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			{{Short description|Number, approximately 1.618}}
	{{About|the number|the pop music album|The Golden Ratio (album)|calendar dates|Golden number (time)}}
			{{Other uses|Golden ratio (disambiguation)|Golden number (disambiguation)}}
	]
			{{pp|small=yes}}
			{{infobox non-integer number
			| title = Golden ratio ({{mvar|φ}})
			| image = Golden ratio line.svg
			| image_alt = two line segments of lengths a and b in the golden ratio: a + b is to a as a is to b
			| decimal = {{val|1.618033988749894}}&nbsp;.&nbsp;.&nbsp;.&nbsp;<ref name=a001622 />
			| continued_fraction = <math>1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}}</math>
			| algebraic = <math>\frac{1 + \sqrt5}{2}</math>
			}}
			] with long side {{math|1=<strong style="color:green;">''a'' + ''b''</strong>}} and short side {{mvar|1=<strong style="color:blue;">a</strong>}} can be divided into two pieces: a ] golden rectangle (shaded red, right) with long side {{mvar|1=<strong style="color:blue;">a</strong>}} and short side {{mvar|1=<strong style="color:red;">b</strong>}} and a ] (shaded blue, left) with sides of length {{mvar|1=<strong style="color:blue;">a</strong>}}. This illustrates the relationship {{math|1={{sfrac|''a'' + ''b''|''a''}} = {{sfrac|''a''|''b''}} = ''φ''.}}]]
			In ], two quantities are in the '''golden ratio''' if their ] is the same as the ratio of their ] to the larger of the two quantities. Expressed algebraically, for quantities {{tmath|a}} and {{tmath|b}} with {{tmath|a > b > 0}}, {{tmath|a}} is in a golden ratio to {{tmath|b}} if
	] 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 ] 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>.]]
			<math display=block> \frac{a+b}{a} = \frac{a}{b} = \varphi,</math>
	In ], two quantities are in the '''golden ratio''' if their ] is the same as the ratio of their ] 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'',
			where the Greek letter ] ({{tmath|\varphi}} or {{tmath|\phi}}) denotes the golden ratio.{{efn|If the constraint on {{tmath|a}} and {{tmath|b}} each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. {{tmath|\varphi}} is defined as the positive solution. The negative solution is {{tmath|1=\textstyle -\varphi^{-1} = \tfrac12\bigl(1 - \sqrt5~\!\bigr)}}. The sum of the two solutions is {{tmath|1}}, and the product of the two solutions is {{tmath|-1}}.}} The constant {{tmath|\varphi}} satisfies the ] {{tmath|1=\textstyle \varphi^2 = \varphi + 1}} and is an ] with a value of<ref name=a001622 />
	:<math> \frac{a+b}{a} = \frac{a}{b} \ \stackrel{\text{def}}{=}\ \varphi,</math>
	where the Greek letter ] (''φ'') represents the golden ratio. Its value is:
	:<math>\varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots.</math><!-- 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.		{{bi |left=1.6 |1=<math>\varphi = \frac{1+\sqrt5}{2} = </math>{{math|{{val|1.618033988749}}....}}}}<!-- PLEASE DO NOT add additional digits to the value of φ in this equation; there is a long-standing consensus that additional digits do not add to understanding. Thank you.-->
	-->
			The golden ratio was called the '''extreme and mean ratio''' by ],<ref name="Elements 6.3" /> and the '''divine proportion''' by ];<ref name=Pacioli /> and also goes by other names.{{Efn|Other names include the ''golden mean'', ''golden section'',{{sfn|Livio|2002|pp=, }} ''golden cut'',<ref name=goldencut /> ''golden proportion'', ''golden number'',{{sfn|Herz-Fischler|1998}} ''medial section'', and ''divine section''.}}
	The golden ratio is also called the '''golden section''' (Latin: ''sectio aurea'') or '''golden mean'''.<ref name="livio">{{Cite book|last=Livio|first=Mario|year=2002|title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number|publisher=Broadway Books|location=New York|isbn=0-7679-0815-5|url=http://books.google.com/books?id=w9dmPwAACAAJ}}</ref><ref name="Sadowski"/><ref name="dunlap">Richard A Dunlap, ''The Golden Ratio and Fibonacci Numbers'', World Scientific Publishing, 1997</ref> Other names include '''extreme and mean ratio''',<ref name="Elements 6.3">Euclid, '''', 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. 37. "And the same applies in architecture, to the ]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. '']'', Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.</ref>
			Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a ]'s diagonal to its side and thus appears in the ] of the ] and ].{{sfn|Herz-Fischler|1998|pp=20–25}} A ]—that is, a rectangle with an aspect ratio of {{tmath|\varphi}}—may be cut into a square and a smaller rectangle with the same ]. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as ]s, in some cases based on dubious fits to data.<ref name="strogatz nytimes" /> The golden ratio appears in some ], including the ] and other parts of vegetation.
	Some twentieth-century ]s and ]s, including ] and ], have proportioned their works to approximate the golden ratio—especially in the form of the ], in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be ] pleasing (see ] below).
	]s since ] have studied the properties of the golden ratio, including its appearance in the dimensions of a ] and in a ], which can be cut into a square and a smaller rectangle with the same ]. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as ]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=]|date=September 24, 2012|url=http://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/}}</ref>
			Some 20th-century ]s and ]s, including ] and ], have proportioned their works to approximate the golden ratio, believing it to be ] pleasing. These uses often appear in the form of a golden rectangle.
	{{TOC limit|3}}		{{TOC limit|3}}
	==Calculation==		==Calculation==
			Two quantities {{tmath|a}} and {{tmath|b}} are in the ''golden ratio'' {{tmath|\varphi}} if<ref name=schielack />
	{| style="float:right; margin:10px" class="wikitable"
			<math display=block>
	| colspan="2" style="text-align:center;"| {{Irrational numbers}}
			\frac{a+b}{a} = \frac{a}{b} = \varphi.
			</math>
			Thus, if we want to find {{tmath|\varphi}}, we may use that the definition above holds for arbitrary {{tmath|b}}; thus, we just set {{tmath|1= b = 1}}, in which case {{tmath|1= \varphi = a}} and we get the equation
			{{tmath|1= (\varphi + 1)/\varphi = \varphi}},
			which becomes a quadratic equation after multiplying by {{tmath|\varphi}}:
			<math display=block>\varphi + 1 = \varphi^2</math>
			which can be rearranged to
			<math display=block>{\varphi}^2 - \varphi - 1 = 0.</math>
			The ] yields two solutions:
			{{bi |left=1.6 |1=<math>\frac{1 + \sqrt5}{2} = 1.618033\dots\ </math> and <math>\ \frac{1 - \sqrt5}{2} = -0.618033\dots.</math>}}
			Because {{tmath|\varphi}} is a ratio between positive quantities, {{tmath|\varphi}} is necessarily the positive root.<ref name=peters /> The negative root is in fact the negative inverse {{tmath|-1/\varphi}}, which shares many properties with the golden ratio.
			==History==
			{{see also|Mathematics and art|Fibonacci number#History}}
			According to ],
			{{blockquote|Some of the greatest mathematical minds of all ages, from ] and ] in ], through the medieval Italian mathematician ] and the Renaissance astronomer ], to present-day scientific figures such as Oxford physicist ], have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.{{sfn|Livio|2002|p=}}|title=''The Golden Ratio: The Story of Phi, the World's Most Astonishing Number''}}
			] mathematicians first studied the golden ratio because of its frequent appearance in ];{{sfn|Livio|2002|p=|ps=: "... line division, which ] 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 ]s and ]s.{{sfn|Livio|2002|pp=}} According to one story, 5th-century BC mathematician ] discovered that the golden ratio was neither a whole number nor a fraction (it is ]), surprising ].{{sfn|Livio|2002|pp=}} ]'s '']'' ({{nowrap|c. 300 BC}}) provides several ] and their proofs employing the golden ratio,{{sfn|Livio|2002|p=}}{{efn|Euclid, '''', Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.}} and contains its first known definition which proceeds as follows:<ref name=hemenway />
			{{blockquote|A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.{{sfn|Livio|2002|p=}}{{efn|"῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."<ref name="fitzpatrick elements" />}}}}
			], the first to write a decimal approximation of the ratio]]
			The golden ratio was studied peripherally over the next millennium. ] (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of ] (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the ]s.{{sfn|Livio|2002|pp=}}
			] named his book '']'' (]) after the ratio; the book, largely plagiarized from ], explored its properties including its appearance in some of the ].<ref name=mackinnon />{{sfn|Livio|2002|pp=}} ], who illustrated Pacioli's book, called the ratio the ''sectio aurea'' ('golden section').<ref name=baravalle /> Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the ] system of rational proportions.{{sfn|Livio|2002|pp=}} Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as ] solved geometric problems using the ratio.{{sfn|Livio|2002|p=}}
			German mathematician Simon Jacob (d.&nbsp;1564) noted that ];<ref name=schreiber /> this was rediscovered by ] in 1608.{{sfn|Livio|2002|pp=}} The first known ] approximation of the (inverse) golden ratio was stated as "about {{tmath|0.6180340}}" in 1597 by ] of the ] in a letter to Kepler, his former student.<ref name=mactutor /> The same year, Kepler wrote to Maestlin of the ], which combines the golden ratio with the ]. Kepler said of these:{{blockquote|Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.<ref name=fink />}}
			Eighteenth-century mathematicians ], ], and ] 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 ], for whom it was named "Binet's formula".<ref name="beutelspacher petri" /> ] first used the German term ''goldener Schnitt'' ('golden section') to describe the ratio in 1835.{{sfn|Herz-Fischler|1998|pp=167–170}} ] used the equivalent English term in 1875.{{sfn|Posamentier|Lehmann|2011|p=8}}
			By 1910, inventor ] began using the ] ] ({{tmath|\varphi}}) as a ] for the golden ratio.{{sfn|Posamentier|Lehmann|2011|p=285}}{{efn|After Classical Greek sculptor ] (c.&nbsp;490–430&nbsp;BC);<ref name=cook /> Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.<ref name=barr />}} It has also been represented by ] ({{tmath|\tau}}), the first letter of the ] τομή ('cut' or 'section').{{sfn|Livio|2002|p=}}
			] demonstrates ]s at the ] in 1985 using a ]toy model.]]
			The ] construction system, developed by ] in the late 1960s, is based on the ] of the ]/], and uses the golden ratio ubiquitously. Between 1973 and 1974, ] developed ], a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.<ref name=gardner /> This gained in interest after ]'s Nobel-winning 1982 discovery of ]s with icosahedral symmetry, which were soon afterwards explained through analogies to the Penrose tiling.<ref name=quasicrystals />
			==Mathematics==
			===Irrationality===
			The golden ratio is an ]. Below are two short proofs of irrationality:
			====Contradiction from an expression in lowest terms====
			], then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the positive integers have a lower bound, so {{mvar|φ}} cannot be rational.]]
			This is a ]. Recall that:
			{{bi |left=1.6 |1=the whole is the longer part plus the shorter part;<br />
			the whole is to the longer part as the longer part is to the shorter part.}}
			If we call the whole {{tmath|n}} and the longer part {{tmath|m}}, then the second statement above becomes
			{{bi |left=1.6 |1={{tmath|n}} is to {{tmath|m}} as {{tmath|m}} is to {{tmath|n-m}}.}}
			To say that the golden ratio {{tmath|\varphi}} is rational means that {{tmath|\varphi}} is a fraction {{tmath|n/m}} where {{tmath|n}} and {{tmath|m}} are integers. We may take {{tmath|n/m}} to be in ] and {{tmath|n}} and {{tmath|m}} to be positive. But if {{tmath|n/m}} is in lowest terms, then the equally valued {{tmath|m/(n-m)}} is in still lower terms. That is a contradiction that follows from the assumption that {{tmath|\varphi}} is rational.
			====By irrationality of the square root of 5 ====
			Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the ] of rational numbers under addition and multiplication. If {{tmath|1= \varphi = \tfrac12\bigl(1 + \sqrt5~\!\bigr)}} is assumed to be rational, then {{tmath|1= 2\varphi - 1 = \sqrt5}}, the ] of {{tmath|5}}, must also be rational. This is a contradiction as the square roots of all non-] ]s are irrational.{{efn|The theorem that non-square natural numbers have irrational square roots can be found in Euclid's ''Elements'', .}}
			===Minimal polynomial===
			] {{math|''x''<sup>2</sup> − ''x'' − 1}}. The golden ratio's negative {{math|−''φ''}} and reciprocal {{math|''φ''<sup>−1</sup>}} are the two roots of the quadratic polynomial {{math|''x''<sup>2</sup> + ''x'' − 1}}.]]
			The golden ratio is also an ] and even an ]. It has ]
			<math display=block>x^2 - x - 1.</math>
			This ] has two ], {{tmath|\varphi}} and {{tmath|1=\textstyle -\varphi^{-1} }}.
			The golden ratio is also closely related to the polynomial {{tmath|\textstyle x^2 + x - 1}}, which has roots {{tmath|-\varphi}} and {{tmath|\textstyle \varphi^{-1} }}. As the root of a quadratic polynomial, the golden ratio is a ].<ref name=constructions />
			===Golden ratio conjugate<!--'Golden ratio conjugate' redirects here--> and powers===
			The ] to the minimal polynomial {{tmath|\textstyle x^2-x-1}} is
			<math display=block>-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt5}{2} = -0.618033\dots.</math>
			The absolute value of this quantity ({{tmath|0.618\ldots}}) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, {{tmath|b/a}}).
			This illustrates the unique property of the golden ratio among positive numbers, that
			<math display=block>\frac1\varphi = \varphi - 1,</math>
			or its inverse,
			<math display=block>\frac1{1/\varphi} = \frac1\varphi + 1.</math>
			The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with {{tmath|\varphi}}:
			<math display=block>\begin{align}
			\varphi^2 &= \varphi + 1 = 2.618033\dots, \\
			\frac1\varphi &= \varphi - 1 = 0.618033\dots.
			\end{align}</math>
			The sequence of powers of {{tmath|\varphi}} contains these values {{tmath|0.618033\ldots}}, {{tmath|1.0}}, {{tmath|1.618033\ldots}}, {{tmath|2.618033\ldots}}; more generally,
			any power of {{tmath|\varphi}} is equal to the sum of the two immediately preceding powers:
			<math display=block>
			\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 {{tmath|\varphi}} into a multiple of {{tmath|\varphi}} and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of {{tmath|\varphi}}:
			If {{tmath|1= \bigl\lfloor \tfrac12n - 1 \bigr\rfloor = m}}, then:
			<math display=block>\begin{align}
			\varphi^n &= \varphi^{n-1} + \varphi^{n-3} + \cdots + \varphi^{n-1-2m} + \varphi^{n-2-2m} \\
			\varphi^n - \varphi^{n-1} &= \varphi^{n-2}.
			\end{align}</math>
			===Continued fraction and square root===
			{{see also|Lucas number#Continued fractions for powers of the golden ratio}}
			]
			The formula {{tmath|1= \varphi = 1 + 1/\varphi}} can be expanded recursively to obtain a ] for the golden ratio:<ref name="Concrete Abstractions" />
			<math display=block>
			\varphi =
			= 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}}
			</math>
			It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
			<math display=block>
			\varphi^{-1} =
			= 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}}
			</math>
			The ]s of these continued fractions, {{tmath|\tfrac11}}, {{tmath|\tfrac21}}, {{tmath|\tfrac32}}, {{tmath|\tfrac53}}, {{tmath|\tfrac85}}, {{nowrap|{{tmath|\tfrac{13}8}}, ...}} or {{tmath|\tfrac11}}, {{tmath|\tfrac12}}, {{tmath|\tfrac23}}, {{tmath|\tfrac35}}, {{tmath|\tfrac58}}, {{nowrap|{{tmath|\tfrac8{13} }}, ...,}} are ratios of successive ]. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the ] for ]s, which states that for every irrational {{tmath|\xi}}, there are infinitely many distinct fractions {{tmath|p/q}} such that,
			<math display=block>\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}.</math>
			This means that the constant {{tmath|\sqrt5}} cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such ].<ref name=hardy />
			A ] form for {{tmath|\varphi}} can be obtained from {{tmath|1=\textstyle \varphi^2 = 1 + \varphi}}, yielding:<ref name=sizer/>
			<math display=block>
			\varphi = \sqrt{1 + \sqrt{\textstyle 1 + \sqrt{ 1 + \cdots \vphantom)}}}.
			</math>
			===Relationship to Fibonacci and Lucas numbers===
			{{further|Fibonacci number#Relation to the golden ratio}}
			{{see also|Lucas number#Relationship to Fibonacci numbers}}
			{{multiple image|align=right|direction=vertical|image1=Fibonacci Spiral.svg|image2=Lucas number spiral.svg|width=250|footer=A ] (top) which approximates the ], using ] square sizes up to {{math|21}}. A different approximation to the golden spiral is generated (bottom) from stacking squares whose lengths of sides are numbers belonging to the sequence of ], here up to {{math|76}}.}}
			]s and ]s have an intricate relationship with the golden ratio. In the Fibonacci sequence, each term <math>F_n</math> is equal to the sum of the preceding two terms <math>F_{n-1}</math> and <math>F_{n-2}</math>, starting with the base sequence {{tmath|0,1}} as the 0th and 1st terms <math>F_0</math> and <math>F_1</math>:
			{{bi |left=1.6 |1={{tmath|0,}} {{tmath|1,}} {{tmath|1,}} {{tmath|2,}} {{tmath|3,}} {{tmath|5,}} {{tmath|8,}} {{tmath|13,}} {{tmath|21,}} {{tmath|34,}} {{tmath|55,}} {{tmath|89,}} {{tmath|\ldots}} ({{OEIS2C|id=A000045}}).}}
			The sequence of Lucas numbers (not to be confused with the generalized ]s, of which this is part) is like the Fibonacci sequence, in that each term <math>L_n</math> is the sum of the previous two terms <math>L_{n-1}</math> and <math>L_{n-2}</math>, however instead starts with {{tmath|2,1}} as the 0th and 1st terms <math>L_0</math> and <math>L_1</math>:
			{{bi |left=1.6 |1={{tmath|2,}} {{tmath|1,}} {{tmath|3,}} {{tmath|4,}} {{tmath|7,}} {{tmath|11,}} {{tmath|18,}} {{tmath|29,}} {{tmath|47,}} {{tmath|76,}} {{tmath|123,}} {{tmath|199,}} {{tmath|\ldots}} ({{OEIS2C|id=A000032}}).}}
			Exceptionally, the golden ratio is equal to the ] of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:<ref name=tattersall />
			<math display=block>
			\lim_{n\to\infty} \frac{F_{n+1}}{F_n}
			= \lim_{n\to\infty} \frac{L_{n+1}}{L_n}
			= \varphi.
			</math>
			In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates {{tmath|\varphi}}. For example,
			{{bi |left=1.6 |1=<math>\frac{F_{16}}{F_{15}} = \frac{987}{610} = 1.6180327\ldots\ </math> and <math>\ \frac{L_{16}}{L_{15}} = \frac{2207}{1364} = 1.6180351\ldots.</math> }}
			These approximations are alternately lower and higher than {{tmath|\varphi}}, and converge to {{tmath|\varphi}} as the Fibonacci and Lucas numbers increase.
			]s for the Fibonacci and Lucas sequences that involve the golden ratio are:
			<math display=block>
			F\left(n\right)
			= \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt5}
			= \frac{\varphi^n - (1 - \varphi)^n}{\sqrt5}
			= \frac{1}{\sqrt5}\left,
			</math>
			<math display=block>
			L\left(n\right)
			= \varphi^n + (- \varphi)^{-n}
			= \varphi^n + (1 - \varphi)^n
			= \left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n .
			</math>
			Combining both formulas above, one obtains a formula for {{tmath|\textstyle \varphi^n}} that involves both Fibonacci and Lucas numbers:
			<math display=block>
			\varphi^n = \tfrac12\bigl(L_n + F_n \sqrt{5}~\!\bigr).
			</math>
			Between Fibonacci and Lucas numbers one can deduce {{tmath|1=\textstyle L_{2n} = 5 F_n^2 + 2(-1)^n = L_n^2 - 2(-1)^n}}, which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the ]:
			<math display=block>\lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}.</math>
			Indeed, much stronger statements are true:
			<math display=block>\begin{align}
			& \bigl\vert L_n - \sqrt5 F_n \bigr\vert = \frac{2}{\varphi^n} \to 0, \\
			& \bigl(\tfrac12 L_{3n}\bigr)^2 = 5 \bigl(\tfrac12 F_{3n}\bigr)^2 + (-1)^n.
			\end{align}</math>
			These values describe {{tmath|\varphi}} as a ] of the ] {{tmath|\mathbb{Q}\bigl(\sqrt5~\!\bigr)}}.
			Successive powers of the golden ratio obey the Fibonacci ], {{tmath|1=\textstyle \varphi^{n+1} = \varphi^n + \varphi^{n-1} }}.
			The reduction to a linear expression can be accomplished in one step by using:
			<math display=block>
			\varphi^n = F_n \varphi + F_{n-1}.
			</math>
			This identity allows any polynomial in {{tmath|\varphi}} to be reduced to a linear expression, as in:
			<math display=block>\begin{align}
			3\varphi^3 - 5\varphi^2 + 4
			&= 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\
			&= 3\bigl((\varphi + 1) + \varphi\bigr) - 5(\varphi + 1) + 4 \\
			&= \varphi + 2 \approx 3.618033.
			\end{align}</math>
			Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by ]:
			<math display=block>\sum_{n=1}^{\infty}\bigl|F_n\varphi-F_{n+1}\bigr| = \varphi.</math>
			In particular, the powers of {{tmath|\varphi}} themselves round to Lucas numbers (in order, except for the first two powers, {{tmath|\textstyle \varphi^0}} and {{tmath|\varphi}}, are in reverse order):
			<math display=block>\begin{align}
			\varphi^0 &= 1, \\
			\varphi^1 &= 1.618033989\ldots \approx 2, \\
			\varphi^2 &= 2.618033989\ldots \approx 3, \\
			\varphi^3 &= 4.236067978\ldots \approx 4, \\
			\varphi^4 &= 6.854101967\ldots \approx 7,
			\end{align}</math>
			and so forth.<ref name=parker4d /> The Lucas numbers also directly generate powers of the golden ratio; for {{tmath|n \ge 2}}:
			<math display=block>
			\varphi^n = L_n - (- \varphi)^{-n}.
			</math>
			Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of ''third'' consecutive Fibonacci numbers equals a Lucas number, that is {{tmath|1=\textstyle L_n = F_{n-1} + F_{n+1}\! }}; and, importantly, that {{tmath|1=\textstyle L_n F_n = F_{2n}\! }}.
			Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the ] (which is a special form of a ]) using quarter-circles with radii from these sequences, differing only slightly from the ''true'' golden logarithmic spiral. ''Fibonacci spiral'' is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.
			===Geometry===
			The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the ], and extends to form part of the coordinates of the vertices of a ], as well as those of a ].<ref name=BurgerStarbird /> It features in the ] and ] too, as well as in various other ].
			====Construction====
			{{multiple image|align=right|direction=vertical|image1=Goldener Schnitt Konstr beliebt.svg|image2=Goldener Schnitt (Äußere Teilung).svg|width=175|footer=Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.}}
			'''Dividing by interior division'''
			# Having a line segment {{tmath|AB}}, construct a perpendicular {{tmath|BC}} at point {{tmath|B}}, with {{tmath|BC}} half the length of {{tmath|AB}}. Draw the ] {{tmath|AC}}.
			# Draw an arc with center {{tmath|C}} and radius {{tmath|BC}}. This arc intersects the hypotenuse {{tmath|AC}} at point {{tmath|D}}.
			# Draw an arc with center {{tmath|A}} and radius {{tmath|AD}}. This arc intersects the original line segment {{tmath|AB}} at point {{tmath|S}}. Point {{tmath|S}} divides the original line segment {{tmath|AB}} into line segments {{tmath|AS}} and {{tmath|SB}} with lengths in the golden ratio.
			'''Dividing by exterior division'''
			# Draw a line segment {{tmath|AS}} and construct off the point {{tmath|S}} a segment {{tmath|SC}} perpendicular to {{tmath|AS}} and with the same length as {{tmath|AS}}.
			# Do bisect the line segment {{tmath|AS}} with {{tmath|M}}.
			# A circular arc around {{tmath|M}} with radius {{tmath|MC}} intersects in point {{tmath|B}} the straight line through points {{tmath|A}} and {{tmath|S}} (also known as the extension of {{tmath|AS}}). The ratio of {{tmath|AS}} to the constructed segment {{tmath|SB}} is the golden ratio.
			Application examples you can see in the articles ], ].
			Both of the above displayed different ]s produce ]s that determine two aligned ]s where the ratio of the longer one to the shorter one is the golden ratio.
			====Golden angle====
			{{main|Golden angle}}
			]
			When two angles that make a full circle have measures in the golden ratio, the smaller is called the ''golden angle'', with measure {{tmath|g}}:
			<math display=block>\begin{align}
			\frac{2\pi - g}{g} &= \frac{2\pi}{2\pi - g} = \varphi, \\
			2\pi - g &= \frac{2\pi}{\varphi} \approx 222.5^\circ\!, \\
			g &= \frac{2\pi}{\varphi^2} \approx 137.5^\circ\!.
			\end{align}</math>
			This angle occurs in ] as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.<ref name=phyllotaxis />
			====Pentagonal symmetry system====
			=====Pentagon and pentagram=====
			[[File:Pentagram-phi.svg|right|thumb|A pentagram colored to distinguish its line segments of different lengths. The four
			lengths are in golden ratio to one another.]]
			In a ] the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying ] to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are {{tmath|a}}, and short edges are {{tmath|b}}, then Ptolemy's theorem gives {{tmath|1=\textstyle a^2 = b^2 + ab}}. Dividing both sides by {{tmath|ab}} yields (see {{slink|#Calculation}} above),
			<math display=block>
			\frac ab = \frac{a + b}{a} = \varphi.
			</math>
			The diagonal segments of a pentagon form a ], or five-pointed ], whose geometry is quintessentially described by {{tmath|\varphi}}. Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is {{tmath|\varphi}}, as the four-color illustration shows.
			Pentagonal and pentagrammic geometry permits us to calculate the following values for {{tmath|\varphi}}:
			<math display=block>\begin{align}
			\varphi &= 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ\!, \\
			\varphi &= \tfrac12\csc(\pi/10) = \tfrac12\csc 18^\circ\!, \\
			\varphi &= 2\cos(\pi/5)=2\cos 36^\circ\!, \\
			\varphi &= 2\sin(3\pi/10)=2\sin 54^\circ\!.
			\end{align}</math>
			=====Golden triangle and golden gnomon=====
			{{main|Golden triangle (mathematics)}}
			] {{mvar|ABC}} can be subdivided by an angle bisector into a smaller golden triangle {{mvar|CXB}} and a golden gnomon {{mvar|XAC}}.]]
			The triangle formed by two diagonals and a side of a regular pentagon is called a ''golden triangle'' or ''sublime triangle''. It is an acute ] with apex angle {{tmath|36^\circ}} and base angles {{tmath|72^\circ\!}}.<ref name=fletcher /> Its two equal sides are in the golden ratio to its base.<ref name=loeb /> The triangle formed by two sides and a diagonal of a regular pentagon is called a ''golden gnomon''. It is an obtuse isosceles triangle with apex angle {{tmath|108^\circ}} and base angle {{tmath|36^\circ\!}}. Its base is in the golden ratio to its two equal sides.<ref name=loeb /> The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a ] are golden triangles,<ref name=loeb /> as are the ten triangles formed by connecting the vertices of a ] to its center point.<ref name=miller />
			Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.<ref name=loeb />
			If the apex angle of the golden gnomon is ], the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.<ref name=loeb />
			=====Penrose tilings=====
			{{main|Penrose tiling}}
			]
			The golden ratio appears prominently in the ''Penrose tiling'', a family of ]s of the plane developed by ], inspired by ]'s remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.<ref name="Tilings and Patterns" /> Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
			*Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.<ref name=pentaplexity />
			*The kite and dart Penrose tiling uses ] with three interior angles of {{tmath|72^\circ}} and one interior angle of {{tmath|144^\circ\!}}, and darts, concave quadrilaterals with two interior angles of {{tmath|36^\circ\!}}, one of {{tmath|72^\circ\!}}, and one non-convex angle of {{tmath|216^\circ\!}}. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.<ref name="Tilings and Patterns" />
			*The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context ''Robinson triangles'', can be used as the prototiles for a form of the Penrose tiling.<ref name="Tilings and Patterns" /><ref name=robinson />
			*The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of {{tmath|36^\circ}} and {{tmath|144^\circ\!}}, and a thick rhombus with angles of {{tmath|72^\circ}} and {{tmath|108^\circ\!}}. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals {{tmath|1\mathbin:\varphi}}, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.<ref name="Tilings and Patterns" />
			{{multiple image
			|align=left
			|image1=Penrose Tiling (P1).svg|caption1=Original four-tile Penrose tiling
			|image2=PenroseTilingFilled4.svg|caption2=Rhombic Penrose tiling
			|total_width=540}}
			{{clear|left}}
			====In triangles and quadrilaterals====
			=====Odom's construction=====
			]
			] found a construction for {{tmath|\varphi}} involving an ]: if the line segment joining the midpoints of two sides is extended to intersect the ], then the two midpoints and the point of intersection with the circle are in golden proportion.<ref name=triangleconstruction />
			=====Kepler triangle=====
			{{main|Kepler triangle}}
			{{multiple image
			|image1=Kepler triangle.svg|caption1=Geometric progression of areas of squares on the sides of a Kepler triangle
			|image2=Kepler and the Deathly Hallows.svg|caption2=An isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length
			|total_width=480}}
			The ''Kepler triangle'', named after ], is the unique ] with sides in ]:
			<math display=block>
			1\mathbin:\sqrt{\varphi\vphantom+}\mathbin:\varphi.</math>
			These side lengths are the three ]s of the two numbers {{tmath|\varphi \pm 1}}. The three squares on its sides have areas in the golden geometric progression {{tmath|\textstyle 1\mathbin:\varphi\mathbin:\varphi^2}}.
			Among isosceles triangles, the ratio of ] to side length is maximized for the triangle formed by two ] of the Kepler triangle, sharing the longer of their two legs.<ref name="Liber mensurationum" /> The same isosceles triangle maximizes the ratio of the radius of a ] on its base to its ].<ref name=bruce />
			For a Kepler triangle with smallest side length {{tmath|s}}, the ] and ] ]s are:
			<math display=block>\begin{align}
			A &= \tfrac12 s^2\sqrt{\varphi\vphantom+}, \\
			\theta &= \sin^{-1}\frac{1}{\varphi}\approx 38.1727^\circ\!, \\
			\theta &= \cos^{-1}\frac{1}{\varphi}\approx 51.8273^\circ\!.
			\end{align}</math>
			=====Golden rectangle=====
			{{main|Golden rectangle}}
			] in four simple steps:
			{|
	|-		|-
			|Draw a square.
	|]
	| 1.1001111000110111011...
	|-		|-
			|Draw a line from the midpoint of one side of the square to an opposite corner.
	| ]
	| 1.6180339887498948482...
	|-		|-
			|Use that line as the radius to draw an arc that defines the height of the rectangle.
	| ]
	| 1.9E3779B97F4A7C15F39...
	|-		|-
			|Complete the golden rectangle.
	| ]
	| <math>1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}}</math>
	|-		|-
	| ]
	| <math>\frac{1 + \sqrt{5}}{2}</math>
	|-
	| ]
	| <math>\frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}</math><br />
	|}		|}
			]]
			The golden ratio proportions the adjacent side lengths of a ''golden rectangle'' in {{tmath|1\mathbin:\varphi}} ratio.{{sfn|Posamentier|Lehmann|2011|p=11}} Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in {{tmath|\varphi}} ratio. They can be generated by ''golden spirals'', through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the ] as well as in the ] (see section below for more detail).<ref name=BurgerStarbird />
	Two quantities ''a'' and ''b'' are said to be in the ''golden ratio'' ''φ'' if
			=====Golden rhombus=====
	:<math> \frac{a+b}{a} = \frac{a}{b} = \varphi.</math>
			{{main|Golden rhombus}}
			A ''golden rhombus'' is a ] whose diagonals are in proportion to the golden ratio, most commonly {{tmath|1\mathbin:\varphi}}.<ref name=hexecontahedron /> For a rhombus of such proportions, its acute angle and obtuse angles are:
			<math display=block>\begin{align}
	One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,
			\alpha &= 2\arctan{1\over\varphi}\approx63.43495^\circ\!, \\
			\beta &= 2\arctan\varphi=\pi-\arctan2 = \arctan1+\arctan3 \approx 116.56505^\circ\!.
			\end{align}</math>
			The lengths of its short and long diagonals {{tmath|d}} and {{tmath|D}}, in terms of side length {{tmath|a}} are:
	:<math>\frac{a+b}{a} = 1 + \frac{b}{a} = 1 + \frac{1}{\varphi}.</math>
			<math display=block>\begin{align}
	Therefore,
			d &= \frac{2a}{\sqrt{2+\varphi}}
			= 2\sqrt{\frac{3-\varphi}{5}}a \approx 1.05146a, \\
			D &= 2\sqrt{\frac{2+\varphi}{5}}a \approx 1.70130a.
			\end{align}</math>
			Its area, in terms of {{tmath|a}} and {{tmath|d}}:
	:<math> 1 + \frac{1}{\varphi} = \varphi. </math>
			<math display=block>\begin{align}
	Multiplying by ''φ'' gives
			A &= \sin(\arctan2) \cdot a^2 = {2\over\sqrt5}~a^2 \approx 0.89443a^2, \\
			A &= {{\varphi}\over2}d^2\approx 0.80902d^2.
			\end{align}</math>
			Its ], in terms of side {{tmath|a}}:
	:<math>\varphi + 1 = \varphi^2</math>
			<math display=block>
	which can be rearranged to
			r = \frac{a}{\sqrt{5}}.
			</math>
			Golden rhombi form the faces of the ], the two ], the ],<ref name="golden rhombohedra" /> and the ].<ref name=hexecontahedron />
	:<math>{\varphi}^2 - \varphi - 1 = 0.</math>
			====Golden spiral====
	Using the ], two solutions are obtained:
			{{main|Golden spiral}}
	:<math>\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots</math>
			] (red) and its approximation by quarter-circles (green), with overlaps shown in yellow]]
			] whose radius grows by the golden ratio per {{math|108°}} of turn, surrounding nested golden isosceles triangles. This is a different spiral from the ], which grows by the golden ratio per {{math|90°}} of turn.<ref name=loeb-varney />]]
			] are ] spirals where distances covered per turn are in ]. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the ]. These spirals can be approximated by quarter-circles that grow by the golden ratio,<ref name=quarter-circles /> or their approximations generated from Fibonacci numbers,<ref name=diedrichs /> often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the ] with {{tmath|(r,\theta)}}:
			<math display=block>r = \varphi^{2\theta/\pi}.</math>
			Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each {{tmath|108^\circ}} that it turns, instead of the {{tmath|90^\circ}} turning angle of the golden spiral.<ref name=loeb-varney /> Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.<ref name=quarter-circles />
	and
			====Dodecahedron and icosahedron====
	:<math>\varphi = \frac{1 - \sqrt{5}}{2} = -0.6180\,339887\dots</math>
			[[File:Dodecahedron vertices.svg|240px|right|thumb|
			{|
			|- valign=top
			| colspan=2 |
			|- valign=top
			|] of the ] :
			<br />
			|- valign=top
			|<span style="color:#dd4400">{{math|(±1, ±1, ±1)}}</span>
			|- valign=top
			|<span style="color:#007722">{{math|(0, ±'''φ''', ±{{sfrac|1|'''φ'''}})}}</span>
			|- valign=top
			|<span style="color:#0011bb">{{math|(±{{sfrac|1|'''φ'''}}, 0, ±'''φ''')}}</span>
			|- valign=top
			| <span style="color:#cc0055">{{math|(±'''φ''', ±{{sfrac|1|'''φ'''}}, 0)}}</span>
			|- valign=top
			|A nested cube inside the dodecahedron is represented with <span style="color:#dd4400">dotted</span> lines.
			| colspan=2 |
			|}
			]]
			The ] and its ] the ] are ]s whose dimensions are related to the golden ratio. A dodecahedron has {{tmath|12}} regular pentagonal faces, whereas an icosahedron has {{tmath|20}} ]s; both have {{tmath|30}} ].<ref name ="Regular dodecahedron">{{harvtxt|Livio|2002|pp=70–72}}</ref>
	Because ''φ'' is the ratio between positive quantities ''φ'' is necessarily positive:
			For a dodecahedron of side {{tmath|a}}, the ] of a circumscribed and inscribed sphere, and ] are ({{tmath|r_u}}, {{tmath|r_i}}, and {{tmath|r_m}}, respectively):
	:<math>\varphi = \frac{1 + \sqrt{5}}{2} = 1.61803\,39887\dots</math> .
			{{bi |left=1.6 |1=<math>r_u = a\, \frac{\sqrt{3}\varphi}{2},</math> <math>r_i = a\, \frac{\varphi^2}{2 \sqrt{3-\varphi}},</math> and <math>r_m = a\, \frac{\varphi^2}{2}.</math>}}
	==History==
	] proposed using the first letter in the name of Greek sculptor ], ''phi'', to symbolize the golden ratio. Usually, the lowercase form (φ) is used. Sometimes, the uppercase form (Φ) is used for the ] of the golden ratio, 1/φ.<ref name="MathWorld GR Conjugate"/>]]
	], first to publish a decimal approximation of the golden ratio, in 1597]]
			While for an icosahedron of side {{tmath|a}}, the radius of a circumscribed and inscribed sphere, and ] are:
	The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. According to ]:
			{{bi |left=1.6 |1=<math>r_u = a\frac{\sqrt{\varphi \sqrt{5}}}{2},</math> <math>r_i = a\frac{\varphi^2}{2 \sqrt{3}},</math> and <math>r_m = a\frac{\varphi}{2}.</math>}}
	{{quote|Some of the greatest mathematical minds of all ages, from ] and ] in ], through the medieval Italian mathematician ] and the Renaissance astronomer ], to present-day scientific figures such as Oxford physicist ], have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. 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.<ref>Mario Livio,''The Golden Ratio: The Story of Phi, The World's Most Astonishing Number'', p.6</ref>}}
			The volume and surface area of the dodecahedron can be expressed in terms of {{tmath|\varphi}}:
	] mathematicians first studied what we now call the golden ratio because of its frequent appearance in ]. The division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular ]s and ]s. ]'s '']'' (]: {{lang|grc|Στοιχεῖα}}) provides the first known written definition of what is now called the golden ratio: "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."<ref>"῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν" as translated in {{cite book |title=Euclid's Elements of Geometry |author=Richard Fitzpatrick (translator) |year=2007 |isbn=978-0615179841}}, p. 156</ref> Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e., the golden ratio.<ref>Euclid, '''', Book 6, Proposition 30.</ref> Throughout the ''Elements'', several propositions (]s in modern terminology) and their proofs employ the golden ratio.<ref>Euclid, '''', Book 2, Proposition 11; Book 4, Propositions 10–11; Book 13, Propositions 1–6, 8–11, 16–18.</ref>
			{{bi |left=1.6 |1=<math>A_d = \frac{15\varphi}{\sqrt{3-\varphi}}</math> and <math>V_d = \frac{5\varphi^3}{6-2\varphi}.</math>}}
	The golden ratio is explored in ]'s book '']'' of 1509.
			As well as for the icosahedron:
	The first known approximation of the (inverse) golden ratio by a ], stated as "about 0.6180340", was written in 1597 by ] of the ] in a letter to 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>
			{{bi|left=1.6|1=<math>A_i = 20\frac{\varphi^{2}}{2}</math> and <math>V_i = \frac{5}{6}(1 + \varphi).</math>}}
