Golden Mean Series"]</ref>

:<math>\varphi=\frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{n+1}(2n+1)!}{4^{2n+3}n!(n+2)!}.</math>

Also:

:<math>\varphi = 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ</math>
:<math>\varphi = {1 \over 2}\csc(\pi/10) = {1 \over 2}\csc 18^\circ</math>
:<math>\varphi = 2\cos(\pi/5)=2\cos 36^\circ</math>
:<math> \varphi = 2\sin(3\pi/10)=2\sin 54^\circ. </math>

These correspond to the fact that the length of the diagonal of a regular pentagon is {{math|''φ''}} times the length of its side, and similar relations in a [[pentagram]].

===Geometry===
[[File:FakeRealLogSpiral.svg|thumb|350px|right|Approximate and true [[golden spiral]]s. The <span style="color:green;">green</span> spiral is made from quarter-circles tangent to the interior of each square, while the <span style="color:maroon;">red</span> spiral is a Golden Spiral, a special type of [[logarithmic spiral]]. Overlapping portions appear <span style="color:olive;">yellow</span>. The length of the side of one square divided by that of the next smaller square is the golden ratio.]]
The number {{math|''φ''}} turns up frequently in [[geometry]], particularly in figures with pentagonal [[symmetry]].
The length of a regular [[pentagon]]'s [[diagonal]] is {{math|''φ''}} times its side.
The vertices of a regular [[icosahedron]] are those of [[3|three]] mutually [[orthogonal]] golden rectangles.

There is no known general [[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]]''). However, a useful approximation results from dividing the sphere into parallel bands of equal [[surface area]] and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/{{math|''φ''}} ≅ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory [[artificial satellite|satellite]] [[STARSHINE|Starshine-3]].<ref>{{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 |accessdate=2007-04-16}}</ref>
{{Clear}}

====Dividing a line segment by interior division====
[[File:Goldener Schnitt Konstr beliebt.svg|right|thumb|250px|Dividing a line segment by interior division according to the golden ratio]]
# Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the [[hypotenuse]] AC.
# Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D.
# Draw an arc with center A and radius AD. This arc intersects the original line segment AB at point S. Point S divides the original line segment AB into line segments AS and SB with lengths in the golden ratio.
{{Clear}}

====Dividing a line segment by exterior division====
[[File:Goldener Schnitt (Äußere Teilung).svg|right|thumb|250px|Dividing a line segment by exterior division according to the golden ratio]]
# Draw a line segment AS and construct off the point S a segment SC perpendicular to AS and with the same length as AS.
# Do bisect the line segment AS with M. 
# A circular arc around M with radius MC intersects in point B the straight line through points A and S (also known as the extension of AS). The ratio of AS to the constructed segment SB 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]]s produce [[geometric construction]]s that determine two aligned [[line segment]]s where the ratio of the longer one to the shorter one is the golden ratio.
{{Clear}}

====Golden triangle, pentagon and pentagram====
[[File:Golden triangle (math).svg|right|thumb|[[Golden triangle (mathematics)|Golden triangle]]. The double-red-arched angle is 36 degrees, or <math>\frac{\pi}{5}</math> radians.]]

=====Golden triangle=====
The [[Golden triangle (mathematics)|golden triangle]] can be characterized as an [[isosceles triangle]] ABC with the property that [[bisection|bisecting]] the angle C produces a new [[triangle]] CXB which is a [[similar triangle]] to the original.

If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.

Suppose XB has length 1, and we call BC length {{math|''φ''}}. Because of the isosceles triangles XC=XA and BC=XC, so these are also length&nbsp;φ. Length AC&nbsp;=&nbsp;AB, therefore equals {{math|''φ''}}&nbsp;+&nbsp;1. But triangle ABC is similar to triangle CXB, so AC/BC&nbsp;=&nbsp;BC/BX, AC/{{math|''φ''}}&nbsp;=&nbsp;φ/1, and so AC also equals {{math|''φ''}}<sup>2</sup>. Thus {{math|''φ''}}<sup>2</sup> =&nbsp;φ&nbsp;+&nbsp;1, confirming that {{math|''φ''}} is indeed the golden ratio.

Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to {{math|''φ''}}, while the [[Multiplicative inverse|inverse]] ratio is&nbsp;φ&nbsp;&minus;&nbsp;1.

=====Pentagon=====
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.<ref name = "Pacioli"/>

=====Odom's construction=====
[[File:Odom.svg|thumb|218 px|Let A and B be midpoints of the sides EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle of DEF at C.<br/><center><math>\tfrac{|AB|}{|BC|}=\tfrac{|AC|}{|AB|}=\phi</math></center>]]
[[George Phillips Odom, Jr|George Odom]] has given a remarkably simple construction for {{φ}} involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the [[intersecting chords theorem]] and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]] who published it in Odom's name as a diagram in the ''[[American Mathematical Monthly]]'' accompanied by the single word "Behold!" <ref>{{cite web|title=Quandaries and Queries|url=http://mathcentral.uregina.ca/qq/database/QQ.09.02/mary1.html|publisher=Math Central|accessdate=23 October 2011|author=Chris and Penny}}</ref>

=====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.]]
The golden ratio plays an important role in the geometry of [[pentagram]]s. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is {{φ}}, as the four-color illustration shows.

