Editing Golden ratio (section)

==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===
{{further|Mathematics and architecture}}
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 name=modulor /><ref name=Frings />

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 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===
{{Further|Mathematics and art|History of aesthetics}}
[[File:Divina proportione - Illustration 13, crop & monochrome.jpg|thumb|right|[[Leonardo da Vinci|Da Vinci]]'s illustration of a dodecahedron from [[Pacioli]]'s ''[[Divina proportione]]'' (1509)]]
[[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 [[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 {{tmath|1.34}}, with averages for individual artists ranging from {{tmath|1.04}} ([[Francisco Goya|Goya]]) to {{tmath|1.46}} ([[Giovanni Bellini|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 {{tmath|\sqrt5}} proportions, and others with proportions like {{tmath|\sqrt2}}, {{tmath|3}}, {{tmath|4}}, and {{tmath|6}}.<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===
{{Main|Canons of page construction}}

According to [[Jan Tschichold]],

<blockquote>There was a time when deviations from the truly beautiful page proportions {{tmath|2\mathbin:3}}, {{tmath|1\mathbin:\sqrt3}}, 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]] uses the golden ratio]]

The [[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===
[[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 (Debussy)|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]], 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, ''[[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|2002|p=[https://archive.org/details/goldenratiostory00livi/page/154 154]}}

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 name=padovan /> 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 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 [[Ising model|Ising]] [[ferromagnet]] <chem display=inline>CoNb2O6</chem> (cobalt niobate) has {{tmath|8}} predicted excitation states (with [[E8 (mathematics)|{{tmath|E_8}} 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 [[Kink (materials science)|kinks]] in its ordered-phase to spin-flips in its [[paramagnetic]] phase; revealing, just below its [[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]] 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]] and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. {{tmath|360^\circ~\!/\varphi \approx 222.5^\circ\!}}. This method was used to arrange the {{tmath|1500}} mirrors of the student-participatory [[artificial satellite|satellite]] [[STARSHINE|Starshine-3]].<ref name=disco />
{{Clear}}

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