	Since the 20th century, the golden ratio has been represented by the ] '''''φ''''' (], after ], a sculptor who is said to have employed it) or less commonly by '''''τ''''' (], the first letter of the ] root τομή—meaning ''cut'').<ref name="livio"/><ref>{{Mathworld|title=Golden Ratio|urlname=GoldenRatio}}</ref>
			].]]
	===Timeline===
	Timeline according to Priya Hemenway:<ref name=Hemenway,P>{{Cite book
	| last = Hemenway
	| first = Priya
	| title = Divine Proportion: Phi In Art, Nature, and Science
	| year = 2005
	| publisher = Sterling
	| location = New York
	| isbn = 1-4027-3522-7
	| pages = 20–21
	}}</ref>
	* ] (490–430 BC) made the ] statues that seem to embody the golden ratio.
	* ] (427–347 BC), in his '']'', describes five possible regular solids (the ]: the ], ], ], ], and ]), some of which are related to the golden ratio.<ref>{{cite web
	| last = Plato
	| authorlink = Plato
	| year = 360 BC) (Benjamin Jowett trans.
	| url = http://classics.mit.edu/Plato/timaeus.html
	| title = Timaeus
	| publisher = The Internet Classics Archive
	| accessdate = 2006-05-30
	}}</ref>
	* ] (c. 325–c. 265 BC), in his '']'', gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: ἄκρος καὶ μέσος λόγος).<ref name="Elements 6.3"/>
	* ] (1170–1250) mentioned the ] now named after him in his '']''; the ratio of sequential elements of the ] approaches the golden ratio asymptotically.
	* ] (1445–1517) defines the golden ratio as the "divine proportion" in his ''Divina Proportione''.
	* ] (1550–1631) publishes the first known approximation of the (inverse) golden ratio as a ].
	* ] (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers,<ref name=tatt>
	{{Cite book
	| title = Elementary number theory in nine chapters
	| edition = 2nd
	| author = James Joseph Tattersall
	| publisher = ]
	| year = 2005
	| isbn = 978-0-521-85014-8
	| page = 28
	| url = http://books.google.com/?id=QGgLbf2oFUYC&pg=PA29&dq=golden-ratio+limit+fibonacci+ratio+kepler&q=golden-ratio%20limit%20fibonacci%20ratio%20kepler
	}}</ref> and describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the ], 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." These two treasures are combined in the ].
	* ] (1720–1793) points out that in the spiral ] of plants going ] and counter-clockwise were frequently two successive Fibonacci series.
	* ] (1792–1872) is believed to be the first to use the term ''goldener Schnitt'' (golden section) to describe this ratio, in 1835.<ref>{{Cite book| title = Die Macht der Zahl: Was die Numerologie uns weismachen will | author = Underwood Dudley | publisher = Springer | year = 1999 | isbn = 3-7643-5978-1 | page = 245 | url = http://books.google.com/?id=r6WpMO_hREYC&pg=PA245&dq=%22goldener+Schnitt%22+ohm }}</ref>
	* ] (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
	* Mark Barr (20th century) suggests the Greek letter phi ('''φ'''), the initial letter of Greek sculptor Phidias's name, as a ] for the golden ratio.<ref>{{Cite book
	| last = Cook
	| first = Theodore Andrea
	| title = The Curves of Life
	| origyear = 1914
	| url = http://books.google.com/?id=ea-TStM-07EC&pg=PA420&dq=phi+mark+barr+intitle:The+intitle:Curves+intitle:of+intitle:Life
	| year = 1979
	| publisher = Dover Publications
	| location = New York
	| isbn = 0-486-23701-X
	}}</ref>
	* ] (b. 1931) discovered in 1974 the ], a pattern that is 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>{{citation|title=The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics|first=Martin|last=Gardner|authorlink=Martin Gardner|publisher=W. W. Norton & Company|year=2001|isbn=9780393020236|page=88|url=http://books.google.com/books?id=orz0SDEakpYC&pg=PA88}}.</ref> This in turn led to new discoveries about ]s.<ref>{{citation|title=Introduction to the Mathematics of Quasicrystals|first=Marko V.|last=Jaric|publisher=Elsevier|year=2012|isbn=9780323159470|page=x|url=http://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>
			These geometric values can be calculated from their ], which also can be given using formulas involving {{tmath|\varphi}}. The coordinates of the dodecahedron are displayed on the figure to the right, while those of the icosahedron are:
	==Applications and observations==
			<math display=block>
	===Aesthetics===
			(0,\pm1,\pm\varphi),\
	{{See also|History of aesthetics (pre-20th-century)}}
			(\pm1,\pm\varphi,0),\
			(\pm\varphi,0,\pm1).
			</math>
			Sets of three golden rectangles intersect ]ly inside dodecahedra and icosahedra, forming ].<ref name=borromean /><ref name=BurgerStarbird /> In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. The three golden rectangles together contain all {{tmath|12}} vertices of the icosahedron, or equivalently, intersect the centers of all {{tmath|12}} of the dodecahedron's faces.<ref name="Regular dodecahedron" />
	''De Divina Proportione'', a three-volume work by ], was published in 1509. Pacioli, a ] ], was known mostly as a mathematician, but he was also trained and keenly interested in art. ''De 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 ] system of rational proportions.<ref name="livio"/> Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. ''De Divina Proportione'' contains illustrations of regular solids by ], Pacioli's longtime friend and collaborator.
			A ] can be ] in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is {{tmath|2/(2+\varphi)}} times that of the dodecahedron's.<ref name=hume /> In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in {{tmath|\textstyle \varphi \mathbin: \varphi^{2} }} ratio. On the other hand, the ], which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's {{tmath|12}} vertices touch the {{tmath|12}} edges of an octahedron at points that divide its edges in golden ratio.<ref name="59 Icosahedra" />
	===Architecture===
	] are alleged to exhibit the golden ratio.]]
			===Other properties===
	The ]'s façade 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, Midhat J. Gazalé says, "It was not until Euclid, however, that the golden ratio's mathematical properties were studied. In the ''Elements'' (308 BC) the Greek mathematician merely regarded that number as an interesting irrational number, in connection with the middle and extreme ratios. Its occurrence in regular pentagons and ]s was duly observed, as well as in the dodecahedron (a ] whose twelve faces are regular pentagons). It is indeed exemplary that the great Euclid, contrary to generations of mystics who followed, would soberly treat that number for what it is, without attaching to it other than its factual properties."<ref>Midhat J. Gazalé , ''Gnomon'', Princeton University Press, 1999. ISBN 0-691-00514-1</ref> And ] 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. The one thing we know for sure is that Euclid, in his famous textbook ''Elements'', written around 300 BC, showed how to calculate its value."<ref>Keith J. Devlin ''The Math Instinct: Why You're A Mathematical Genius (Along With Lobsters, Birds, Cats, And Dogs)'', . New York: Thunder's Mouth Press, 2005, ISBN 1-56025-672-9</ref> Near-contemporary sources like ] exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.
			The golden ratio's ''decimal expansion'' can be calculated via root-finding methods, such as ] or ], on the equation {{tmath|1=\textstyle x^2-x-1=0}} or on {{tmath|1=\textstyle x^2-5=0}} (to compute {{tmath|\sqrt5}} first). The time needed to compute {{tmath|n}} digits of the golden ratio using Newton's method is essentially {{tmath|O(M(n))}}, where {{tmath|M(n)}} is ] two {{tmath|n}}-digit numbers.<ref name=muller /> This is considerably faster than known algorithms for ] and ]. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers {{tmath|F_{25001} }} and {{tmath|F_{25000} }}, each over {{tmath|5000}} digits, yields over {{tmath|10{,}000}} significant digits of the golden ratio. The decimal expansion of the golden ratio {{tmath|\varphi}}<ref name=a001622 /> has been calculated to an accuracy of ten trillion ({{tmath|1=\textstyle 1 \times 10^{13} = 10{,}000{,}000{,}000{,}000}}) digits.<ref name=ycruncher />
	A 2004 geometrical analysis of earlier research into the ] reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.<ref>Boussora, Kenza and Mazouz, Said, ''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 proportion of the plan and in the dimensioning of the prayer space, the court, and the ]. The authors note, however, that the areas where ratios close to the golden ratio were found are not part of the original construction, and theorize that these elements were added in a reconstruction.
			In the ], the fifth ] {{tmath|1=\textstyle z = e^{2\pi k i/5} }} (for an integer {{tmath|k}}) satisfying {{tmath|1=\textstyle z^5 = 1}} are the vertices of a pentagon. They do not form a ] of ]s, however the sum of any fifth root of unity and its ], {{tmath|z + \bar z}}, ''is'' a quadratic integer, an element of {{tmath|\Z}}. Specifically,
	The Swiss ] ], famous for his contributions to the ] ], 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. 25, as cited in Padovan, Richard, ''Proportion: Science, Philosophy, Architecture'' (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6</ref>
			<math display=block>\begin{align}
	Le Corbusier explicitly used the golden ratio in his ] system for the ] of ]. He saw this system as a continuation of the long tradition of ], Leonardo da Vinci's "]", the work of ], and others who used the proportions of the human body to improve the appearance and function of ]. In addition to the golden ratio, Le Corbusier based the system on ], ], 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 ] system. Le Corbusier's 1927 Villa Stein in ] 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. 35, as cited in Padovan, Richard, ''Proportion: Science, Philosophy, Architecture'' (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: "Both the paintings and the architectural designs make use of the golden section".</ref>
			e^{0} + e^{-0} &= 2, \\
			e^{2\pi i / 5} + e^{-2\pi i / 5} &= \varphi^{-1} = -1 + \varphi, \\
			e^{4\pi i / 5} + e^{-4\pi i / 5} &= -\varphi.
			\end{align}</math>
			This also holds for the remaining tenth roots of unity satisfying {{tmath|1=\textstyle z^{10} = 1}},
	Another Swiss architect, ], 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 ], the golden ratio is the proportion between the central section and the side sections of the house.<ref>Urwin, Simon. ''Analysing Architecture'' (2003) pp. 154-5, ISBN 0-415-30685-X</ref>
			<math display=block>\begin{align}
	In a recent book, author Jason Elliot speculated that the golden ratio was used by the designers of the ] and the adjacent Lotfollah mosque.<ref>
			e^{\pi i} + e^{-\pi i} &= -2, \\
	{{Cite book
			e^{\pi i / 5} + e^{-\pi i / 5} &= \varphi, \\
	| author = Jason Elliot
			e^{3\pi i / 5} + e^{-3\pi i / 5} &= -\varphi^{-1} = 1 - \varphi.
	| title = Mirrors of the Unseen: Journeys in Iran
			\end{align}</math>
	| year = 2006
	| pages = 277, 284
	| publisher = Macmillan
	| isbn = 978-0-312-30191-0
	| url = http://books.google.com/?id=Gcs4IjUx3-4C&pg=PA284&dq=intitle:%22Mirrors+of+the+Unseen%22+golden-ratio+maidan
	}}</ref>
			For the ] {{tmath|\Gamma}}, the only solutions to the equation {{tmath|1= \Gamma(z-1) = \Gamma(z+1)}} are {{tmath|1= z = \varphi}} and {{tmath|1=\textstyle z = -\varphi^{-1} }}.
	===Painting===
	]
			When the golden ratio is used as the base of a ] (see ], sometimes dubbed ''phinary'' or {{tmath|\varphi}}''-nary''), ]s in the ring {{tmath|\Z}} – that is, numbers of the form {{tmath|a + b\varphi}} for {{tmath|a}} and {{tmath|b}} in {{tmath|\Z}} – have ] representations, but rational fractions have non-terminating representations.
	The 16th-century philosopher ] drew a man over a ] inside a circle, implying a relationship to the golden ratio.<ref name="Sadowski">
	{{cite book
	| title = The knight on his quest: symbolic patterns of transition in Sir Gawain and the Green Knight
	| author = Piotr Sadowski
	| publisher = University of Delaware Press
	| year = 1996
	| isbn = 978-0-87413-580-0
	| page = 124
	| url = http://books.google.com/books?id=RNFqRs3Ccp4C&pg=PA124
	}}</ref>
			The golden ratio also appears in ], as the maximum distance from a point on one side of an ] to the closer of the other two sides: this distance, the side length of the ] formed by the points of tangency of a circle inscribed within the ideal triangle, is {{tmath|4\log(\varphi)}}.<ref name=horocycle />
	]'s illustrations of ] in '']'' (''On the Divine Proportion'') and his views that some bodily proportions exhibit the golden ratio have led some scholars to speculate that he incorporated the golden ratio in his paintings.<ref>''Leonardo da Vinci's Polyhedra'', by ]</ref> But the suggestion that his '']'', for example, employs golden ratio proportions, is not supported by anything in Leonardo's own writings.<ref>{{cite web|url=http://plus.maths.org/issue22/features/golden/|author=Livio, Mario|accessdate=2008-03-21|title=The golden ratio and aesthetics}}</ref> Similarly, although the '']'' is often<ref>"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." {{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}}</ref> 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=Donald E. Simanek|url=http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm|title=Fibonacci Flim-Flam|accessdate=April 9, 2013}}</ref>
			The golden ratio appears in the theory of ] as well. For <math>|q|<1,</math> let
	], influenced by the works of ],<ref>{{cite video |people=Salvador Dalí |date=2008 |title=The Dali Dimension: Decoding the Mind of a Genius |url= |format=DVD |language=English |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 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/><ref>Hunt, Carla Herndon and Gilkey, Susan Nicodemus. ''Teaching Mathematics in the Block'' pp. 44, 47, ISBN 1-883001-51-X</ref>
			<math display=block>
			R(q) = \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+{ \vphantom{1} \atop \ddots}}}}}.
			</math>
			Then
			<math display=block>
			R(e^{-2\pi})
			= \sqrt{\varphi\sqrt5}-\varphi ,\quad R(-e^{-\pi})
			= \varphi^{-1}-\sqrt{2-\varphi^{-1}}
			</math>
			and
			<math display=block>
			R(e^{-2\pi i/\tau})=\frac{1-\varphi R(e^{2\pi i\tau})}{\varphi+R(e^{2\pi i\tau})}
			</math>
			where {{tmath|\operatorname{Im}\tau>0}} and {{tmath|\textstyle (e^z)^{1/5} }} in the continued fraction should be evaluated as {{tmath|\textstyle e^{z/5} }}. The function {{tmath|\textstyle \tau\mapsto R(e^{2\pi i\tau})}} is invariant under {{tmath|\Gamma(5)}}, a ]. Also for ] {{tmath|a}} and {{tmath|b}} such that {{tmath|1=\textstyle ab = \pi^2,}} <ref name=rrcf />
			<math display=block>\begin{align}
	] 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.247-8, Harcourt, Brace & World, ISBN 0-87817-259-9</ref> though other experts (including critic ]) have disputed this claim.<ref name=livio/>
			\Bigl(\varphi+R{\bigl(e^{-2a}\bigr)}\Bigr)\Bigl(\varphi+R{\bigl(e^{-2b}\bigr)}\Bigr)&=\varphi\sqrt5, \\
			\Bigl(\varphi^{-1}-R{\bigl({-e^{-a}}\bigr)}\Bigr)\Bigl(\varphi^{-1}-R{\bigl({-e^{-b}}\bigr)}\Bigr)&=\varphi^{-1}\sqrt5.
			\end{align}</math>
			{{tmath|\varphi}} is a ].<ref name=duffin />
	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'' </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, p. 134–144</ref>
	{{-}}
			==Applications and observations==
	===Book design===
			]
	]: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."<ref>]. ''The Form of the Book'', pp.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>]]
	{{Main|Canons of page construction}}
			===Architecture===
	According to ],<ref>], ''The Form of the Book'', Hartley & Marks (1991), ISBN 0-88179-116-4.</ref>
			{{further|Mathematics and architecture}}
			The Swiss ] ], famous for his contributions to the ] ], 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 ] system for the ] of ]. He saw this system as a continuation of the long tradition of ], Leonardo da Vinci's "]", the work of ], and others who used the proportions of the human body to improve the appearance and function of ].
	<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.</blockquote>
			In addition to the golden ratio, Le Corbusier based the system on ], ], 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 ] system. Le Corbusier's 1927 ] in ] exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.<ref name=modulor2 />
	===Design===
	Some sources claim that the golden ratio is commonly used in everyday design, for example in the shapes of postcards, playing cards, posters, wide-screen televisions, photographs, light switch plates and cars.<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 = Art Johnson
	| publisher = Libraries Unlimited
	| year = 1999
	| isbn = 978-1-56308-446-1
	| page = 45
	| url = http://books.google.com/?id=STKX4qadFTkC&pg=PA45&dq=switch+%22golden+ratio%22#v=onepage&q=switch%20%22golden%20ratio%22&f=false
	| quote = The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
	}}</ref><ref>
	{{cite book
	| title = The mathematics of harmony: from Euclid to contemporary mathematics and computer science
	| edition =
	| author = Alexey Stakhov, Scott Olsen, Scott Anthony Olsen
	| publisher = World Scientific
	| year = 2009
	| isbn =978-981-277-582-5
	| page = 21
	| url = http://books.google.com/?id=K6fac9RxXREC&pg=PA21&dq=%22credit+card%22+%22golden+ratio%22+rectangle#v=onepage&q=%22credit%20card%22%20%22golden%20ratio%22%20rectangle&f=false
	| quote = A credit card has a form of the golden rectangle.
	}}</ref><ref>
	{{cite book
	| title = Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown's bestselling novel
	| author = Simon Cox
	| publisher = Barnes & Noble Books
	| year = 2004
	| isbn = 978-0-7607-5931-8
	| url = http://books.google.com/?id=TbjwhwLCEeAC&q=%22golden+ratio%22+postcard&dq=%22golden+ratio%22+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.
	}}</ref><ref>
	{{cite web
	| title = THE NEW RAPIDE S : Design
	| url = http://www.astonmartin.com/cars/rapide-s/rapide-s-design
	| quote = The ‘Golden Ratio’ sits at the heart of every Aston Martin.
	}}</ref>
			Another Swiss architect, ], 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 ], the golden ratio is the proportion between the central section and the side sections of the house.<ref name=urwin />
	===Music===
	] analyzes ]'s works as being based on two opposing systems, that of the golden ratio and the ],<ref>Lendvai, Ernő (1971). ''Béla Bartók: An Analysis of His Music''. London: Kahn and Averill.</ref> though other music scholars reject that analysis.<ref name="livio"/> French composer ] 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 ]'s ''] (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. '''' (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X</ref>
			===Art===
	The musicologist ] has observed that the formal boundaries of '']'' correspond exactly to the golden section.<ref>{{Cite book| title = Debussy in Proportion: A Musical Analysis | author = Roy Howat | url = http://books.google.com/?id=4bwKykNp24wC&pg=PA169&dq=intitle:Debussy+intitle:in+intitle:Proportion+golden+la-mer | publisher = Cambridge University Press | year = 1983 | isbn = 0-521-31145-4 }}</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 = 0-521-44656-2 | page = 53 | url = http://books.google.com/?id=THD1nge_UzcC&pg=PA53&dq=inauthor:Trezise+golden+evidence }}</ref>
			{{Further|Mathematics and art|History of aesthetics}}
			]'s illustration of a dodecahedron from ]'s '']'' (1509)]]