The pentagram includes ten [[isosceles triangle]]s: five [[acute triangle|acute]] and five [[obtuse triangle|obtuse]] isosceles triangles. In all of them, the ratio of the longer side to the shorter side is {{φ}}. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.

=====Ptolemy's theorem=====
[[File:Ptolemy Pentagon.svg|thumb|The golden ratio in a regular pentagon can be computed using [[Ptolemy's theorem]].]]

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 ''b'', and short edges are ''a'', then Ptolemy's theorem gives ''b''<sup>2</sup>&nbsp;=&nbsp;''a''<sup>2</sup>&nbsp;+&nbsp;''ab'' which yields

:<math>{b \over a}={{1+\sqrt{5}}\over 2}.</math>

====Scalenity of triangles====
Consider a [[triangle]] with sides of lengths ''a'', ''b'', and ''c'' in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios ''a''/''b'' and ''b''/''c''. The scalenity is always less than {{φ}} and can be made as close as desired to {{φ}}.<ref>''[[American Mathematical Monthly]]'', pp.&nbsp;49–50, 1954.</ref>

====Triangle whose sides form a geometric progression====
If the side lengths of a triangle form a [[geometric progression]] and are in the ratio 1 : ''r'' : ''r''<sup>2</sup>, where ''r'' is the common ratio, then ''r'' must lie in the range {{φ}}−1 < ''r'' < {{φ}}, which is a consequence of the [[triangle inequality]] (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If ''r'' = {{φ}} then the shorter two sides are 1 and {{φ}} but their sum is {{φ}}<sup>2</sup>, thus ''r'' < {{φ}}. A similar calculation shows that ''r'' > {{φ}}−1. A triangle whose sides are in the ratio 1 : {{radic|{{φ}}}} : {{φ}} is a right triangle (because 1 + {{φ}} = {{φ}}<sup>2</sup>) known as a [[Kepler triangle]].<ref name=herz/>

====Golden triangle, rhombus, and rhombic triacontahedron====
[[File:GoldenRhombus.svg|thumb|One of the rhombic triacontahedron's rhombi]]

[[File:Rhombictriacontahedron.svg|thumb|All of the faces of the rhombic triacontahedron are golden rhombi]]

A [[golden rhombus]] is a [[rhombus]] whose diagonals are in the golden ratio. The [[rhombic triacontahedron]] is a [[convex polytope]] that has a very special property: all of its faces are golden rhombi.  In the [[rhombic triacontahedron]] the [[dihedral angle]] between any two adjacent rhombi is 144°, which is twice the isosceles angle of a golden triangle and four times its most acute angle.<ref>{{citation|title=Catalan solids derived from three-dimensional-root systems and quaternions|first1=Mehmet|last1=Koca|first2=Nazife Ozdes|last2=Koca|first3=Ramazan|last3=Koç|journal=	Journal of Mathematical Physics|volume=51|issue=4|year=2010|page=043501|doi=10.1063/1.3356985|arxiv=0908.3272|bibcode=2010JMP....51d3501K}}.</ref>

===Relationship to Fibonacci sequence===
The mathematics of the golden ratio and of the [[Fibonacci number|Fibonacci sequence]] are intimately interconnected. The Fibonacci sequence is:

:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

A [[closed-form expression]] for the Fibonacci sequence involves the golden ratio:

:<math>F\left(n\right)
 = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}
 = {{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}}.</math>

[[File:Fibonacci spiral 34.svg|thumb|250px|right|A [[Fibonacci sequence|Fibonacci spiral]] which approximates the golden spiral, using Fibonacci sequence square sizes up to 34. The spiral is drawn starting from the inner 1×1 square and continues outwards to successively larger squares.]]
[[File:Golden Square fractal with T-branching 8.svg|thumb|Golden squares with [[T-square (fractal)|T-branching]]]]
[[File:Golden Square fractal 6.svg|thumb|Golden square fractal]]

The golden ratio is the [[limit of a sequence|limit]] of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as shown by [[Kepler]]:<ref>{{Cite book | title = Elementary number theory in nine chapters | edition = 2nd | last = Tattersall | first = James Joseph | publisher = [[Cambridge University Press]] | year = 2005 | isbn = 978-0-521-85014-8 | page = 28 | url = https://books.google.com/?id=QGgLbf2oFUYC&pg=PA29 }}</ref>