			]'s illustrations of ] in Pacioli's ''Divina proportione'' have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his '']'', for example, employs golden ratio proportions, is not supported by Leonardo's own writings.<ref name="livio plus"/> Similarly, although Leonardo's '']'' 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 />
			], influenced by the works of ],<ref name=dalidimension /> explicitly used the golden ratio in his masterpiece, '']''. 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 ] and dominates the composition.<ref name="livio plus" /><ref name="hunt gilkey" />
	] 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 ] 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
	}}</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 {{tmath|1.34}}, with averages for individual artists ranging from {{tmath|1.04}} (]) to {{tmath|1.46}} (]).<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 />
	Though ] proposed the non-octave-repeating ] based on ]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>"", ''Huygens-Fokker.org''. Accessed December 1, 2012.</ref>
			]: "Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section."<ref name=tschichold />]]
	===Nature===
	]'' in ], ]]]
			===Books and design===
	], whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the ]s of plants and of ] in leaves. He extended his research to the ]s of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of ]s, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.<ref>{{Cite book
			{{Main|Canons of page construction}}
	| title = Proportion
	| author = Richard Padovan
	| publisher = Taylor & Francis
	| year = 1999
	| isbn = 978-0-419-22780-9
	| pages = 305–306
	| url = http://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> In connection with his scheme for golden-ratio-based human body proportions, Zeising wrote in 1854 of a universal law "in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all ]s, ]s and ], whether cosmic or individual, ] or ], ] or ]; which finds its fullest realization, however, in the human form."<ref>{{cite book|first=Adolf|last=Zeising|title=Neue Lehre van den Proportionen des meschlischen Körpers|year=1854|page=preface|nopp=true}}</ref>
			According to ],
	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>
			<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>
	Since 1991, several researchers have proposed connections between the golden ratio and ] ].<ref>
	J.C. Perez (1991), , in ''Speculations in Science and Technology'' vol. 14 no. 4, {{ISSN|0155-7785}}.
	</ref><ref>
	Yamagishi, Michel E.B., and Shimabukuro, Alex I. (2007), , in ''Bulletin of Mathematical Biology,'' {{ISSN|0092-8240}} (print), {{ISSN|1522-9602}} (online).
	</ref><ref>{{cite journal |author= Perez, J.-C. |title= Codon populations in single-stranded whole human genome DNA are fractal and fine-tuned by the Golden Ratio 1.618 |journal= Interdisciplinary Sciences: Computational Life Science |date=September 2010 |volume= 2 |issue= 3 |pages= 228–240 |pmid= 20658335 |doi= 10.1007/s12539-010-0022-0 }}
	</ref><ref>{{cite journal |author= Perez, J.-C. |title= The "3 Genomic Numbers" Discovery: How Our Genome Single-Stranded DNA Sequence Is "Self-Designed" as a Numerical Whole |journal= Applied Mathematics, Biomathematics issue |date=October 2013 |volume= 4 |issue= 10B |pages= 37–53 |doi= 10.4236/am.2013.410A2004 }} </ref>
			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 />
	However, some have argued that many of the apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are in fact fictitious.<ref>Pommersheim, James E., Tim K. Marks, and ], eds. 2010. Number Theory: A Lively Introduction with Proofs, Applications, and Stories. John Wiley and Sons: 82.</ref>
	===Optimization===		===Flags===
			], whose ] uses the golden ratio]]
	The golden ratio is key to the ].
			The ] (width to height ratio) of the ] 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>
	===Perceptual studies===
	Studies by psychologists, starting with ], have been devised to test the idea that the golden ratio plays a role in human perception of ]. 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.<ref name="livio" /><ref>, by Mario Livio.</ref>
	==Mathematics==		===Music===
			] analyzes ]'s works as being based on two opposing systems, that of the golden ratio and the ],<ref name=lendvai /> though other music scholars reject that analysis.{{sfn|Livio|2002|p=}} French composer ] used the golden ratio in several of his pieces, including '']''. The golden ratio is also apparent in the organization of the sections in the music of ]'s ''] (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 ] has observed that the formal boundaries of Debussy's '']'' 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 />
	===Golden ratio conjugate===
	The negative root of the quadratic equation for '''φ''' (the "conjugate root") is
			Music theorists including ] and ] have experimented with the ], a musical scale based on using the golden ratio as its fundamental ]. When measured in ], a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.<ref name=833cents />
	:<math>-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt{5}}{2} = -0.61803\,39887\dots.</math>
			===Nature===
	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 ('''Φ'''):
			]'', showing the multiple spiral arrangement (])]]
			{{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=}}
	:<math>\Phi = {1 \over \varphi} = \varphi^{-1} = 0.61803\,39887\ldots.</math>
			The psychologist ] noted that the golden ratio appeared in ] and argued from these ] that the golden ratio was a universal law.<ref name=padovan /> Zeising wrote in 1854 of a universal ] law of "striving for beauty and completeness in the realms of both nature and art".<ref name=zeising />
	Alternatively, '''Φ''' can be expressed as
			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 />
	:<math>\Phi = \varphi -1 = 1.61803\,39887\ldots -1 = 0.61803\,39887\ldots.</math>
			===Physics===
	This illustrates the unique property of the golden ratio among positive numbers, that
			The quasi-one-dimensional ] ] <chem display=inline>CoNb2O6</chem> (cobalt niobate) has {{tmath|8}} predicted excitation states (with ]), 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 ] in its ordered-phase to spin-flips in its ] phase; revealing, just below its ], a spin dynamics with sharp modes at low energies approaching the golden mean.<ref name=ising />
			===Optimization===
	:<math>{1 \over \varphi} = \varphi - 1,</math>
			There is no known general ] to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, '']'' or '']''). However, a useful approximation results from dividing the sphere into parallel bands of equal ] 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 ] ].<ref name=disco />
			{{Clear}}
			The golden ratio is a critical element to ] as well.
	or its inverse:
			==Disputed observations==
	:<math>{1 \over \Phi} = \Phi + 1.</math>
			Examples of disputed observations of the golden ratio include the following:
			] shells are often erroneously claimed to be golden-proportioned.]]
			* Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive ] and ] (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.<ref name=pheasant /><ref name=vanLaack />
			* The shells of mollusks such as the ] are often claimed to be in the golden ratio.<ref name=dunlap /> The growth of nautilus shells follows a ], and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio,<ref name=falbo /> or sometimes claimed that each new chamber is golden-proportioned relative to the previous one.<ref name=moscovich /> However, measurements of nautilus shells do not support this claim.<ref name=shellspirals />
			* Historian ] states that both the pages and text area of the ] were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is {{tmath|1.45}}.<ref name=gutenberg />
			* Studies by psychologists, starting with ] {{Circa|1876}},<ref name=Fechner /> have been devised to test the idea that the golden ratio plays a role in human perception of ]. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.{{sfn|Livio|2002|p=}}<ref name="livio plus" />
			* In investing, some practitioners of ] use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.<ref name=osler/> The use of the golden ratio in investing is also related to more complicated patterns described by ] (e.g. ] and ]). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.<ref name=magicdow />
			===Egyptian pyramids===
	This means 0.61803...:1 = 1:1.61803....
			]]]
			The ] (also known as the Pyramid of Cheops or Khufu) has been analyzed by ] as having a doubled ] as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on ] or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.<ref name=greatpyramid />
			===The Parthenon===
	===Short proofs of irrationality===
			] are alleged to exhibit the golden ratio, but this has largely been discredited.{{sfn|Livio|2002|pp=}}]]
			The ]'s façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles.<ref name=Polemic /> Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, ] says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation."<ref name=mathinstinct /> ] affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."<ref name=gazalé />
	====Contradiction from an expression in lowest terms====
	], then it would be the ratio of sides of a rectangle with integer sides. But it is also a ratio of sides, which are also integers, of the smaller rectangle obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely, so &phi; cannot be rational.]]
			From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries.<ref name=foutakis />
	Recall that:
			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===
	: the whole is the longer part plus the shorter part;
			], '']'' (1912)]]
	: the whole is to the longer part as the longer part is to the shorter part.
			The ] ('Golden Section') was a collective of ], sculptors, poets and critics associated with ] and ].<ref name=centrepompidou1 /> Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with ].<ref name=centrepompidou2 /> (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat's writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.)<ref name=seuratclaims /> The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception".<ref name=herbert /> However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 ] exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not,{{sfn|Livio|2002|p=}} and ] said as much in an interview.<ref name=camfield /> On the other hand, an analysis suggests that ] made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition.<ref name=camfield /><ref name=juangris /> Art historian ] 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 ] and other former members of the ] had been involved.<ref name=allard />
			] has been said to have used the golden section extensively in his geometrical paintings,<ref name=bouleau /> though other experts (including critic ]) have discredited these claims.<ref name="livio plus" />{{sfn|Livio|2002|pp=}}
	If we call the whole ''n'' and the longer part ''m'', then the second statement above becomes
			==See also==
	: ''n'' is to ''m'' as ''m'' is to ''n''&nbsp;−&nbsp;''m'',
			* ]
			* ]
			* ]
			* ]
			* ]
			* ]
			==References==
	or, algebraically
			=== Explanatory footnotes ===
			{{notelist}}
			=== Citations ===
	: <math> \frac nm = \frac{m}{n-m}.\qquad (*) </math>
			{{reflist|30em|refs=
			<ref name="a001622">{{cite OEIS
	To say that ''φ'' is rational means that ''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 ''φ'' is rational.
			|A001622 |2=Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2}}
			</ref>
			<ref name=Pacioli>
	====Derivation from irrationality of √5====
			{{cite book
	Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the ] 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-] ] is irrational.
			| last = Pacioli
			| first = Luca
			| title = ]
			| publisher = Luca Paganinem de Paganinus de Brescia (Antonio Capella)
			| year = 1509
			| publication-place = Venice
			}}
			</ref>
			<ref name="Elements 6.3">
	===Alternative forms===
			{{cite book
	]
			| last = Euclid
	The formula ''φ'' = 1 + 1/''φ'' can be expanded recursively to obtain a ] for the golden ratio:<ref>{{Cite book| title = Concrete Abstractions: An Introduction to Computer Science Using Scheme
			| title = ]
	| author = Max. Hailperin, Barbara K. Kaiser, and Karl W. Knight | publisher = Brooks/Cole Pub. Co | year = 1998 | isbn = 0-534-95211-9 | url = http://books.google.com/?id=yYyVRueWlZ8C&pg=PA63&dq=continued-fraction+substitute+golden-ratio }}</ref>
			| orig-year = c. 300 BCE
			| chapter = Book 6, Definition 3
			}}
			</ref>
			<ref name=goldencut>
	:<math>\varphi = = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>
			{{cite book
			| last = Summerson
			| first = John
			| year = 1963
			| title = Heavenly Mansions and Other Essays on Architecture
			| publication-place = New York
			| publisher = W.W. Norton
			| page = 37
			| url = https://archive.org/details/heavenlymansions0000summ/page/37/
			| quote = And the same applies in architecture, to the ]s representing these and other ratios (e.g., the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.
			}}
			</ref>
			<ref name="strogatz nytimes">
	and its reciprocal:
			{{Cite news
			| first = Steven
			| last = Strogatz
			| author-link = Steven Strogatz
			| title = Me, Myself, and Math: Proportion Control
			| newspaper = ]
			| date = 2012-09-24
			| url = http://opinionator.blogs.nytimes.com/2012/09/24/proportion-control/
			}}
			</ref>
			<ref name=schielack>
	:<math>\varphi^{-1} = = 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}</math>
			{{Cite journal
			| last = Schielack
			| first = Vincent P.
			| year = 1987
			| title = The Fibonacci Sequence and the Golden Ratio
			| journal = The Mathematics Teacher
			| volume = 80 | issue = 5 | pages = 357–358 | doi = 10.5951/MT.80.5.0357
			| jstor = 27965402
			}} This source contains an elementary derivation of the golden ratio's value.
			</ref>
			<ref name=peters>
	The ]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 ].
			{{Cite journal
			| last = Peters
			| first = J. M. H.
			| year = 1978
			| title = An Approximate Relation between π and the Golden Ratio
			| journal = The Mathematical Gazette
			| volume = 62
			| issue = 421
			| pages = 197–198
			| doi = 10.2307/3616690
			| jstor = 3616690
			| s2cid = 125919525
			}}</ref>
			<ref name="fitzpatrick elements">
	The equation ''φ''<sup>2</sup> = 1 + ''φ'' likewise produces the continued ], or infinite surd, form:
			{{cite book
			| last = Euclid
			| translator-last = Fitzpatrick
			| translator-first = Richard
			| year = 2007
			| title = Euclid's Elements of Geometry
			| isbn = 978-0615179841
			| page = 156
			| publisher = Lulu.com
			}}
			</ref>
			<ref name=hemenway>
	:<math>\varphi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}.</math>
			{{cite book
			| last = Hemenway
			| first = Priya
			| title = Divine Proportion: Phi In Art, Nature, and Science
			| url = https://archive.org/details/divineproportion0000heme/page/20/
			| url-access = limited
			| year = 2005
			| publisher = Sterling
			| location = New York
			| pages = 20–21
			| isbn = 9781402735226
			}}
			</ref>
			<ref name=mackinnon>
	An infinite series can be derived to express phi:<ref>Brian Roselle, </ref><br />
			{{cite journal
	:<math>\varphi=\frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}.</math>
			| last = Mackinnon
			| first = Nick
			| year = 1993
			| title = The Portrait of Fra Luca Pacioli
			| volume = 77
			| issue = 479
			| journal = The Mathematical Gazette
			| pages = 130–219
			| doi = 10.2307/3619717
			| jstor = 3619717
			| s2cid = 195006163
			}}
			</ref>
			<ref name=baravalle>
	Also:
			{{cite journal
			| last = Baravalle
			| first = H. V.
			| title = The geometry of the pentagon and the golden section
			| journal = Mathematics Teacher
			| volume = 41
			| year = 1948
			| pages = 22–31
			| doi = 10.5951/MT.41.1.0022
			}}
			</ref>
			<ref name=schreiber>
	:<math>\varphi = 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ</math>
			{{cite journal
	:<math>\varphi = {1 \over 2}\csc(\pi/10) = {1 \over 2}\csc 18^\circ</math>
			| last = Schreiber
	:<math>\varphi = 2\cos(\pi/5)=2\cos 36^\circ</math>
			| first = Peter
	:<math> \varphi = 2\sin(3\pi/10)=2\sin 54^\circ. </math>
			| year = 1995
			| title = A Supplement to J. Shallit's Paper 'Origins of the Analysis of the Euclidean Algorithm'
			| journal = ]
			| volume = 22
			| issue = 4
			| pages = 422–424
			| doi = 10.1006/hmat.1995.1033
			| doi-access=free
			}}
			</ref>
			<ref name=mactutor>
	These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a ].
			{{cite web
			| last1 = O'Connor
			| first1 = John J.
			| last2 = Robertson
			| first2 = Edmund F.
			| authorlink2 = Edmund F. Robertson
			| year = 2001
			| title = The Golden Ratio
			| work = ]
			| url = http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html
			| access-date = 2007-09-18
			}}
			</ref>
			<ref name="beutelspacher petri">
	===Geometry===
			{{cite book
	]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 ]. 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.]]
			| last1 = Beutelspacher
	The number φ turns up frequently in ], particularly in figures with pentagonal ].
			| first1 = Albrecht
	The length of a regular ]'s ] is φ times its side.
			| last2 = Petri
	The vertices of a regular ] are those of ] mutually ] ]s.
			| first2 = Bernhard
			| year = 1996
			| contribution = Fibonacci-Zahlen
			| title = Der Goldene Schnitt
			| series = Einblick in die Wissenschaft
			| publisher = Vieweg+Teubner Verlag
			| doi = 10.1007/978-3-322-85165-9_6
			| language = de
			| pages = 87–98
			| isbn = 978-3-8154-2511-4
			}}
			</ref>
			<ref name=fink>
	There is no known general ] to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, '']''). However, a useful approximation results from dividing the sphere into parallel bands of equal ] and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ {{Unicode|≅}} 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory ] ].<ref>{{cite web|url=http://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>
			{{cite book
	{{-}}
			| title = A Brief History of Mathematics
			| last1 = Fink
			| first1 = Karl
			| translator-last1 = Beman
			| translator-first1 = Wooster Woodruff
			| translator-last2 = Smith
			| translator-first2 = David Eugene
			| translator-link2 = David Eugene Smith
			| year = 1903
			| publisher = Open Court
			| location = Chicago
			| edition = 2nd
			| page = 223
			| url = https://archive.org/details/bub_gb_3hkPAAAAIAAJ/page/n238
			}} (Originally published as ''Geschichte der Elementar-Mathematik''.)
			</ref>
			<ref name=cook>
	====Dividing a line segment====
			{{Cite book
	]
			| last = Cook
	The following ] produces a ] that divides a ] into two line segments where the ratio of the longer to the shorter line segment is the golden ratio:
			| first = Theodore Andrea
	# Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the ] AC.
			| author-link = Theodore Andrea Cook
	# Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D.
			| title = The Curves of Life
	# 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 segment AB into line segments AS and SB with lengths in the golden ratio.
			| year = 1914
	{{-}}
			| page = 420
			| publisher = Constable
			| location = London
			| url = https://archive.org/details/cu31924028937179/page/n455
			}}
			</ref>