:<math>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>

In other words, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates {{φ}}; e.g., 987/610&nbsp;{{math|≈}}&nbsp;1.6180327868852. These approximations are alternately lower and higher than {{φ}}, and converge to {{φ}} as the Fibonacci numbers increase, and:

:<math>\sum_{n=1}^{\infty}|F_n\varphi-F_{n+1}|

= \varphi.</math>

More generally:
:<math>\lim_{n\to\infty}\frac{F_{n+a}}{F_n} = \varphi^a,</math>

where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when <math>a = 1.</math>

Furthermore, the successive powers of {{φ}} obey the Fibonacci [[recurrence relation|recurrence]]:

:<math>\varphi^{n+1} = \varphi^n + \varphi^{n-1}.</math>

This identity allows any polynomial in {{φ}} to be reduced to a linear expression. For example:
: <math>
\begin{align}
3\varphi^3 - 5\varphi^2 + 4 & = 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\
& = 3[(\varphi + 1) + \varphi] - 5(\varphi + 1) + 4 \\
& = \varphi + 2 \approx 3.618.
\end{align}
</math>

The reduction to a linear expression can be accomplished in one step by using the relationship
:<math>\varphi^k=F_k \varphi +F_{k-1},</math>
where <math>F_k</math> is the ''k''th Fibonacci number.

However, this is no special property of {{φ}}, because polynomials in any solution ''x'' to a [[quadratic equation]] can be reduced in an analogous manner, by applying:
:<math>x^2=ax+b</math>
for given coefficients ''a'', ''b'' such that ''x'' satisfies the equation. Even more generally, any [[rational function]] (with rational coefficients) of the root of an irreducible ''n''th-degree polynomial over the rationals can be reduced to a polynomial of degree {{nowrap|''n'' ‒ 1.}}  Phrased in terms of [[field theory (mathematics)|field theory]], if α is a root of an irreducible ''n''th-degree polynomial, then <math>\Q(\alpha)</math> has degree ''n'' over <math>\Q</math>, with basis <math>\{1, \alpha, \dots, \alpha^{n-1}\}.</math>

===Symmetries===
The golden ratio and inverse golden ratio <math>\varphi_\pm = (1\pm \sqrt{5})/2</math> have a set of symmetries that preserve and interrelate them. They are both preserved by the [[fractional linear transformation]]s <math>x, 1/(1-x), (x-1)/x,</math> – this fact corresponds to the identity and the definition quadratic equation.
Further, they are interchanged by the three maps <math>1/x, 1-x, x/(x-1)</math> – they are reciprocals, symmetric about <math>1/2</math>, and (projectively) symmetric about 2.

More deeply, these maps form a subgroup of the [[modular group]] <math>\operatorname{PSL}(2,\mathbf{Z})</math> isomorphic to the [[symmetric group]] on 3 letters, <math>S_3,</math> corresponding to the [[stabilizer subgroup|stabilizer]] of the set <math>\{0,1,\infty\}</math> of 3 standard points on the [[projective line]], and the symmetries correspond to the quotient map <math>S_3 \to S_2</math> – the subgroup <math>C_3 < S_3</math> consisting of the 3-cycles and the identity <math>() (0 1 \infty) (0 \infty 1)</math> fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.

===Other properties===
The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see ''Alternate forms'' above). It is, for that reason, one of the [[continued fraction#A property of the golden ratio φ|worst cases]] of [[Lagrange's approximation theorem]] and it is an extremal case of the [[Hurwitz's theorem (number theory)|Hurwitz inequality]] for [[Diophantine approximation]]s. This may be why angles close to the golden ratio often show up in [[phyllotaxis]] (the growth of plants).<ref>[http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html Fibonacci Numbers and Nature – Part 2 : Why is the Golden section the "best" arrangement?], from [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/ Dr. Ron Knott's] [http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/ Fibonacci Numbers and the Golden Section], retrieved 2012-11-29.</ref>

The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with {{φ}}:

:<math>\varphi^2 = \varphi + 1 = 2.618\dots</math>

:<math>{1 \over \varphi} = \varphi - 1 = 0.618\dots.</math>

The sequence of powers of {{φ}} contains these values 0.618..., 1.0, 1.618..., 2.618...; more generally,
any power of {{φ}} is equal to the sum of the two immediately preceding powers:

: <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 {{φ}} into a multiple of {{φ}} and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of {{φ}}:

If <math> \lfloor n/2 - 1 \rfloor = m </math>, then:

:<math> \!\ \varphi^n = \varphi^{n-1} + \varphi^{n-3} + \cdots + \varphi^{n-1-2m} + \varphi^{n-2-2m} </math>

:<math> \!\ \varphi^n - \varphi^{n-1} = \varphi^{n-2} . </math>

When the golden ratio is used as the base of a [[numeral system]] (see [[Golden ratio base]], sometimes dubbed ''phinary'' or {{φ}}''-nary''), every integer has a terminating representation, despite {{φ}} being irrational, but every fraction has a non-terminating representation.

The golden ratio is a [[Fundamental unit (number theory)|fundamental unit]] of the [[algebraic number field]] <math>\mathbb{Q}(\sqrt{5})</math> and is a [[Pisot–Vijayaraghavan number]].<ref>{{MathWorld|urlname=PisotNumber |title=Pisot Number}}</ref> In the field <math>\mathbb{Q}(\sqrt{5})</math> we have <math>\varphi^n = {{L_n + F_n \sqrt{5}} \over 2}</math>, where <math>L_n</math> is the <math>n</math>-th [[Lucas number]].

The golden ratio also appears in [[hyperbolic geometry]], as the maximum distance from a point on one side of an [[ideal triangle]] to the closer of the other two sides: this distance, the side length of the [[equilateral triangle]] formed by the points of tangency of a circle inscribed within the ideal triangle, is <math>4\log(\varphi)</math>.<ref>[http://www.cabri.net/abracadabri/GeoNonE/GeoHyper/KBModele/Biss3KB.html Horocycles exinscrits : une propriété hyperbolique remarquable], cabri.net, retrieved 2009-07-21.</ref>

===Decimal expansion===
The golden ratio's decimal expansion can be calculated directly from the expression
:<math>\varphi = {1+\sqrt{5} \over 2}</math>

with {{radic|5}} ≈ 2.2360679774997896964 {{OEIS2C|id=A002163}}. The [[square root of 5]] can be calculated with the [[Babylonian method]], starting with an initial estimate such as ''x''{{φ}} = 2 and [[iterative method|iterating]]

:<math>x_{n+1} = \frac{(x_n + 5/x_n)}{2}</math>

for ''n'' = 1, 2, 3, ..., until the difference between ''x''<sub>''n''</sub> and ''x''<sub>''n''−1</sub> becomes zero, to the desired number of digits.

The Babylonian algorithm for {{radic|5}} is equivalent to [[Newton's method]] for solving the equation ''x''<sup>2</sup>&nbsp;−&nbsp;5 = 0. In its more general form, Newton's method can be applied directly to any [[algebraic equation]], including the equation ''x''<sup>2</sup>&nbsp;−&nbsp;x&nbsp;−&nbsp;1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,

:<math>x_{n+1} = \frac{x_n^2 + 1}{2x_n - 1},</math>

for an appropriate initial estimate ''x''{{φ}} such as ''x''{{φ}} = 1. A slightly faster method is to rewrite the equation as ''x''&nbsp;−&nbsp;1&nbsp;−&nbsp;1/''x'' = 0, in which case the Newton iteration becomes

:<math>x_{n+1} = \frac{x_n^2 + 2x_n}{x_n^2 + 1}.</math>

These iterations all [[quadratic convergence|converge quadratically]]; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with [[arbitrary-precision arithmetic|arbitrary precision]]. The time needed to compute ''n'' digits of the golden ratio is proportional to the time needed to divide two ''n''-digit numbers. This is considerably faster than known algorithms for the [[transcendental number]]s '''[[pi|{{pi}}]]''' and [[e (mathematical constant)|{{mvar|'''e'''}}]].

An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers ''F'' <sub>25001</sub> and ''F'' <sub>25000</sub>, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.

The decimal expansion of the golden ratio {{φ}}<ref name="a001622"/> has been calculated to an accuracy of two trillion ({{val|2|e=12}} = 2,000,000,000,000) digits.<ref>{{Cite web |url=http://www.numberworld.org/digits/GoldenRatio/ |title=Golden Ratio |first= Alexander J. |last=Yee |date=17 August 2015 |website=numberword.org}}  Independent computations done by Ron Watkins and Dustin Kirkland.</ref>

==Pyramids==
{{Further|mathematics and art}}
[[File:Mathematical Pyramid.svg|right|thumb|250px|A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a of <math>1:\sqrt{\varphi}:\varphi</math> and <math>3:4:5</math> and <math>1:4/\pi:1.61899</math> are of particular interest in relation to Egyptian pyramids.]]
Both Egyptian pyramids and the regular [[square pyramid]]s that resemble them can be analyzed with respect to the golden ratio and other ratios.

===Mathematical pyramids===
A pyramid in which the apothem (slant height along the bisector of a face) is equal to {{φ}} times the semi-base (half the base width) is sometimes called a ''golden pyramid''. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is <math>\sqrt{\varphi}</math> times the semi-base (that is, the slope of the face is <math>\sqrt{\varphi}</math>); the square of the height is equal to the area of a face, {{φ}} times the square of the semi-base.