			<ref name=barr>
	====Golden triangle, pentagon and pentagram====
			{{cite magazine
	]]]
			|last=Barr
			|first=Mark
			|title=Parameters of beauty
			|magazine=] (NY)
			|volume=60
			|page=325
			|year=1929
			}} Reprinted: {{cite magazine
			| title = Parameters of beauty
			| magazine = Think
			| volume = 10–11
			| publisher = ]
			| year = 1944
			}}
			</ref>
			<ref name=gardner>
	=====Golden triangle=====
			{{cite book
	The ] can be characterized as an ] ABC with the property that ] the angle C produces a new ] CXB which is a ] to the original.
			| last = Gardner
			| first = Martin
			| author-link = Martin Gardner
			| title = The Colossal Book of Mathematics
			| publisher = Norton
			| year = 2001
			| chapter = 7. Penrose Tiles
			| pages=73–93
			| chapter-url = https://archive.org/details/martingardnerthecolossalbookofmathematics/page/n89
			}}
			</ref>
			<ref name=quasicrystals>
	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°.
			{{harvnb|Livio|2002|pp=}}
			<br>
			{{cite journal
			| last1 = Gratias
			| first1 = Denis
			| author1-link = Denis Gratias
			| last2 = Quiquandon
			| first2 = Marianne
			| author2-link = Marianne Quiquandon
			| year = 2019
			| title = Discovery of quasicrystals: The early days
			| issue = 7–8
			| journal = ]
			| pages = 803–816
			| url = https://hal.archives-ouvertes.fr/hal-02379494
			| volume = 20
			| doi = 10.1016/j.crhy.2019.05.009
			| bibcode = 2019CRPhy..20..803G
			| s2cid = 182005594
			| doi-access = free
			}}
			<br>
			{{cite book
			| last = Jaric
			| first = Marko V.
			| year = 1989
			| title = Introduction to the Mathematics of Quasicrystals
			| publisher = Academic Press
			| page = x
			| isbn = 9780120406029
			| url = https://archive.org/details/introductiontoma0000unse_b9i6/page/n13/
			| url-access = limited
			| quote = Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by ], that provided the major influence on the new field.
			}}
			<br>
			{{cite journal
			| last1 = Goldman | first1 = Alan I.
			| last2 = Anderegg | first2 = James W.
			| last3 = Besser | first3 = Matthew F.
			| last4 = Chang | first4 = Sheng-Liang
			| last5 = Delaney | first5 = Drew W.
			| last6 = Jenks | first6 = Cynthia J. | author6-link = Cynthia Jenks
			| last7 = Kramer | first7 = Matthew J.
			| last8 = Lograsso | first8 = Thomas A.
			| last9 = Lynch | first9 = David W.
			| last10 = McCallum | first10 = R. William
			| last11 = Shield | first11 = Jeffrey E.
			| last12 = Sordelet | first12 = Daniel J.
			| last13 = Thiel | first13 = Patricia A. | author13-link = Patricia Thiel
			| year = 1996
			| volume = 84
			| issue = 3
			| journal = American Scientist
			| jstor = 29775669
			| pages = 230–241
			| title = Quasicrystalline materials
			}}
			</ref>
			<ref name=constructions>
	Suppose XB has length 1, and we call BC length 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 φ&nbsp;+&nbsp;1. But triangle ABC is similar to triangle CXB, so AC/BC&nbsp;=&nbsp;BC/BX, and so AC also equals φ<sup>2</sup>. Thus φ<sup>2</sup> =&nbsp;φ&nbsp;+&nbsp;1, confirming that φ is indeed the golden ratio.
			{{ cite book
			| last = Martin
			| first = George E.
			| year = 1998
			| title = Geometric Constructions
			| title-link = Geometric Constructions
			| series = Undergraduate Texts in Mathematics
			| publisher = Springer
			| pages = 13–14
			| doi = 10.1007/978-1-4612-0629-3
			| isbn = 978-1-4612-6845-1
			}}
			</ref>
			<ref name=sizer>{{cite journal
	Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the ] ratio is&nbsp;φ&nbsp;&minus;&nbsp;1.
			| last = Sizer | first = Walter S.
			| doi = 10.1080/0025570X.1986.11977215
			| issue = 1
			| journal = ]
			| jstor = 2690013
			| mr = 828417
			| pages = 23–27
			| title = Continued roots
			| volume = 59
			| year = 1986}}</ref>
			<ref name="Concrete Abstractions">
	=====Pentagon=====
			{{cite book
	In a regular pentagon the ratio between a side and a diagonal is <math>\Phi</math> (i.e. 1/φ), while intersecting diagonals section each other in the golden ratio.<ref name = "Pacioli"/>
			| last1 = Hailperin
			| first1 = Max
			| last2 = Kaiser
			| first2 = Barbara K.
			| last3 = Knight
			| first3 = Karl W.
			| year = 1999
			| title = Concrete Abstractions: An Introduction to Computer Science Using Scheme
			| publisher = Brooks/Cole
			| page = 63
			| url = https://archive.org/details/ost-computer-science-concreteabstractions/page/n78
			}}
			</ref>
			<ref name=hardy>
	=====Odom's construction=====
			{{cite book
	]
			| last1 = Hardy
	] 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 ] and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer ] who published it in Odom's name as a diagram in the '']'' 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>
			| first1 = G. H.
			| authorlink1 = G. H. Hardy
			| last2 = Wright
			| first2 = E. M.
			| authorlink2 = E. M. Wright
			| year = 1960
			| orig-year = 1938
			| chapter = §11.8. The measure of the closest approximations to an arbitrary irrational
			| title = An Introduction to the Theory of Numbers
			| edition = 4th
			| publisher = Oxford University Press
			| chapter-url = https://archive.org/details/introductiontoth0000hard/page/163
			| pages = 163–164
			| isbn = 978-0-19-853310-8
			| chapter-url-access = limited
			}}
			</ref>
			<ref name=tattersall>
	=====Pentagram=====
			{{Cite book
	[[File:Pentagram-phi.svg|right|thumb|A pentagram colored to distinguish its line segments of different lengths. The four
			| last = Tattersall
	lengths are in golden ratio to one another.]]
			| first = James Joseph
	The golden ratio plays an important role in the geometry of ]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.
			| year = 1999
			| title = Elementary number theory in nine chapters
			| publisher = Cambridge University Press
			| page = 28
			| url = https://archive.org/details/elementarynumber0000tatt/page/28/
			| url-access = limited
			}}
			</ref>
			<ref name=parker4d>
	The pentagram includes ten ]s: five ] and five ] isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are ]s. The obtuse isosceles triangles are ].
			{{cite book
			| last1 = Parker
			| first1 = Matt
			| date = 2014
			| title = Things to Make and Do in the Fourth Dimension
			| publisher = Farrar, Straus and Giroux
			| page = 284
			| isbn = 9780374275655
			| url = https://archive.org/details/thingstomakedoin0000park_r9y6/page/284/
			| url-access = limited
			}}
			</ref>
			<ref name=phyllotaxis>{{cite journal
	=====Ptolemy's theorem=====
			| last1 = King
	].]]
			| first1 = S.
			| last2 = Beck
			| first2 = F.
			| last3 = Lüttge
			| first3 = U.
			| year = 2004
			| title = On the mystery of the golden angle in phyllotaxis
			| journal = Plant, Cell and Environment
			| volume = 27
			| issue = 6
			| pages = 685–695
			| doi = 10.1111/j.1365-3040.2004.01185.x
			| bibcode = 2004PCEnv..27..685K
			}}
			</ref>
			<ref name=triangleconstruction>
	The golden ratio properties of a regular pentagon can be confirmed by applying ] 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
			{{cite journal
			| last1 = Odom
			| first1 = George
			| last2 = van de Craats
			| first2 = Jan
			| date = 1986
			| title = E3007: The golden ratio from an equilateral triangle and its circumcircle
			| department = Problems and solutions
			| journal = The American Mathematical Monthly
			| volume = 93
			| issue = 7
			| page = 572
			| doi = 10.2307/2323047
			| jstor = 2323047
			}}
			</ref>
			<ref name="Liber mensurationum">
	:<math>{b \over a}={{1+\sqrt{5}}\over 2}.</math>
			{{cite journal
			| last = Busard
			| first = Hubert L. L.
			| date = 1968
			| title = L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr
			| journal = ]
			| volume = 1968
			| issue = 2
			| language = fr,la
			| pages = 65–124
			| url = https://www.persee.fr/doc/jds_0021-8103_1968_num_2_1_1175
			| doi = 10.3406/jds.1968.1175
			}} See problem 51, reproduced on p. 98
			</ref>
			<ref name=bruce>
	====Scalenity of triangles====
			{{cite journal
	Consider a ] 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>'']'', pp. 49-50, 1954.</ref>
			| last = Bruce
			| first = Ian
			| year = 1994
			| title = Another instance of the golden right triangle
			| journal = ]
			| volume = 32
			| issue = 3
			| pages = 232–233
			| doi = 10.1080/00150517.1994.12429219
			| url = https://www.mathstat.dal.ca/FQ/Scanned/32-3/bruce.pdf
			}}
			</ref>
			<ref name=fletcher>
	====Triangle whose sides form a geometric progression====
			{{cite journal
	If the side lengths of a triangle form a ] 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 ] (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 : √φ : φ is a right triangle (because 1 + φ = φ<sup>2</sup>) known as a ].<ref name=herz/>
			| last = Fletcher
			| first = Rachel
			| year = 2006
			| title = The golden section
			| journal = Nexus Network Journal
			| volume = 8
			| issue = 1
			| pages = 67–89
			| doi = 10.1007/s00004-006-0004-z
			| s2cid = 120991151
			| doi-access = free
			}}
			</ref>
			<ref name=loeb>
	====Golden triangle, rhombus, and rhombic triacontahedron====
			{{cite book
	]
			| last = Loeb
			| first = Arthur
			| year = 1992
			| chapter = The Golden Triangle
			| title = Concepts & Images: Visual Mathematics
			| publisher = Birkhäuser
			| pages = 179–192
			| doi = 10.1007/978-1-4612-0343-8_20
			| isbn = 978-1-4612-6716-4
			}}
			</ref>
			<ref name=hexecontahedron>
	]
			{{cite journal
			| last = Grünbaum
			| first = Branko
			| author-link = Branko Grünbaum
			| year = 1996
			| journal = Geombinatorics
			| volume = 6
			| issue = 1
			| pages = 15–18
			| title = A new rhombic hexecontahedron
			| url = https://faculty.washington.edu/moishe/branko/BG207.New%20rhombic%20hexecontah.pdf
			}}
			</ref>
			<ref name=quarter-circles>{{cite conference
	A ] is a ] whose diagonals are in the golden ratio. The ] is a ] that has a very special property: all of its faces are golden rhombi. In the ] the ] between any two adjacent rhombi is 144°, which is twice the isosceles angle of a ] 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|year=2010|page=043501|doi=10.1063/1.3356985|arxiv=0908.3272}}.</ref>
			| last1 = Reitebuch | first1 = Ulrich
			| last2 = Skrodzki | first2 = Martin
			| last3 = Polthier | first3 = Konrad
			| editor1-last = Swart | editor1-first = David
			| editor2-last = Farris | editor2-first = Frank
			| editor3-last = Torrence | editor3-first = Eve
			| contribution = Approximating logarithmic spirals by quarter circles
			| contribution-url = https://archive.bridgesmathart.org/2021/bridges2021-95.html
			| isbn = 978-1-938664-39-7
			| location = Phoenix, Arizona
			| pages = 95–102
			| publisher = Tessellations Publishing
			| title = Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture
			| year = 2021}}</ref>
			<ref name=diedrichs>{{cite journal
	===Relationship to Fibonacci sequence===
			| last = Diedrichs | first = Danilo R.
	The mathematics of the golden ratio and of the ] are intimately interconnected. The Fibonacci sequence is:
			| date = February 2019
			| doi = 10.1017/mag.2019.7
			| issue = 556
			| journal = ]
			| pages = 52–64
			| title = Archimedean, Logarithmic and Euler spirals – intriguing and ubiquitous patterns in nature
			| volume = 103| s2cid = 127189159
			}}</ref>
			<ref name=loeb-varney>{{cite book
	:1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....
			| last1 = Loeb | first1 = Arthur L.
			| last2 = Varney | first2 = William
			| editor1-last = Hargittai | editor1-first = István
			| editor2-last = Pickover | editor2-first = Clifford A.
			| contribution = Does the golden spiral exist, and if not, where is its center?
			| contribution-url = https://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA47
			| date = March 1992
			| doi = 10.1142/9789814343084_0002
			| pages = 47–61
			| publisher = World Scientific
			| title = Spiral Symmetry| isbn = 978-981-02-0615-4
			}}</ref>
			<ref name=BurgerStarbird>
	The ] (known as ]'s formula, even though it was already known by ]) for the Fibonacci sequence involves the golden ratio:
			{{cite book
			| first1 = Edward B.
			| last1 = Burger
			| first2 = Michael P.
			| last2 = Starbird
			| year = 2005
			| orig-year = 2000
			| edition = 2nd
			| title = The Heart of Mathematics: An Invitation to Effective Thinking
			| publisher = Springer
			| page = 382
			| isbn = 9781931914413
			| url = https://archive.org/details/heartofmathemati0000burg_z4x5/page/382/
			}}
			</ref>
			<ref name="golden rhombohedra">
	:<math>F\left(n\right)
			{{cite book
			| last = Senechal
			| first = Marjorie
			| authorlink = Marjorie Senechal
			| year = 2006
			| contribution = Donald and the golden rhombohedra
			| editor1-last = Davis
			| editor1-first = Chandler
			| editor2-last = Ellers
			| editor2-first = Erich W.
			| title = The Coxeter Legacy
			| isbn = 0-8218-3722-2
			| pages = 159–177
			| publisher = American Mathematical Society
			}}
			</ref>
			<ref name="Tilings and Patterns">
	= {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}
			{{cite book
			| last1 = Grünbaum
			| first1 = Branko
			| author1-link = Branko Grünbaum
			| last2 = Shephard
			| first2 = G. C.
			| title = Tilings and Patterns
			| location = New York
			| publisher = W. H. Freeman
			| year = 1987
			| pages = 537–547
			| isbn = 9780716711933
			| url = https://archive.org/details/isbn_0716711931/page/537/
			}}
			</ref>
			<ref name=pentaplexity>
	= {{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}}.</math>
			{{cite magazine
			| first = Roger
			| last = Penrose
			| author-link = Roger Penrose
			| year = 1978
			| title = Pentaplexity
			| magazine = ]
			| volume = 39
			| pages = 32
			| url = https://archive.org/details/eureka-39/page/16/
			}} ()
			</ref>
			<ref name=borromean>
	] which approximates the ], using Fibonacci sequence square sizes up to 34.]]
			{{cite conference
			| last1 = Gunn
			| first1 = Charles
			| last2 = Sullivan
			| first2 = John M.
			| author2-link = John M. Sullivan (mathematician)
			| year = 2008
			| title = The Borromean Rings: A video about the New IMU logo
			| url = https://archive.bridgesmathart.org/2008/bridges2008-63.html
			| editor1-last = Sarhangi
			| editor1-first = Reza
			| editor2-last = Séquin
			| editor2-first = Carlo H.
			| editor2-link = Carlo H. Séquin
			| book-title = Proceedings of ] 2008
			| conference = Leeuwarden, the Netherlands
			| pages = 63–70
			| publisher = Tarquin Publications
			}}; Video at {{cite web
			| title = The Borromean Rings: A new logo for the IMU
			| website = International Mathematical Union
			| url = http://torus.math.uiuc.edu/jms/Videos/imu/
			| archive-url = https://web.archive.org/web/20210308065039/http://torus.math.uiuc.edu/jms/Videos/imu/
			| archive-date = 2021-03-08
			}}
			</ref>
			<ref name=hume>
	The golden ratio is the ] of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by ]:<ref name="tatt"/>
			{{cite journal
			| last = Hume
			| first = Alfred
			| year = 1900
			| title = Some propositions on the regular dodecahedron
			| journal = ]
			| volume = 7
			| issue = 12
			| pages = 293–295
			| doi = 10.2307/2969130
			| jstor = 2969130
			}}
			</ref>
			<ref name="59 Icosahedra">
	:<math>\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi.</math>
			{{Cite book
			| last1 = Coxeter
			| first1 = H.S.M.
			| author1-link = Harold Scott MacDonald Coxeter
			| last2 = du Val
			| first2 = Patrick
			| author2-link = Patrick du Val
			| last3 = Flather
			| first3 = H.T.
			| last4 = Petrie
			| author4-link = John Flinders Petrie
			| first4 = J.F.
			| year = 1938
			| title = The Fifty-Nine Icosahedra
			| title-link = The Fifty-Nine Icosahedra
			| publisher = University of Toronto Studies
			| volume = 6
			| page = 4
			| quote = Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section.
			}}
			</ref>
			<ref name=muller>
	Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610&nbsp;≈&nbsp;1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:
			{{cite book
			| last1 = Muller
			| first1 = J. M.
			| date = 2006
			| title = Elementary functions : algorithms and implementation
			| edition = 2nd
			| publisher = Birkhäuser
			| location = Boston
			| isbn = 978-0817643720
			| page = 93
			}}
			</ref>
			<ref name=ycruncher>
	:<math>\sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)|
			{{Cite web
			| first = Alexander J.
			| last = Yee
			| date = 2021-03-13
			| title = Records Set by y-cruncher
			| website = numberword.org
			| url = http://www.numberworld.org/y-cruncher/records.html
			}} Two independent computations done by Clifford Spielman.
			</ref>
			<ref name=horocycle>
	= \varphi.</math>
			, cabri.net, retrieved 2009-07-21.
			</ref>
			<ref name=rrcf>{{cite journal
	More generally:
			| last1 = Berndt
	:<math>\lim_{n\to\infty}\frac{F(n+a)}{F(n)}={\varphi}^a,</math>
			| first1 = Bruce C.
			| last2 = Chan
			| first2 = Heng Huat
			| last3 = Huang
			| first3 = Sen-Shan
			| last4 = Kang
			| first4 = Soon-Yi
			| last5 = Sohn
			| first5 = Jaebum
			| last6 = Son
			| first6 = Seung Hwan
			| year = 1999
			| title = The Rogers–Ramanujan Continued Fraction
			| journal = Journal of Computational and Applied Mathematics
			| volume = 105
			| number = 1–2
			| pages = 9–24
			| url = https://faculty.math.illinois.edu/~berndt/articles/rrcf.pdf
			| doi = 10.1016/S0377-0427(99)00033-3
			| access-date = 2022-11-29
			| archive-date = 2022-10-06
			| archive-url = https://web.archive.org/web/20221006212209/https://faculty.math.illinois.edu/~berndt/articles/rrcf.pdf
			| url-status = dead
			}}</ref>
			<ref name=duffin>
	where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when <math>a = 1</math>.
			{{cite journal
			| last = Duffin
			| first = Richard J.
			| author-link = Richard Duffin
			| year = 1978
			| title = Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers
			| journal = ]
			| volume = 74
			| issue = 1
			| pages = 47–56
			| url = https://projecteuclid.org/euclid.pjm/1102810434
			| doi = 10.2140/pjm.1978.74.47
			| doi-access = free
			}}
			</ref>
			<ref name=modulor>Le Corbusier, ''The Modulor'', {{nobr|p. 25}}, as cited in
	Furthermore, the successive powers of φ obey the Fibonacci ]:
			{{cite book
			| last = Padovan
			| first = Richard
			| year = 1999
			| title = Proportion: Science, Philosophy, Architecture
			| page = 316
			| publisher = Taylor & Francis
			| doi = 10.4324/9780203477465
			| isbn = 9781135811112
			}}</ref>
			<ref name=modulor2>Le Corbusier, ''The Modulor'', {{nobr|p. 35}}, as cited in
	:<math>\varphi^{n+1}
			{{cite book
			| last = Padovan
			| first = Richard
			| year = 1999
			| title = Proportion: Science, Philosophy, Architecture
			| page = 320
			| publisher = Taylor & Francis
			| doi = 10.4324/9780203477465
			| isbn = 9781135811112
			| quote = Both the paintings and the architectural designs make use of the golden section
			}}</ref>
			<ref name=Frings>
	= \varphi^n + \varphi^{n-1}.</math>
			{{cite journal
			| last = Frings
			| first = Marcus
			| year = 2002
			| title = The Golden Section in Architectural Theory
			| journal = Nexus Network Journal
			| volume = 4
			| number = 1
			| pages = 9–32
			| doi = 10.1007/s00004-001-0002-0
			| s2cid = 123500957
			| url = http://belveduto.de/text-nnj.htm
			| doi-access = free
			}}
			</ref>
			<ref name=urwin>
	This identity allows any polynomial in φ to be reduced to a linear expression. For example:
			{{cite book
	: <math>
			| last = Urwin
	\begin{align}
			| first = Simon
	3\varphi^3 - 5\varphi^2 + 4 & = 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\
			| year = 2003
	& = 3 - 5(\varphi + 1) + 4 \\
			| title = Analysing Architecture
	& = \varphi + 2 \approx 3.618.
			| edition = 2nd
	\end{align}
			| publisher = Routledge
	</math>
			| pages = 154–155
			| url = https://archive.org/details/analysingarchite0000unwi/page/154/
			| url-access = limited
			}}
			</ref>
			<ref name=devlin>{{cite web
	However, this is no special property of φ, because polynomials in any solution ''x'' to a ] can be reduced in an analogous manner, by applying:
			| first = Keith
	:<math>x^2=ax+b</math>
			| last = Devlin
	for given coefficients ''a'', ''b'' such that ''x'' satisfies the equation. Even more generally, any ] (with rational coefficients) of the root of an irreducible ''n''th-degree polynomial over the rationals can be reduced to a polynomial of degree ''n'' ‒ 1. Phrased in terms of ], 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>.
			| year = 2007
			| title = The Myth That Will Not Go Away
			| access-date = September 26, 2013
			| url = http://www.maa.org/external_archive/devlin/devlin_05_07.html
			| quote = Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.
			| archive-date = November 12, 2020
			| archive-url = https://web.archive.org/web/20201112034625/https://www.maa.org/external_archive/devlin/devlin_05_07.html
			| url-status = dead
			}}</ref>
			<ref name="livio plus">
	===Symmetries===
			{{cite web
	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 ]s <math>x, 1/(1-x), (x-1)/x,</math> – this fact corresponds to the identity and the definition quadratic equation.
			| last = Livio
	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.
			| first = Mario
			| year = 2002
			| title = The golden ratio and aesthetics
			| url = https://plus.maths.org/content/os/issue22/features/golden/index
			| ref = none
			| website = Plus Magazine
			| access-date = November 26, 2018
			}}
			</ref>
			<ref name=simanek>
	More deeply, these maps form a subgroup of the ] <math>\operatorname{PSL}(2,\mathbf{Z})</math> isomorphic to the ] on 3 letters, <math>S_3,</math> corresponding to the ] of the set <math>\{0,1,\infty\}</math> of 3 standard points on the ], 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.
			{{cite web
			| first = Donald E.
			| last = Simanek
			| title = Fibonacci Flim-Flam
			| url = http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
			| access-date = April 9, 2013
			| url-status = dead
			| archive-url = https://web.archive.org/web/20100109045556/http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm
			| archive-date = January 9, 2010
			}}
			</ref>
			<ref name=dalidimension>
	===Other properties===
			{{cite video
	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 ] of ] and it is an extremal case of the ] for ]s. This may be why angles close to the golden ratio often show up in ] (the growth of plants).<ref>, from , retrieved 2012-11-29.</ref>
			| people = Salvador Dalí
			| date = 2008
			| title = The Dali Dimension: Decoding the Mind of a Genius
			| format = DVD
			| publisher = Media 3.14-TVC-FGSD-IRL-AVRO
			| url = http://www.dalidimension.com/eng/index.html
			}}
			</ref>
			<ref name="hunt gilkey">
	The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with φ:
			{{cite book
			| last1 = Hunt
			| first1 = Carla Herndon
			| last2 = Gilkey
			| first2 = Susan Nicodemus
			| year = 1998
			| title = Teaching Mathematics in the Block
			| pages = 44, 47
			| publisher = Eye On Education
			| isbn = 1-883001-51-X
			}}
			</ref>
			<ref name=olariu>
	:<math>\varphi^2 = \varphi + 1 = 2.618\dots</math>
			{{cite arXiv
			| last = Olariu
			| first = Agata
			| year = 1999
			| title = Golden Section and the Art of Painting
			| eprint = physics/9908036
			}}
			</ref>
			<ref name=tosto>
	:<math>{1 \over \varphi} = \varphi - 1 = 0.618\dots.</math>
			{{cite book
			| last = Tosto
			| first = Pablo
			| year = 1969
			| title = La composición áurea en las artes plásticas
			| language = es
			| trans-title = The golden composition in the plastic arts
			| publisher = Hachette
			| pages = 134–144
			}}
			</ref>
			<ref name=tschichold>
	The sequence of powers of φ contains these values 0.618..., 1.0, 1.618..., 2.618...; more generally,
			{{cite book
	any power of φ is equal to the sum of the two immediately preceding powers:
			| last = Tschichold
			| first = Jan
			| author-link = Jan Tschichold
			| year = 1991
			| title = The Form of the Book
			| publisher = Hartley & Marks
			| isbn = 0-88179-116-4
			| page = 43 Fig 4
			| url = https://archive.org/details/the-form-of-the-book-jan-tschichold/page/n59/
			| quote = Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well.
			}}
			</ref>
			<ref name=tschichold2>
	: <math>\varphi^n = \varphi^{n-1} + \varphi^{n-2} = \varphi \cdot \operatorname{F}_n + \operatorname{F}_{n-1}.</math>
			{{cite book
			| last = Tschichold
			| first = Jan
			| year = 1991
			| title = The Form of the Book
			| publisher = Hartley & Marks
			| isbn = 0-88179-116-4
			| pages = 27–28
			| url = https://archive.org/details/the-form-of-the-book-jan-tschichold/page/n43/
			}}
			</ref>
			<ref name=miscellany>
	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:
			{{Cite journal
			| last = Jones
			| first = Ronald
			| title = The golden section: A most remarkable measure
			| year = 1971
			| journal = The Structurist
			| volume = 11
			| pages = 44–52
			| quote = Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
			}}
			{{pb}}
			{{cite book
			| last = Johnson
			| first = Art
			| year = 1999
			| title = Famous problems and their mathematicians
			| publisher = Teacher Ideas Press
			| page = 45
			| isbn = 9781563084461
			| url = https://archive.org/details/famousproblemsth0000john/page/45
			| url-access = limited
			| quote = The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
			}}
			{{pb}}
			{{cite book
			| last1 = Stakhov
			| first1 = Alexey P.
			| author1-link = Alexey Stakhov
			| last2 = Olsen
			| first2 = Scott
			| year = 2009
			| title = The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science
			| chapter = §1.4.1 A Golden Rectangle with a Side Ratio of ''τ''
			| publisher = World Scientific
			| pages = 20–21
			| quote = A credit card has a form of the golden rectangle
			| chapter-url = https://archive.org/details/alexy-stakhov-the-mathematics-of-harmony-from-euclid-to-contemporary-mathematics/page/21/
			}}
			{{pb}}
			{{cite book
			| last = Cox
			| first = Simon
			| year = 2004
			| title = Cracking the Da Vinci Code
			| publisher = Barnes & Noble
			| page = 62
			| isbn = 978-1-84317-103-4
			| url = https://archive.org/details/crackingdavincic00coxs/page/62/
			| quote = The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.
			}}
			</ref>
			<ref name=lendvai>
	If <math> \lfloor n/2 - 1 \rfloor = m </math>, then:
			{{cite book
			| last = Lendvai
			| first = Ernő
			| year = 1971
			| title = Béla Bartók: An Analysis of His Music
			| publication-place = London
			| publisher = Kahn and Averill
			}}
			</ref>
			<ref name=Smith>
	:<math> \!\ \varphi^n = \varphi^{n-1} + \varphi^{n-3} + \cdots + \varphi^{n-1-2m} + \varphi^{n-2-2m} </math>
			{{cite book
			| last = Smith
			| first = Peter F.
			| year = 2003
			| title = The Dynamics of Delight: Architecture and Aesthetics
			| url = https://books.google.com/books?id=ZgftUKoMnpkC&pg=PA83
			| publisher = Routledge
			| page = 83
			| isbn = 9780415300100
			}}
			</ref>
			<ref name=howat>
	:<math> \!\ \varphi^n - \varphi^{n-1} = \varphi^{n-2} . </math>
			{{cite book
			| last= Howat
			| first = Roy
			| year = 1983
			| title = Debussy in Proportion: A Musical Analysis
			| chapter = 1. Proportional structure and the Golden Section
			| chapter-url = https://archive.org/details/debussyinproport0000howa/page/1
			| publisher = Cambridge University Press
			| pages = 1–10
			}}
			</ref>
			<ref name=trezise>
	When the golden ratio is used as the base of a ] (see ], sometimes dubbed ''phinary'' or ''φ-nary''), every integer has a terminating representation, despite φ being irrational, but every fraction has a non-terminating representation.
			{{cite book
			| last = Trezise
			| first = Simon
			| year = 1994
			| title = Debussy: La Mer
			| publisher = Cambridge University Press
			| page = 53
			| isbn = 9780521446563
			| url = https://books.google.com/books?id=THD1nge_UzcC&pg=PA53
			}}
			</ref>
			<ref name=833cents>
	The golden ratio is a ] of the ] <math>\mathbb{Q}(\sqrt{5})</math> and is a ].<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 ].
			{{cite journal
			| last = Mongoven
			| first = Casey
			| year = 2010
			| title = A style of music characterized by Fibonacci and the golden ratio
			| url = https://www.mat.ucsb.edu/Publications/mongoven_CongressusNumerantium2010.pdf
			| journal = Congressus Numerantium
			| volume = 201
			| pages = 127–138
			}}
			{{pb}}
			{{cite journal
			| last = Hasegawa
			| first = Robert
			| year = 2011
			| title = ''Gegenstrebige Harmonik'' in the Music of Hans Zender
			| journal = Perspectives of New Music
			| volume = 49
			| issue = 1
			| doi = 10.1353/pnm.2011.0000
			| publisher = Project Muse
			| jstor = 10.7757/persnewmusi.49.1.0207
			| pages = 207–234
			}}
			{{pb}}
			{{cite conference
			| last = Smethurst
			| first = Reilly
			| year = 2016
			| title = Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal
			| url = https://archive.bridgesmathart.org/2016/bridges2016-519.html
			| editor1-last = Torrence
			| editor1-first = Eve
			| editor2-last = Torrence
			| editor2-first = Bruce
			| editor3-last = Séquin
			| editor3-first = Carlo
			| editor4-last = McKenna
			| editor4-first = Douglas
			| editor5-last = Fenyvesi
			| editor5-first = Kristóf
			| editor6-last = Sarhangi
			| editor6-first = Reza
			| display-editors = 1
			| book-title = Proceedings of ] 2016
			| conference = Jyväskylä, Finland
			| pages = 519–522
			| publisher = Tessellations Publishing
			}}
	The golden ratio also appears in ], as the maximum distance from a point on one side of an ] to the closer of the other two sides: this distance, the side length of the ] formed by the points of tangency of a circle inscribed within the ideal triangle, is 4&nbsp;ln&nbsp;φ.<ref>, cabri.net, retrieved 2009-07-21.</ref>
			</ref>
			<ref name=padovan>
	===Decimal expansion===
			{{Cite book
	The golden ratio's decimal expansion can be calculated directly from the expression
			| last = Padovan
	:<math>\varphi = {1+\sqrt{5} \over 2}</math>
			| first = Richard
			| year = 1999
			| title = Proportion: Science, Philosophy, Architecture
			| publisher = Taylor & Francis
			| pages = 305–306
			| doi = 10.4324/9780203477465
			| isbn = 9781135811112
			}}
			{{pb}}
			{{cite journal
			| last = Padovan
			| first = Richard
			| year=2002
			| title = Proportion: Science, Philosophy, Architecture
			| journal = Nexus Network Journal
			| volume = 4
			| issue=1
			| pages = 113–122
			| doi=10.1007/s00004-001-0008-7
			| doi-access=free
			}}
			</ref>
			<ref name=zeising>
	with √5 ≈ 2.2360679774997896964. The ] can be calculated with the ], starting with an initial estimate such as ''x''φ = 2 and ]
			{{cite book
			| first = Adolf
			| last = Zeising
			| author-link = Adolf Zeising
			| title = Neue Lehre von den Proportionen des menschlichen Körpers
			| trans-title = New doctrine of the proportions of the human body
			| language = de
			| year = 1854
			| publisher = Weigel
			| chapter = Einleitung
			| pages = 1–10
			| chapter-url = https://archive.org/details/neuelehrevondenp00zeis/page/n25/
			}}
			</ref>
			<ref name=pommersheim>
	:<math>x_{n+1} = \frac{(x_n + 5/x_n)}{2}</math>
			{{cite book
			| editor1-last = Pommersheim
			| editor1-first = James E.
			| editor2-last = Marks
			| editor2-first = Tim K.
			| editor3-last = Flapan
			| editor3-first = Erica L.
			| editor3-link = Erica Flapan
			| year = 2010
			| title = Number Theory: A Lively Introduction with Proofs, Applications, and Stories
			| publisher = Wiley
			| page = 82
			}}
			</ref>
			<ref name=ising>
	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.
			{{cite journal
			| last1 = Coldea
			| first1 = R.
			| last2 = Tennant
			| first2 = D.A.
			| last3 = Wheeler
			| first3 = E.M.
			| last4 = Wawrzynksa
			| first4 = E.
			| last5 = Prabhakaran
			| first5 = D.
			| last6 = Telling
			| first6 = M.
			| last7 = Habicht
			| first7 = K.
			| last8 = Smeibidl
			| first8 = P.
			| last9 = Keifer
			| first9 = K.
			| year = 2010
			| title = Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry
			| journal = Science
			| volume = 327
			| issue = 5962
			| pages = 177–180
			| doi = 10.1126/science.1180085
			| pmid = 20056884
			| arxiv = 1103.3694
			| bibcode = 2010Sci...327..177C
			| s2cid = 206522808
			}}
			</ref>
			<ref name=disco>{{cite web
	The Babylonian algorithm for √5 is equivalent to ] 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 ], 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,
			| url = https://science.nasa.gov/science-news/science-at-nasa/2001/ast09oct_1/
			| title = A Disco Ball in Space
			| publisher = NASA
			| date = 2001-10-09
			| access-date = 2007-04-16
			| archive-date = 2020-12-22
			| archive-url = https://web.archive.org/web/20201222224745/https://science.nasa.gov/science-news/science-at-nasa/2001/ast09oct_1/
			| url-status = dead
			}}</ref>
			<ref name=pheasant>
	:<math>x_{n+1} = \frac{x_n^2 + 1}{2x_n - 1},</math>
			{{cite book
			| last = Pheasant
			| first = Stephen
			| year = 1986
			| title = Bodyspace
			| publisher = Taylor & Francis
			| isbn = 9780850663402
			| url = https://archive.org/details/bodyspaceanthrop0000phea/
			| url-access = limited
			}}
			</ref>
			<ref name=vanLaack>
	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
			{{cite book
			| last = van Laack
			| first = Walter
			| title = A Better History Of Our World: Volume 1 The Universe
			| location = Aachen
			| publisher = van Laach
			| year = 2001
			}}
			</ref>
			<ref name=dunlap>
	:<math>x_{n+1} = \frac{x_n^2 + 2x_n}{x_n^2 + 1}.</math>
			{{cite book
			| last = Dunlap
			| first = Richard A.
			| year = 1997
			| title = The Golden Ratio and Fibonacci Numbers
			| publisher = World Scientific
			| page =
			}}
			</ref>
			<ref name=falbo>
	These iterations all ]; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with ]. 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 ]s ''']''' and ].
			{{cite journal
			| last = Falbo
			| first = Clement
			| date = March 2005
			| title = The golden ratio—a contrary viewpoint
			| journal = The College Mathematics Journal
			| volume = 36
			| issue = 2
			| pages = 123–134
			| doi = 10.1080/07468342.2005.11922119
			| s2cid = 14816926
			}}
			</ref>
			<ref name=miller>
	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.
			{{cite journal
			| last = Miller
			| first = William
			| year = 1996
			| title = Pentagons and Golden Triangles
			| journal = Mathematics in School
			| volume = 25
			| issue = 4
			| pages = 2–4
			| jstor = 30216571
			}}
			</ref>
			<ref name=robinson>
	The golden ratio ''φ'' has been calculated to an accuracy of several millions of decimal digits {{OEIS|id=A001622}}. Alexis Irlande performed computations and verification of the first 17,000,000,000 digits.<ref>{{Cite book| title = The golden number to 17 000 000 000 digits | url = http://www.matematicas.unal.edu.co/airlande/phi.html.en | publisher = Universidad Nacional de Colombia | year = 2008 }}{{dead link|date=June 2011}}</ref>
			{{cite web
			| last1 = Frettlöh
			| first1 = D.
			| last2 = Harriss
			| first2 = E.
			| last3 = Gähler
			| first3 = F.
			| title = Robinson Triangle
			| website = Tilings Encyclopedia
			| url = https://tilings.math.uni-bielefeld.de/substitution/robinson-triangle/
			}}
			{{pb}}
			{{cite journal
			| last = Clason
			| first = Robert G
			| year = 1994
			| title = A family of golden triangle tile patterns.
			| journal = The Mathematical Gazette
			| volume = 78
			| number = 482
			| pages = 130–148
			| doi = 10.2307/3618569
			| jstor = 3618569
			| s2cid = 126206189
			}}
			</ref>
			<ref name=moscovich>
	==Pyramids==
			{{cite book
	]
			| last = Moscovich
	Both Egyptian pyramids and those mathematical regular ]s that resemble them can be analyzed with respect to the golden ratio and other ratios.
			| first = Ivan
			| author-link = Ivan Moscovich
			| year = 2004
			| title = The Hinged Square & Other Puzzles
			| location = New York
			| publisher = Sterling
			| url = https://books.google.com/books?id=4n7PvKC1dfwC&pg=PA122
			| page = 122
			| isbn = 9781402716669
			}}
			</ref>
			<ref name=shellspirals>
	===Mathematical pyramids and triangles===
			{{cite journal
	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.
			| last = Peterson
			| first = Ivars
			| author-link = Ivars Peterson
			| date = 1 April 2005
			| title = Sea shell spirals
			| journal = ]
			| url = http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
			| access-date = 10 November 2008
			| archive-date = 3 October 2012
			| archive-url = https://web.archive.org/web/20121003045834/http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals
			| url-status = dead
			}}
			</ref>
			<ref name=gutenberg>
	The medial ] of this "golden" pyramid (see diagram), with sides <math>1:\sqrt{\varphi}:\varphi</math> is interesting in its own right, demonstrating via the ] the relationship <math>\sqrt{\varphi} = \sqrt{\varphi^2 - 1}</math> or <math>\varphi = \sqrt{1 + \varphi}</math>. This "]"<ref>{{Cite book| title = The Best of Astraea: 17 Articles on Science, History and Philosophy | url = http://books.google.com/?id=LDTPvbXLxgQC&pg=PA93&dq=kepler-triangle | publisher = Astrea Web Radio | isbn = 1-4259-7040-0 | year = 2006 | author1 = Radio, Astraea Web }}</ref>
			{{cite book
	is the only right triangle proportion with edge lengths in ],<ref name=herz>{{Cite book| title = The Shape of the Great Pyramid | author = Roger Herz-Fischler | publisher = Wilfrid Laurier University Press | year = 2000 | isbn = 0-88920-324-5 | url = http://books.google.com/?id=066T3YLuhA0C&pg=PA81&dq=kepler-triangle+geometric }}</ref> just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in ]. The angle with 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>
			| last = Man
			| first = John
			| year = 2002
			| title = Gutenberg: How One Man Remade the World with Word
			| pages = 166–167
			| publisher = Wiley
			| isbn = 9780471218234
			| url = https://archive.org/details/gutenberghowonem00john/page/166
			| quote = The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.
			}}
			</ref>
			<ref name=Fechner>
	A nearly similar pyramid shape, but with rational proportions, is described in the ] (the source of a large part of modern knowledge of ancient ]), based on the 3:4:5 triangle;<ref name = "maor"/> the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).<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}}</ref> 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, ''Mathematics for the Million'', London: Allen & Unwin, 1942, p. 63., as cited by Dick Teresi, ''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"/>
			{{cite book
			| title = Vorschule der Ästhetik
			| language = de
			| trans-title = Preschool of Aesthetics
			| last = Fechner
			| first = Gustav
			| year = 1876
			| publisher = Breitkopf & Härtel
			| location = Leipzig
			| pages = 190–202
			| url = https://archive.org/details/vorschulederaest12fechuoft/page/n203/
			}}
			</ref>
			<ref name=osler>
	Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51'), very close to the 51.827° of the ]. This pyramid relationship corresponds to the ] <math>\sqrt{\varphi} \approx 4/\pi</math>.
			{{cite journal
			| last = Osler
			| first = Carol
			| title = Support for Resistance: Technical Analysis and Intraday Exchange Rates
			| journal = Federal Reserve Bank of New York Economic Policy Review
			| year = 2000
			| volume = 6
			| issue = 2
			| pages= 53–68
			| quote = 38.2 percent and 61.8 percent retracements of recent rises or declines are common,
			| url= http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf
			| archive-url = https://web.archive.org/web/20070512155447/http://ftp.ny.frb.org/research/epr/00v06n2/0007osle.pdf
			| archive-date = 2007-05-12
			| url-status = live
			}}
			</ref>
			<ref name=magicdow>
	Egyptian pyramids very close in proportion to these mathematical pyramids are known.<ref name=Herkommer/>
			{{cite report
			| last1 = Batchelor
			| first1 = Roy
			| author1-link = Roy Batchelor
			| last2 = Ramyar
			| first2 = Richard
			| year = 2005
			| title = Magic numbers in the Dow
			| publisher = Cass Business School
			| url = http://openaccess.city.ac.uk/16276/
			| pages = 13, 31
			}}
			Popular press summaries can be found in:
			{{cite news
			| last = Stevenson
			| first = Tom
			| date = 2006-04-10
			| title = Not since the 'big is beautiful' days have giants looked better
			| newspaper = The Daily Telegraph
			| url = https://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html
			}}
			{{cite news
			| date = 2006-09-23
			| title = Technical failure
			| magazine = The Economist
			| url = https://www.telegraph.co.uk/finance/2947908/Not-since-the-big-is-beautiful-days-have-giants-looked-better.html
			}}
			</ref>
			<ref name=greatpyramid>
	===Egyptian pyramids===
			{{cite book
	In the mid-nineteenth century, Röber studied various Egyptian pyramids including Khafre, Menkaure and some of the Giza, Sakkara, and Abusir groups, and was interpreted as saying that half the base of the side of the pyramid is the middle mean of the side, forming what other authors identified as the ]; many other mathematical theories of the shape of the pyramids have also been explored.<ref name=herz/>
			| last = Herz-Fischler
			| first = Roger
			| year = 2000
			| isbn = 0-88920-324-5
			| publisher = Wilfrid Laurier University Press
			| title = The Shape of the Great Pyramid
			}} The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
			{{pb}}
			{{cite book
			| last = Rossi
			| first = Corinna
			| author-link = Corinna Rossi
			| title = Architecture and Mathematics in Ancient Egypt
			| year = 2004
			| publisher = Cambridge University Press
			| pages = 67–68
			| url = https://archive.org/details/architechture-and-mathematics-in-ancient-egypt-corianna-rossi-2003/page/67/
			| quote = there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to {{tmath|\varphi}}, and {{tmath|\varphi}} itself as a number, do not fit with the extant Middle Kingdom mathematical sources
			}}; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
			{{pb}}
			{{cite journal
			| last1 = Rossi
			| first1 = Corinna
			| last2 = Tout
			| first2 = Christopher A.
			| year = 2002
			| title = Were the Fibonacci series and the Golden Section known in ancient Egypt?
			| journal = ]
			| volume = 29
			| issue = 2
			| doi = 10.1006/hmat.2001.2334
			| pages = 101–113
			| hdl = 11311/997099
			| hdl-access = free
			}}
			{{pb}}
			{{cite journal
			| last = Markowsky
			| first = George
			| year = 1992
			| title = Misconceptions about the Golden Ratio
			| journal = ]
			| volume = 23
			| issue = 1
			| doi = 10.2307/2686193
			| url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
			| publisher = Mathematical Association of America
			| pages = 2–19
			| jstor = 2686193
			| quote = It does not appear that the Egyptians even knew of the existence of {{tmath|\varphi}} much less incorporated it in their buildings
			| access-date = 2012-06-29
			}}
			</ref>
			<ref name=Polemic>
	One Egyptian pyramid is remarkably close to a "golden pyramid"—the ] (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52' is extremely close to the "golden" pyramid inclination of 51° 50' and the π-based pyramid inclination of 51° 51'; other pyramids at Giza (Chephren, 52° 20', and Mycerinus, 50° 47')<ref name="maor">], ''Trigonometric Delights'', Princeton Univ. Press, 2000</ref> are also quite close. Whether the relationship to the golden ratio in these pyramids is by design or by accident remains open to speculation.<ref>{{cite book
			{{cite journal
	|title=The history of mathematics: an introduction
			| last = Van Mersbergen
	|edition=4
	|first1=David M.		| first = Audrey M.
			| year = 1998
	|last1=Burton
			| title = Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic
	|publisher=WCB McGraw-Hill
			| journal = Communication Quarterly
	|year=1999
			| volume = 46
	|isbn=0-07-009468-3
			| number = 2
	|page=56
			| pages = 194–213
	|url=http://books.google.com/books?id=GKtFAAAAYAAJ}}</ref> Several other Egyptian pyramids are very close to the rational 3:4:5 shape.<ref name=Herkommer/>
			| doi = 10.1080/01463379809370095
			}}
			</ref>
			<ref name=mathinstinct>
	Adding fuel to controversy over the architectural authorship of the Great Pyramid, ], 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 π or φ.{{Disputed-inline|Egyptian Pyramids|date=November 2013}}<ref>{{Cite book |last=Bell |first=Eric Temple |title=The Development of Mathematics |location=New York |publisher=Dover |year=1940 |page=40 |url=http://books.google.com/books?id=_5KAnw3QMC8C&pg=PA40#v=onepage&q&f=false}}</ref>
			{{cite book
			| last = Devlin
			| first = Keith J.
			| year = 2005
			| title = The Math Instinct
			| url = https://archive.org/details/isbn_9781560258391/page/108
			| location = New York
			| publisher = Thunder's Mouth Press
			| page = 108
			}}
			</ref>
			<ref name=gazalé>
	Michael Rice<ref>Rice, Michael, ''Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C'' pp. 24 Routledge, 2003, ISBN 0-415-26876-1</ref> asserts that principal authorities on the history of ] have argued that the Egyptians were well acquainted with the golden ratio and that it is part of 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'' pp.24 Routledge, 2003</ref> Historians of science have always debated whether the Egyptians had any such knowledge or not, contending rather that its appearance in an Egyptian building is the result of chance.<ref>{{Cite journal
			{{cite book
	| last = Markowsky
	| first = George		| last = Gazalé
	| date = January 1992		| first = Midhat J.
			| author-link = Midhat J. Gazalé
	| title = Misconceptions about the Golden Ratio
			| year = 1999
	| journal = College Mathematics Journal
			| title = Gnomon: From Pharaohs to Fractals
	| volume = 23
	| issue = 1		| publisher = Princeton
			| page = 125
	| doi = 10.2307/2686193
			| isbn = 9780691005140
	| url = http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
			| url = https://archive.org/details/gnomonfrompharao0000gaza/page/125/
	| format = PDF
			| url-access = limited
	| accessdate =
			}}
	| jstor = 2686193
			</ref>
	| publisher = Mathematical Association of America
	| pages = 2–19
	}}</ref>
			<ref name=foutakis>
	In 1859, the ] ] claimed that, in the ], the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle ] to the ground, to half the length of the side of the square base, equivalent to the ] of the angle θ.<ref>Taylor, ''The Great Pyramid: Why Was It Built and Who Built It?'', 1859</ref> The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, ], according to Matila Ghyka,<ref>Matila Ghyka ''The Geometry of Art and Life'', New York: Dover, 1977</ref> reported the great pyramid height 148.2 m, and half-base 116.4 m, yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.
			{{cite journal
			| last = Foutakis
			| first = Patrice
			| year = 2014
			| title = Did the Greeks Build According to the Golden Ratio?
			| journal = Cambridge Archaeological Journal
			| volume = 24
			| number = 1
			| pages = 71–86
			| doi = 10.1017/S0959774314000201
			| s2cid = 162767334
			}}
			</ref>
			<ref name=centrepompidou1>
	==Disputed observations==
			, October 1912, Mediation Centre Pompidou
	Examples of disputed observations of the golden ratio include the following:
			</ref>
	*Historian John Man states that the pages of the ] were "based on the golden section shape". However, according to Man's own measurements, the ratio of height to width was 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>
	*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=0-7484-0067-2 }}</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 ] shell, the construction of which proceeds in a ], 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 proportioned by the golden ratio relative to the previous one;<ref name=moscovich>], ''Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles,'' New York: Sterling, 2004</ref> however, measurements of nautilus shells do not support this claim.<ref>{{Cite journal|title=Sea shell spirals|last=Peterson|first=Ivars|journal=Science News|url=http://www.sciencenews.org/view/generic/id/6030/title/Sea_Shell_Spirals}}</ref>
	*In investing, some practitioners of ] 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|format=PDF}}</ref> The use of the golden ratio in investing is also related to more complicated patterns described by ] (e.g. ] and ]). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.<ref>] and Richard Ramyar, "," 25th International Symposium on Forecasting, 2005, p. 13, 31. "", Tom Stevenson, ], Apr. 10, 2006, and "Technical failure", ], Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.</ref>
			<ref name=centrepompidou2>
	==See also==
			{{Webarchive
	{{div col|colwidth=20em}}
			| url = https://web.archive.org/web/20201030080014/http://bibliothequekandinsky.centrepompidou.fr/imagesbk/RP225%5C001/M5050_X0031_PER_P02251912001.pdf
	* ]
			| date = 2020-10-30
	* ]
			}}, Bibliothèque Kandinsky
	* ]
			</ref>
	* ]
	* ]
	* ]
	{{div col end}}
			<ref name=seuratclaims>
	==References and footnotes==
			{{cite journal
	{{reflist|30em}}
			| last = Herz-Fischler
			| first = Roger
			| year = 1983
			| title = An Examination of Claims Concerning Seurat and the Golden Number
			| journal = Gazette des Beaux-Arts
			| volume = 101
			| pages = 109–112
			| url = https://people.math.carleton.ca/~rhfischl/PUBLICATIONS/seurat_GN.pdf
			}}</ref>
			<ref name=herbert>
			{{cite book
			| last = Herbert
			| first = Robert
			| year = 1968
			| title = Neo-Impressionism
			| page = 24
			| publisher = Guggenheim Foundation
			| url = https://archive.org/details/neoimpressionism0000herb
			| url-access = limited
			}}
			</ref>
			<ref name=camfield>
			{{cite journal
			| last = Camfield
			| first = William A.
			| date = March 1965
			| doi = 10.1080/00043079.1965.10788819
			| issue = 1
			| journal = The Art Bulletin
			| pages = 128–134
			| title = Juan Gris and the golden section
			| volume = 47
			}}
			</ref>
			<ref name=juangris>
			{{cite book
			| last = Green
			| first = Christopher
			| year = 1992
			| title = Juan Gris
			| publisher = Yale
			| pages = 37–38
			| isbn = 9780300053746
			| url = https://archive.org/details/juangris0000gree/page/37/
			| url-access = limited
			}}
			{{pb}}
			{{cite book
			| last = Cottington
			| first = David
			| year = 2004
			| title = Cubism and Its Histories
			| publisher = Manchester University Press
			| pages = 112, 142
			| url = https://books.google.com/books?isbn=0719050049
			}}
			</ref>
			<ref name=allard>
			{{cite journal
			| last = Allard
			| first = Roger
			| date = June 1911
			| title = Sur quelques peintres
			| journal = Les Marches du Sud-Ouest
			| pages = 57–64
			}}
			Reprinted in
			{{cite book
			| editor1-last = Antliff
			| editor1-first = Mark
			| editor2-last = Leighten
			| editor2-first = Patricia
			| title = A Cubism Reader, Documents and Criticism, 1906–1914
			| year = 2008
			| publisher = The University of Chicago Press
			| pages = 178–191
			| url = https://archive.org/details/cubismreaderdocu0008unse/page/178/
			}}
			</ref>
			<ref name=bouleau>
			{{cite book
			| last = Bouleau
			| first = Charles
			| year = 1963
			| title = The Painter's Secret Geometry: A Study of Composition in Art
			| publisher = Harcourt, Brace & World
			| pages = 247–248
			| url = https://archive.org/details/painterssecretge0000char/page/247/
			}}
			</ref>
			<!-- END REFLIST --> }}
			===Works cited===
			{{refbegin|30em}}
			* {{Cite book
			| last = Herz-Fischler
			| first = Roger
			| year = 1998
			| orig-year = 1987
			| title = A Mathematical History of the Golden Number
			| publisher = Dover
			| isbn = 9780486400075
			| url-access = registration
			| url = https://archive.org/details/mathematicalhist0000herz
			}} (Originally titled ''A Mathematical History of Division in Extreme and Mean Ratio''.)
			* {{cite book
			| last = Livio
			| first = Mario
			| author-link = Mario Livio
			| year = 2002
			| title = The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
			| url = https://archive.org/details/goldenratiostory00livi/
			| url-access = registration
			| publisher = Broadway Books
			| location = New York
			| isbn = 9780767908153
			}}
			* {{cite book
			| last1 = Posamentier
			| first1 = Alfred S.
			| author-link1 = Alfred S. Posamentier
			| last2 = Lehmann
			| first2 = Ingmar
			| year = 2011
			| title = The Glorious Golden Ratio
			| url = https://books.google.com/books?id=Gw-lqvE6fNgC
			| publisher = ]
			| isbn = 9-781-61614-424-1
			}}
			{{refend}}
	==Further reading==		==Further reading==
	{{refbegin|30em}}		{{refbegin|30em}}
	*{{Cite book		* {{Cite book
	| last = Doczi		| last = Doczi
	| first = György		| first = György
	| title = The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture		| title = The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture
	| origyear = 1981		| year = 1981
	| year = 2005		| publisher = Shambhala
	| publisher = Shambhala Publications
	| location = Boston		| location = Boston
			| url = https://archive.org/details/poweroflimits00gyeo/
	| isbn = 1-59030-259-1
			| url-access = registration
	}}		}}
	*{{Cite book		* {{cite book
			| editor-last = Hargittai
			| editor-first = István
			| title = Fivefold Symmetry
			| publisher = World Scientific
			| year = 1992
			| isbn = 9789810206000
			| url = https://archive.org/details/fivefoldsymmetry0000unse/
			| url-access = registration
			| ref = CITEREFHargittai2002
			}}
			* {{Cite book
	| last = Huntley		| last = Huntley
	| first = H. E.		| first = H. E.
	| title = The Divine Proportion: A Study in Mathematical Beauty		| title = The Divine Proportion: A Study in Mathematical Beauty
	| url =
	| year = 1970		| year = 1970
	| publisher = Dover Publications		| publisher = Dover
	| location = New York		| location = New York
	| isbn = 0-486-22254-3		| isbn = 978-0-486-22254-7
	}}		}}
	*{{Cite book		* {{cite book
	| last = Livio		| editor-last=Schaaf
	| first = Mario		| editor-first=William L.
			| year=1967
	| title = The Golden Ratio: The Story of PHI, the World's Most Astonishing Number
			| title=The Golden Measure
	| origyear = 2002
			| publisher=Stanford University
	| url =
			| series=California School Mathematics Study Group Reprint Series
	| edition = Hardback
			| url=https://files.eric.ed.gov/fulltext/ED175696.pdf |archive-url=https://web.archive.org/web/20150425083400/http://files.eric.ed.gov/fulltext/ED175696.pdf |archive-date=2015-04-25 |url-status=live
	| year = 2002
	| publisher = Broadway (Random House)
	| location = NYC
	| isbn = 0-7679-0815-5
	}}		}}
	*{{Cite book		* {{Cite book
	| last = Joseph		| last = Scimone
	| first = George G.		| first = Aldo
	| title = The Crest of the Peacock: The Non-European Roots of Mathematics		| title = La Sezione Aurea. Storia culturale di un leitmotiv della Matematica
	| origyear = 1991		| year = 1997
			| publisher = Sigma Edizioni
	| url =
	| edition = New		| location = Palermo
			| isbn = 978-88-7231-025-0
	| year = 2000
	| publisher = Princeton University Press
	| location = Princeton, NJ
	| isbn = 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 = 1-4196-2157-2
	}}		}}
	*{{Cite book		* {{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 = 0-06-016939-7
	}}
	*{{Cite book
	| last = Stakhov
	| first = A. P.
	| title = The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science
	| url =
	| year = 2009
	| publisher = World Scientific Publishing
	| location = Singapore
	| isbn = 978-981-277-582-5
	}}
	*{{Cite book
	| last = Walser		| last = Walser
	| first = Hans		| first = Hans
	| others = Peter Hilton trans.		| others = Peter Hilton trans.
	| title = The Golden Section		| title = The Golden Section
	| origyear = ''Der Goldene Schnitt'' 1993		| orig-year = ''Der Goldene Schnitt'' 1993
	| url =
	| year = 2001		| year = 2001
	| publisher = The Mathematical Association of America		| publisher = The Mathematical Association of America
	| location = Washington, DC		| location = Washington, DC
	| isbn = 0-88385-534-8		| isbn = 978-0-88385-534-8
	}}		}}
	*{{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}}
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	*{{springer|title=Golden ratio|id=p/g044570}}
	* by Michael Schreiber, ], 2007.
			* {{MathWorld|title=Golden Ratio|urlname=GoldenRatio}}
	*
			* {{cite web
	*{{MathWorld|title=Golden Ratio|urlname=GoldenRatio}}
			| last = Bogomolny
	*{{cite web|url=http://www.physorg.com/news180531747.html |title=Researcher explains mystery of golden ratio |work=] |date=December 21, 2009 |postscript=<!--None--> }}.
			| first = Alexander
	*{{cite web
			| author-link = Alexander Bogomolny
	| url = http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html
			| year = 2018
	| title = The Golden section ratio: Phi
			| title = Golden Ratio in Geometry
	| last = Knott
			| website = ]
	| first = Ron
			| url = http://www.cut-the-knot.org/do_you_know/GoldenRatio.shtml
	| date =
			}}
	}} Information and activities by a mathematics professor.
			* {{cite web | url = http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html| title = The Golden section ratio: Phi| last = Knott| first = Ron}} Information and activities by a mathematics professor.
	*. Green, Thomas M. Updated June 2005. Archived November 2007. Geometry instruction with problems to solve.
			* {{Webarchive|url=https://web.archive.org/web/20201112034625/https://www.maa.org/external_archive/devlin/devlin_05_07.html |date=2020-11-12 }}, by ], addressing multiple allegations about the use of the golden ratio in culture.
	*{{cite arXiv
			* collected by ]
	|first1=Robert P.
			*
	|last1=Schneider
	|eprint=1109.3216
			{{Algebraic numbers}}
	|title=A Golden Pair of Identities in the Theory of Numbers
			{{Irrational number}}
	|class=math.HO
			{{Metallic ratios}}
	|year=2011}} Proves formulas that involve the golden mean and the ] and ]s.
			{{Ancient Greek mathematics}}
	* , by ], addressing multiple allegations about the use of the golden ratio in culture.
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Latest revision as of 01:39, 28 December 2024