The medial [[right triangle]] of this "golden" pyramid (see diagram), with sides <math>1:\sqrt{\varphi}:\varphi</math> is interesting in its own right, demonstrating via the [[Pythagorean theorem]] the relationship <math>\sqrt{\varphi} = \sqrt{\varphi^2 - 1}</math> or <math>\varphi = \sqrt{1 + \varphi}</math>. This [[Kepler triangle]]<ref>{{Cite book|last=Alison|first=Jim|editor1-last=Nixon|editor1-first=Steve|title=The Best of Astraea: 17 Articles on Science, History and Philosophy|url=|publisher=Astrea Web Radio|isbn=978-1-4259-7040-6|year=2006|pages=92–93}}</ref>
is the only right triangle proportion with edge lengths in [[geometric progression]],<ref name=Ghyka>{{cite book|last=Ghyka|first=Matila|authorlink=Matila Ghyka|title=The Geometry of Art and Life|location=New York|publisher=Dover|year=1977|url=https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA22|pages=22–24|isbn=9780486235424}}</ref><ref name=herz>{{Cite book|title=The Shape of the Great Pyramid|last=Herz-Fischler|first=Roger|publisher=Wilfrid Laurier University Press|year=2000|isbn=978-0-88920-324-2|page=|url=}}{{page needed|date=January 2019}}</ref> just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in [[arithmetic progression]]. The angle with [[trigonometric functions#tangent|tangent]] <math>\sqrt{\varphi}</math> corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827... degrees (51° 49' 38").<ref>Midhat Gazale, ''Gnomon: From Pharaohs to Fractals'', Princeton Univ. Press, 1999</ref>

A nearly similar pyramid shape, but with rational proportions, is described in the [[Rhind Mathematical Papyrus]] (the source of a large part of modern knowledge of ancient [[Egyptian mathematics]]), based on the 3:4:5 triangle;<ref name = "maor"/> the face slope corresponding to the angle with tangent 4/3 is, to two decimal places, 53.13 degrees (53 degrees and 8 minutes). The slant height or apothem is 5/3 or 1.666... times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers,<ref>[[Lancelot Hogben|Hogben, Lancelot]], ''Mathematics for the Million'', London: Allen & Unwin, 1942, p.&nbsp;63., as cited by [[Dick Teresi|Teresi, Dick]], ''Lost Discoveries: The Ancient Roots of Modern Science—from the Babylonians to the Maya'', New York: Simon & Schuster, 2003, p.56</ref> and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.<ref name="maor">[[Eli Maor]], ''Trigonometric Delights'', Princeton Univ. Press, 2000</ref>

Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2{{pi}} times the height, or h:b = 4:{{pi}}. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the Kepler triangle. This pyramid relationship corresponds to the [[mathematical coincidences|coincidental relationship]] <math>\sqrt{\varphi} \approx 4/\pi</math>.

Egyptian pyramids very close in proportion to these mathematical pyramids are known.<ref name=Herkommer>{{cite web|url=http://www.petrospec-technologies.com/Herkommer/pyramid/pyramid.htm |title=The Great Pyramid, The Great Discovery, and The Great Coincidence |accessdate=2007-11-25 |url-status=dead |archiveurl=https://web.archive.org/web/20140102194148/http://www.petrospec-technologies.com/Herkommer/pyramid/pyramid.htm |archivedate=2014-01-02 |df= }}</ref><ref name=herz/>

===Egyptian pyramids===
[[File:Egypt. Pyramids. Pyramids and palm grove reflections LOC matpc.23063.jpg|thumb|left|The [[Great Pyramid of Giza]]]]
One Egyptian pyramid that is close to a "golden pyramid" is the [[Great Pyramid of Giza]] (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is close to the "golden" pyramid inclination of 51° 50' – and even closer to the {{pi}}-based pyramid inclination of 51° 51'.  However, several other mathematical theories of the shape of the great pyramid, based on rational slopes, have been found to be both more accurate and more plausible explanations for the 51° 52' slope.<ref name=herz/>

In the mid-nineteenth century, Friedrich Röber studied various Egyptian pyramids including those of [[Pyramid of Khafre|Khafre]], [[Pyramid of Menkaure|Menkaure]], and some of the [[Giza pyramid complex|Giza]], [[Saqqara]], and [[Abusir]] groups. He did not apply the golden ratio to the Great Pyramid of Giza, but instead agreed with [[John Shae Perring]] that its side-to-height ratio is 8:5. For all the other pyramids he applied measurements related to the Kepler triangle, and claimed that either their whole or half-side lengths are related to their heights by the golden ratio.<ref>{{Cite book|title=The Shape of the Great Pyramid|last=Herz-Fischler|first=Roger|publisher=Wilfrid Laurier University Press|year=2000|isbn=978-0-88920-324-2|pages=80–89|url=https://books.google.com/?id=066T3YLuhA0C&pg=PA81}}</ref>

In 1859, the [[Pyramidology|pyramidologist]] [[John Taylor (1781–1864)|John Taylor]] misinterpreted [[Herodotus]] ({{nowrap|c. 440 BC}}) as indicating that the Great Pyramid's height squared equals the area of one of its face triangles.{{efn|Taylor translated Herodotus: "this Pyramid, which is four-sided, each face is, on every side 8 [[plethra]], and the height equal." He interpreted this imaginatively, and in 1860, [[John Herschel]] was the first of many authors to repeat his false claim. In 2000, Roger Herz-Fischler traced the error back to Taylor.{{sfn|Livio|2003|pp=55–58}}}} This led Taylor to claim that, in the Great Pyramid, the golden ratio is represented by the ratio of the length of the face (the slope height, inclined at an angle [[θ]] to the ground) to half the length of the side of the square base (equivalent to the [[Cosecant|secant]] of the angle θ).<ref>Taylor, ''The Great Pyramid: Why Was It Built and Who Built It?'', 1859</ref> The above two lengths are about {{convert|186.4|m|ft}} and {{convert|115.2|m|ft}}, respectively.{{sfn|Livio|2003|pp=55–58}} The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, [[Richard William Howard Vyse|Howard Vyse]] reported the great pyramid height {{convert|148.2|m|ft}}, and half-base {{convert|116.4|m|ft}}, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.<ref name=Ghyka/>

[[Eric Temple Bell]], mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as {{pi}} or {{φ}}.<ref>{{Cite book |last=Bell |first=Eric Temple |title=The Development of Mathematics |location=New York |publisher=Dover |year=1940 |page=40 |url=https://books.google.com/books?id=_5KAnw3QMC8C&pg=PA40#v=onepage|isbn=9780486272399 }}</ref> Example geometric problems of pyramid design in the Rhind papyrus correspond to various rational slopes.<ref name=herz/>

Michael Rice<ref>Rice, Michael, ''Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C'' p. 24 Routledge, 2003, {{isbn|0-415-26876-1}}</ref> asserts that principal authorities on the history of [[Egyptian architecture]] have argued that the Egyptians were well acquainted with the golden ratio and that it is part of the mathematics of the pyramids, citing Giedon (1957).<ref>S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Lectures in the Fine Arts, 457, as cited in Rice, Michael, ''Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C'' p. 24 Routledge, 2003</ref> Historians of science have long debated whether the Egyptians had any such knowledge, contending that its appearance in the Great Pyramid is the result of chance.<ref name=Markowsky>{{Cite journal
 | last = Markowsky
 | first = George
 | date = January 1992
 | title = Misconceptions about the Golden Ratio
 | journal = College Mathematics Journal
 | volume = 23
 | issue = 1
 | doi = 10.2307/2686193
 | url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
 | accessdate =
 | jstor = 2686193
 | publisher = Mathematical Association of America
 | pages = 2–19
 }}</ref>

==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.]]
*Some specific proportions in the bodies of many animals (including humans)<ref name=pheasant>{{Cite book |first=Stephen |last=Pheasant |title=Bodyspace |location=London |publisher=Taylor & Francis |year=1998 |isbn=978-0-7484-0067-6 }}</ref><ref name=vanLaack>{{Cite book |first=Walter |last=van Laack |title=A Better History Of Our World: Volume 1 The Universe |location=Aachen |publisher=van Laach GmbH |year=2001 }}</ref> and parts of the shells of mollusks<ref name="dunlap"/> are often claimed to be in 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/> The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.<ref name=vanLaack/> The [[nautilus]] shell, the construction of which proceeds in a [[logarithmic spiral]], is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is golden-proportioned relative to the previous one.<ref name=moscovich>[[Ivan Moscovich|Moscovich, Ivan]], ''Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles,'' New York: Sterling, 2004{{page needed|date=February 2019}}</ref> However, measurements of nautilus shells do not support this claim.<ref>{{Cite journal|title=Sea shell spirals|last=Peterson|first=Ivars|authorlink=Ivars Peterson|date=1 April 2005|journal=[[Science News]]|url=http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals}}</ref>
*Historian [[John Man (author)|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 1.45.<ref>Man, John, ''Gutenberg: How One Man Remade the World with Word'' (2002) pp.  166–167, Wiley, {{isbn|0-471-21823-5}}. "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>
*Studies by psychologists, starting with [[Gustav Fechner]] c. 1876,<ref>{{Cite book|title=Vorschule der Ästhetik|last=Fechner|first=Gustav|publisher=Breitkopf & Härtel|year=1876|isbn=|location=Leipzig|pp=190–202}}</ref> have been devised to test the idea that the golden ratio plays a role in human perception of [[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|2003|p=7}}<ref name="livio plus"/>
*In investing, some practitioners of [[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>For instance, Osler writes that "38.2 percent and 61.8 percent retracements of recent rises or declines are common," in {{Cite journal|author=Osler, 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| url=http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf}}</ref> The use of the golden ratio in investing is also related to more complicated patterns described by [[Fibonacci numbers]] (e.g. [[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>[[Roy Batchelor]] and Richard Ramyar, "[https://web.archive.org/web/20090918053822/http://www.cass.city.ac.uk/media/stories/resources/Magic_Numbers_in_the_Dow.pdf Magic numbers in the Dow]," 25th International Symposium on Forecasting, 2005, p.  13, 31. "[https://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html Not since the 'big is beautiful' days have giants looked better]", Tom Stevenson, [[The Daily Telegraph]], Apr. 10, 2006, and "Technical failure", [[The Economist]], Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.</ref>

===The Parthenon===
[[File:7234014 Parthenonas (cropped).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|2003|pp=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>Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic", ''Communication Quarterly'', Vol. 46 No. 2, 1998, pp 194–213.</ref> Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, [[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>{{cite book|last=Devlin|first=Keith J.|title=The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs)|url=https://books.google.com/books?id=T4zbCwAAQBAJ&pg=PT54|location=New York|publisher=[[Basic Books]]|year=2009|origyear=2005|isbn=978-1-56025-672-4|page=54}}</ref> [[Midhat J. Gazalé]] affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."<ref>[[Midhat J. Gazalé|Gazalé, Midhat J.]], ''Gnomon: From Pharaohs to Fractals'', Princeton University Press, 1999, p.&nbsp;125. {{isbn|0-691-00514-1}}</ref>

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>Patrice Foutakis, "Did the Greeks Build According to the Golden Ratio?", ''Cambridge Archaeological Journal'', vol. 24, n° 1, February 2014, pp.&nbsp;71–86.</ref>
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|left|[[Albert Gleizes]], ''[[Les Baigneuses (Gleizes)|Les Baigneuses]]'' (1912)]]
The [[Section d'Or]] ('Golden Section') was a collective of [[Painting|painters]], sculptors, poets and critics associated with [[Cubism]] and [[Orphism (art)|Orphism]].<ref>[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> 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>[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], Bibliothèque Kandinsky</ref> 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>Herbert, Robert, ''Neo-Impressionism'', New York: The Solomon R. Guggenheim Foundation, 1968{{page needed|date=February 2019}}</ref> However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 [[Section d'Or#Salon de la Section d'Or, 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|2003|p=169}} and [[Marcel Duchamp]] said as much in an interview.<ref name=tandf>[https://www.tandfonline.com/doi/abs/10.1080/00043079.1965.10788819 Camfield, William A., ''Juan Gris and the Golden Section''], Art Bulletin, 47, no. 1, March 1965, 128–134. 68</ref> 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=tandf/><ref>[https://books.google.com/books?isbn=0300053746 Green, Christopher, ''Juan Gris'', Whitechapel Art Gallery, London, 18 September–29 November 1992 ; Staatsgalerie Stuttgart 18 December 1992–14 February 1993 ; Rijksmuseum Kröller-Müller, Otterlo, 6 March–2 May 1993], Yale University Press, 1992, pp.&nbsp;37–38, {{ISBN|0300053746}}</ref><ref>[https://books.google.com/books?isbn=0719050049 Cottington, David, ''Cubism and Its Histories, Barber Institute's critical perspectives in art history series, Critical Perspectives in Art History''], Manchester University Press, 2004, pp.&nbsp;112, 142, {{ISBN|0719050049}}</ref> Art historian [[Daniel Robbins (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">Roger Allard, ''Sur quelques peintre'', Les Marches du Sud-Ouest, June 1911, pp. 57–64. In Mark Antliff and Patricia Leighten, ''A Cubism Reader, Documents and Criticism, 1906-1914'', The University of Chicago Press, 2008, pp.&nbsp;178–191, 330.</ref>

[[Piet Mondrian]] has been said to have used the golden section extensively in his geometrical paintings,<ref>Bouleau, Charles, ''The Painter's Secret Geometry: A Study of Composition in Art'' (1963) pp. 247–248, Harcourt, Brace & World, {{isbn|0-87817-259-9}}</ref> though other experts (including critic [[Yve-Alain Bois]]) have discredited these claims.<ref name="livio plus"/>{{sfn|Livio|2003|pp=177–78}}

==See also==
{{div col|colwidth=20em}}
* [[Golden angle]]
* [[List of works designed with the golden ratio]]
* [[Metallic mean]]
* [[Plastic number]]
* [[Sacred geometry]]
* [[Silver ratio]]
* [[Supergolden ratio]]
{{div col end}}

==References==

'''Footnotes'''
{{notelist}}

'''Citations'''
{{reflist}}

===Works cited===

*{{Cite book | last = Livio | first = Mario | authorlink = Mario Livio | title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number | url = https://books.google.com/books?id=bUARfgWRH14C | origyear = 2002 | edition = First trade paperback | year = 2003 | publisher = [[Random House|Broadway Books]] | location = New York City | isbn = 978-0-7679-0816-0 | ref = harv}}
*{{Cite book | last1 = Stakhov | first1 = Alexey P. | authorlink1 = Alexey Stakhov | last2 = Olsen | first2 = Scott | title = The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science | url = https://books.google.com/books?id=K6fac9RxXREC | year = 2009 | publisher = [[World Scientific Publishing]] | location = Singapore | isbn = 978-981-277-582-5 | ref = harv}}

==Further reading==
{{refbegin|30em}}
*{{Cite book
| last = Doczi
| first = György
| title = The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture
| origyear = 1981
| year = 2005
| publisher = Shambhala Publications
| location = Boston
| isbn = 978-1-59030-259-0
}}
*{{cite book
 | last = Hemenway
 | first = Priya
 | title = Divine Proportion: Phi In Art, Nature, and Science
 | year = 2005
 | publisher = Sterling
 | location = New York
 | isbn = 978-1-4027-3522-6
 }}
*{{Cite book
| last = Huntley
| first = H. E.
| title = The Divine Proportion: A Study in Mathematical Beauty
| url =
| year = 1970
| publisher = Dover Publications
| location = New York
| isbn = 978-0-486-22254-7
}}
*{{Cite book
| last = Joseph
| first = George G.
| title = The Crest of the Peacock: The Non-European Roots of Mathematics
| origyear = 1991
| url = https://archive.org/details/crestofpeacockno00jose
| edition = New
| year = 2000
| publisher = Princeton University Press
| location = Princeton, NJ
| isbn = 978-0-691-00659-8
}}
*{{Cite book
| last = Sahlqvist
| first = Leif
| title = Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design
| edition = 3rd Rev.
| year = 2008
| publisher = BookSurge
| location = Charleston, SC
| isbn = 978-1-4196-2157-4
}}
*{{Cite book
| last = Schneider
| first = Michael S.
| title = A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science
| url =
| year = 1994
| publisher = HarperCollins
| location = New York
| isbn = 978-0-06-016939-8
}}
*{{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
| origyear = ''Der Goldene Schnitt'' 1993
| url =
| 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. |
    |                                                                     |
    |           Excessive or inappropriate links WILL BE DELETED.         |
    | See [[Wikipedia:External links]] & [[Wikipedia:Spam]] for details.  |
    |                                                                     |
    | If there are already plentiful links, please propose additions or   |
    | replacements on this article's discussion page, or submit your link |
    | to the relevant category at the Open Directory Project (dmoz.org)   |
    | and link back to that category using the {{dmoz}} template.         |
==={{No more links}}=========-->
{{Commons category|Golden ratio}}
*{{springer|title=Golden ratio|id=p/g044570}}
* [http://demonstrations.wolfram.com/GoldenSection/ "Golden Section"] by Michael Schreiber, [[Wolfram Demonstrations Project]], 2007.
* [http://photoinf.com/Golden_Mean/Eugene_Ilchenko/GoldenSection.html Golden Section in Photography: Golden Ratio, Golden Triangles, Golden Spiral]
*{{MathWorld|title=Golden Ratio|urlname=GoldenRatio}}
* [https://www.goodreads.com/quotes/tag/golden-ratio Quotes about the Golden Ratio]
*{{cite web|url=http://www.physorg.com/news180531747.html |title=Researcher explains mystery of golden ratio |work=[[PhysOrg]] |date=December 21, 2009}}.
*{{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| date =}} Information and activities by a mathematics professor.
*[https://web.archive.org/web/20071105084747/http://www.contracosta.cc.ca.us/math/pentagrm.htm The Pentagram & The Golden Ratio]. Green, Thomas M. Updated June 2005. Archived November 2007. Geometry instruction with problems to solve. 
*{{cite arXiv |first1=Robert P. |last1=Schneider |eprint=1109.3216 |title=A Golden Pair of Identities in the Theory of Numbers |class=math.HO |year=2011}} Proves formulas that involve the golden mean and the [[Euler totient]] and [[Möbius function]]s.
* [http://www.maa.org/external_archive/devlin/devlin_05_07.html The Myth That Will Not Go Away], by [[Keith Devlin]], addressing multiple allegations about the use of the golden ratio in culture.

{{Algebraic numbers}}
{{Irrational number}}
{{Metallic ratios}}
{{Mathematical art}}

{{Authority control}}

{{DEFAULTSORT:Golden Ratio}}
[[Category:Golden ratio| ]]
[[Category:Euclidean plane geometry]]
<!--[[Category:Irrational numbers]]-->[[Category:Quadratic irrational numbers]]
[[Category:Periodic continued fractions]]
[[Category:Mathematical constants]]
[[Category:History of geometry]]
[[Category:Visual arts theory]]
[[Category:Composition in visual art]]