Number, approximately 1.618 For other uses, see Golden ratio (disambiguation) and Golden number (disambiguation).

Golden ratio (φ)

Representations
Decimal	1.618033988749894 . . .
Algebraic form	${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$
Continued fraction	${\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{{\vphantom {1}} \atop \ddots }}}}}}}}$

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ with ⁠${\displaystyle a>b>0}$⁠, ⁠${\displaystyle a}$⁠ is in a golden ratio to ⁠${\displaystyle b}$⁠ if

$${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi ,}$$

where the Greek letter phi (⁠${\displaystyle \varphi }$⁠ or ⁠${\displaystyle \phi }$⁠) denotes the golden ratio. The constant ⁠${\displaystyle \varphi }$⁠ satisfies the quadratic equation ⁠${\displaystyle \textstyle \varphi ^{2}=\varphi +1}$⁠ and is an irrational number with a value of

${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=}$1.618033988749....

The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; and also goes by other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of ⁠${\displaystyle \varphi }$⁠—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

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

Calculation

Two quantities ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ are in the golden ratio ⁠${\displaystyle \varphi }$⁠ if $${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .}$$

Thus, if we want to find ⁠${\displaystyle \varphi }$⁠, we may use that the definition above holds for arbitrary ⁠${\displaystyle b}$⁠; thus, we just set ⁠${\displaystyle b=1}$⁠, in which case ⁠${\displaystyle \varphi =a}$⁠ and we get the equation ⁠${\displaystyle (\varphi +1)/\varphi =\varphi }$⁠, which becomes a quadratic equation after multiplying by ⁠${\displaystyle \varphi }$⁠: $${\displaystyle \varphi +1=\varphi ^{2}}$$ which can be rearranged to $${\displaystyle {\varphi }^{2}-\varphi -1=0.}$$

The quadratic formula yields two solutions:

${\displaystyle {\frac {1+{\sqrt {5}}}{2}}=1.618033\dots \ }$ and ${\displaystyle \ {\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .}$

Because ⁠${\displaystyle \varphi }$⁠ is a ratio between positive quantities, ⁠${\displaystyle \varphi }$⁠ is necessarily the positive root. The negative root is in fact the negative inverse ⁠${\displaystyle -1/\varphi }$⁠, which shares many properties with the golden ratio.

History

See also: Mathematics and art and Fibonacci number § History

According to Mario Livio,

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician 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.

— The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreans. Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

The golden ratio 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 but did not observe that it was connected to the Fibonacci numbers.

Luca Pacioli named his book Divina proportione (1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section'). Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.

German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about ⁠${\displaystyle 0.6180340}$⁠" in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.

Eighteenth-century mathematicians Abraham de Moivre, Nicolaus I 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". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875.

By 1910, inventor Mark Barr began using the Greek letter phi (⁠${\displaystyle \varphi }$⁠) as a symbol for the golden ratio. It has also been represented by tau (⁠${\displaystyle \tau }$⁠), the first letter of the ancient Greek τομή ('cut' or 'section').

The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. 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. This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterwards explained through analogies to the Penrose tiling.

Mathematics

Irrationality

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

Contradiction from an expression in lowest terms

This is a proof by infinite descent. 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 ⁠${\displaystyle n}$⁠ and the longer part ⁠${\displaystyle m}$⁠, then the second statement above becomes

⁠${\displaystyle n}$⁠ is to ⁠${\displaystyle m}$⁠ as ⁠${\displaystyle m}$⁠ is to ⁠${\displaystyle n-m}$⁠.

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

By irrationality of the square root of 5

Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If ⁠${\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}$⁠ is assumed to be rational, then ⁠${\displaystyle 2\varphi -1={\sqrt {5}}}$⁠, the square root of ⁠${\displaystyle 5}$⁠, must also be rational. This is a contradiction as the square roots of all non-square natural numbers are irrational.

Minimal polynomial

The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial $${\displaystyle x^{2}-x-1.}$$

This quadratic polynomial has two roots, ⁠${\displaystyle \varphi }$⁠ and ⁠${\displaystyle \textstyle -\varphi ^{-1}}$⁠.

The golden ratio is also closely related to the polynomial ⁠${\displaystyle \textstyle x^{2}+x-1}$⁠, which has roots ⁠${\displaystyle -\varphi }$⁠ and ⁠${\displaystyle \textstyle \varphi ^{-1}}$⁠. As the root of a quadratic polynomial, the golden ratio is a constructible number.

Golden ratio conjugate and powers

The conjugate root to the minimal polynomial ⁠${\displaystyle \textstyle x^{2}-x-1}$⁠ is

$${\displaystyle -{\frac {1}{\varphi }}=1-\varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .}$$

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

This illustrates the unique property of the golden ratio among positive numbers, that $${\displaystyle {\frac {1}{\varphi }}=\varphi -1,}$$

or its inverse, $${\displaystyle {\frac {1}{1/\varphi }}={\frac {1}{\varphi }}+1.}$$

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

$${\displaystyle {\begin{aligned}\varphi ^{2}&=\varphi +1=2.618033\dots ,\\{\frac {1}{\varphi }}&=\varphi -1=0.618033\dots .\end{aligned}}}$$

The sequence of powers of ⁠${\displaystyle \varphi }$⁠ contains these values ⁠${\displaystyle 0.618033\ldots }$⁠, ⁠${\displaystyle 1.0}$⁠, ⁠${\displaystyle 1.618033\ldots }$⁠, ⁠${\displaystyle 2.618033\ldots }$⁠; more generally, any power of ⁠${\displaystyle \varphi }$⁠ is equal to the sum of the two immediately preceding powers: $${\displaystyle \varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}=\varphi \cdot \operatorname {F} _{n}+\operatorname {F} _{n-1}.}$$

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

If ⁠${\displaystyle {\bigl \lfloor }{\tfrac {1}{2}}n-1{\bigr \rfloor }=m}$⁠, then: $${\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-3}+\cdots +\varphi ^{n-1-2m}+\varphi ^{n-2-2m}\\\varphi ^{n}-\varphi ^{n-1}&=\varphi ^{n-2}.\end{aligned}}}$$

Continued fraction and square root

See also: Lucas number § Continued fractions for powers of the golden ratio

The formula ⁠${\displaystyle \varphi =1+1/\varphi }$⁠ can be expanded recursively to obtain a simple continued fraction for the golden ratio: $${\displaystyle \varphi ==1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{{\vphantom {1}} \atop \ddots }}}}}}}}$$

It is in fact the simplest form of a continued fraction, alongside its reciprocal form: $${\displaystyle \varphi ^{-1}==0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{{\vphantom {1}} \atop \ddots }}}}}}}}$$

The convergents of these continued fractions, ⁠${\displaystyle {\tfrac {1}{1}}}$⁠, ⁠${\displaystyle {\tfrac {2}{1}}}$⁠, ⁠${\displaystyle {\tfrac {3}{2}}}$⁠, ⁠${\displaystyle {\tfrac {5}{3}}}$⁠, ⁠${\displaystyle {\tfrac {8}{5}}}$⁠, ⁠${\displaystyle {\tfrac {13}{8}}}$⁠, ... or ⁠${\displaystyle {\tfrac {1}{1}}}$⁠, ⁠${\displaystyle {\tfrac {1}{2}}}$⁠, ⁠${\displaystyle {\tfrac {2}{3}}}$⁠, ⁠${\displaystyle {\tfrac {3}{5}}}$⁠, ⁠${\displaystyle {\tfrac {5}{8}}}$⁠, ⁠${\displaystyle {\tfrac {8}{13}}}$⁠, ..., are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational ⁠${\displaystyle \xi }$⁠, there are infinitely many distinct fractions ⁠${\displaystyle p/q}$⁠ such that, $${\displaystyle \left|\xi -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}.}$$

This means that the constant ⁠${\displaystyle {\sqrt {5}}}$⁠ cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.

A continued square root form for ⁠${\displaystyle \varphi }$⁠ can be obtained from ⁠${\displaystyle \textstyle \varphi ^{2}=1+\varphi }$⁠, yielding: $${\displaystyle \varphi ={\sqrt {1+{\sqrt {\textstyle 1+{\sqrt {1+\cdots {\vphantom {)}}}}}}}}.}$$

Relationship to Fibonacci and Lucas numbers

Further information: Fibonacci number § Relation to the golden ratio See also: Lucas number § Relationship to Fibonacci numbers A Fibonacci spiral (top) which approximates the golden spiral, using Fibonacci sequence square sizes up to 21. A different approximation to the golden spiral is generated (bottom) from stacking squares whose lengths of sides are numbers belonging to the sequence of Lucas numbers, here up to 76.

Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each term ${\displaystyle F_{n}}$ is equal to the sum of the preceding two terms ${\displaystyle F_{n-1}}$ and ${\displaystyle F_{n-2}}$, starting with the base sequence ⁠${\displaystyle 0,1}$⁠ as the 0th and 1st terms ${\displaystyle F_{0}}$ and ${\displaystyle F_{1}}$:

⁠${\displaystyle 0,}$⁠ ⁠${\displaystyle 1,}$⁠ ⁠${\displaystyle 1,}$⁠ ⁠${\displaystyle 2,}$⁠ ⁠${\displaystyle 3,}$⁠ ⁠${\displaystyle 5,}$⁠ ⁠${\displaystyle 8,}$⁠ ⁠${\displaystyle 13,}$⁠ ⁠${\displaystyle 21,}$⁠ ⁠${\displaystyle 34,}$⁠ ⁠${\displaystyle 55,}$⁠ ⁠${\displaystyle 89,}$⁠ ⁠${\displaystyle \ldots }$⁠ (OEIS: A000045).

The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in that each term ${\displaystyle L_{n}}$ is the sum of the previous two terms ${\displaystyle L_{n-1}}$ and ${\displaystyle L_{n-2}}$, however instead starts with ⁠${\displaystyle 2,1}$⁠ as the 0th and 1st terms ${\displaystyle L_{0}}$ and ${\displaystyle L_{1}}$:

⁠${\displaystyle 2,}$⁠ ⁠${\displaystyle 1,}$⁠ ⁠${\displaystyle 3,}$⁠ ⁠${\displaystyle 4,}$⁠ ⁠${\displaystyle 7,}$⁠ ⁠${\displaystyle 11,}$⁠ ⁠${\displaystyle 18,}$⁠ ⁠${\displaystyle 29,}$⁠ ⁠${\displaystyle 47,}$⁠ ⁠${\displaystyle 76,}$⁠ ⁠${\displaystyle 123,}$⁠ ⁠${\displaystyle 199,}$⁠ ⁠${\displaystyle \ldots }$⁠ (OEIS: A000032).

Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: $${\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\lim _{n\to \infty }{\frac {L_{n+1}}{L_{n}}}=\varphi .}$$

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates ⁠${\displaystyle \varphi }$⁠. For example,

${\displaystyle {\frac {F_{16}}{F_{15}}}={\frac {987}{610}}=1.6180327\ldots \ }$ and ${\displaystyle \ {\frac {L_{16}}{L_{15}}}={\frac {2207}{1364}}=1.6180351\ldots .}$

These approximations are alternately lower and higher than ⁠${\displaystyle \varphi }$⁠, and converge to ⁠${\displaystyle \varphi }$⁠ as the Fibonacci and Lucas numbers increase.

Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:

$${\displaystyle F\left(n\right)={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}={\frac {1}{\sqrt {5}}}\left,}$$ $${\displaystyle L\left(n\right)=\varphi ^{n}+(-\varphi )^{-n}=\varphi ^{n}+(1-\varphi )^{n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n}.}$$

Combining both formulas above, one obtains a formula for ⁠${\displaystyle \textstyle \varphi ^{n}}$⁠ that involves both Fibonacci and Lucas numbers: $${\displaystyle \varphi ^{n}={\tfrac {1}{2}}{\bigl (}L_{n}+F_{n}{\sqrt {5}}~\!{\bigr )}.}$$

Between Fibonacci and Lucas numbers one can deduce ⁠${\displaystyle \textstyle L_{2n}=5F_{n}^{2}+2(-1)^{n}=L_{n}^{2}-2(-1)^{n}}$⁠, which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five: $${\displaystyle \lim _{n\to \infty }{\frac {L_{n}}{F_{n}}}={\sqrt {5}}.}$$

Indeed, much stronger statements are true: $${\displaystyle {\begin{aligned}&{\bigl \vert }L_{n}-{\sqrt {5}}F_{n}{\bigr \vert }={\frac {2}{\varphi ^{n}}}\to 0,\\&{\bigl (}{\tfrac {1}{2}}L_{3n}{\bigr )}^{2}=5{\bigl (}{\tfrac {1}{2}}F_{3n}{\bigr )}^{2}+(-1)^{n}.\end{aligned}}}$$

These values describe ⁠${\displaystyle \varphi }$⁠ as a fundamental unit of the algebraic number field ⁠${\displaystyle \mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}}$⁠.

Successive powers of the golden ratio obey the Fibonacci recurrence, ⁠${\displaystyle \textstyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}}$⁠.

The reduction to a linear expression can be accomplished in one step by using: $${\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}$$

This identity allows any polynomial in ⁠${\displaystyle \varphi }$⁠ to be reduced to a linear expression, as in:

$${\displaystyle {\begin{aligned}3\varphi ^{3}-5\varphi ^{2}+4&=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4\\&=3{\bigl (}(\varphi +1)+\varphi {\bigr )}-5(\varphi +1)+4\\&=\varphi +2\approx 3.618033.\end{aligned}}}$$

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation: $${\displaystyle \sum _{n=1}^{\infty }{\bigl |}F_{n}\varphi -F_{n+1}{\bigr |}=\varphi .}$$

In particular, the powers of ⁠${\displaystyle \varphi }$⁠ themselves round to Lucas numbers (in order, except for the first two powers, ⁠${\displaystyle \textstyle \varphi ^{0}}$⁠ and ⁠${\displaystyle \varphi }$⁠, are in reverse order):

$${\displaystyle {\begin{aligned}\varphi ^{0}&=1,\\\varphi ^{1}&=1.618033989\ldots \approx 2,\\\varphi ^{2}&=2.618033989\ldots \approx 3,\\\varphi ^{3}&=4.236067978\ldots \approx 4,\\\varphi ^{4}&=6.854101967\ldots \approx 7,\end{aligned}}}$$

and so forth. The Lucas numbers also directly generate powers of the golden ratio; for ⁠${\displaystyle n\geq 2}$⁠: $${\displaystyle \varphi ^{n}=L_{n}-(-\varphi )^{-n}.}$$

Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of third consecutive Fibonacci numbers equals a Lucas number, that is ⁠${\displaystyle \textstyle L_{n}=F_{n-1}+F_{n+1}\!}$⁠; and, importantly, that ⁠${\displaystyle \textstyle L_{n}F_{n}=F_{2n}\!}$⁠.

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

Geometry

The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a regular icosahedron. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.

Construction

Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.

Dividing by interior division

1. Having a line segment ⁠${\displaystyle AB}$⁠, construct a perpendicular ⁠${\displaystyle BC}$⁠ at point ⁠${\displaystyle B}$⁠, with ⁠${\displaystyle BC}$⁠ half the length of ⁠${\displaystyle AB}$⁠. Draw the hypotenuse ⁠${\displaystyle AC}$⁠.
2. Draw an arc with center ⁠${\displaystyle C}$⁠ and radius ⁠${\displaystyle BC}$⁠. This arc intersects the hypotenuse ⁠${\displaystyle AC}$⁠ at point ⁠${\displaystyle D}$⁠.
3. Draw an arc with center ⁠${\displaystyle A}$⁠ and radius ⁠${\displaystyle AD}$⁠. This arc intersects the original line segment ⁠${\displaystyle AB}$⁠ at point ⁠${\displaystyle S}$⁠. Point ⁠${\displaystyle S}$⁠ divides the original line segment ⁠${\displaystyle AB}$⁠ into line segments ⁠${\displaystyle AS}$⁠ and ⁠${\displaystyle SB}$⁠ with lengths in the golden ratio.

Dividing by exterior division

1. Draw a line segment ⁠${\displaystyle AS}$⁠ and construct off the point ⁠${\displaystyle S}$⁠ a segment ⁠${\displaystyle SC}$⁠ perpendicular to ⁠${\displaystyle AS}$⁠ and with the same length as ⁠${\displaystyle AS}$⁠.
2. Do bisect the line segment ⁠${\displaystyle AS}$⁠ with ⁠${\displaystyle M}$⁠.
3. A circular arc around ⁠${\displaystyle M}$⁠ with radius ⁠${\displaystyle MC}$⁠ intersects in point ⁠${\displaystyle B}$⁠ the straight line through points ⁠${\displaystyle A}$⁠ and ⁠${\displaystyle S}$⁠ (also known as the extension of ⁠${\displaystyle AS}$⁠). The ratio of ⁠${\displaystyle AS}$⁠ to the constructed segment ⁠${\displaystyle SB}$⁠ is the golden ratio.

Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.

Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.

Golden angle

Main article: Golden angle

When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure ⁠${\displaystyle g}$⁠:

$${\displaystyle {\begin{aligned}{\frac {2\pi -g}{g}}&={\frac {2\pi }{2\pi -g}}=\varphi ,\\2\pi -g&={\frac {2\pi }{\varphi }}\approx 222.5^{\circ }\!,\\g&={\frac {2\pi }{\varphi ^{2}}}\approx 137.5^{\circ }\!.\end{aligned}}}$$

This angle occurs in patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.

Pentagonal symmetry system

Pentagon and pentagram

In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are ⁠${\displaystyle a}$⁠, and short edges are ⁠${\displaystyle b}$⁠, then Ptolemy's theorem gives ⁠${\displaystyle \textstyle a^{2}=b^{2}+ab}$⁠. Dividing both sides by ⁠${\displaystyle ab}$⁠ yields (see § Calculation above), $${\displaystyle {\frac {a}{b}}={\frac {a+b}{a}}=\varphi .}$$

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

Pentagonal and pentagrammic geometry permits us to calculate the following values for ⁠${\displaystyle \varphi }$⁠: $${\displaystyle {\begin{aligned}\varphi &=1+2\sin(\pi /10)=1+2\sin 18^{\circ }\!,\\\varphi &={\tfrac {1}{2}}\csc(\pi /10)={\tfrac {1}{2}}\csc 18^{\circ }\!,\\\varphi &=2\cos(\pi /5)=2\cos 36^{\circ }\!,\\\varphi &=2\sin(3\pi /10)=2\sin 54^{\circ }\!.\end{aligned}}}$$

Golden triangle and golden gnomon

Main article: Golden triangle (mathematics)

The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with apex angle ⁠${\displaystyle 36^{\circ }}$⁠ and base angles ⁠${\displaystyle 72^{\circ }\!}$⁠. Its two equal sides are in the golden ratio to its base. The triangle formed by two sides and a diagonal of a regular pentagon is called a golden gnomon. It is an obtuse isosceles triangle with apex angle ⁠${\displaystyle 108^{\circ }}$⁠ and base angle ⁠${\displaystyle 36^{\circ }\!}$⁠. Its base is in the golden ratio to its two equal sides. The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a regular pentagram are golden triangles, as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.

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

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

Penrose tilings

Main article: Penrose tiling

The golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together. Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:

• Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.
• The kite and dart Penrose tiling uses kites with three interior angles of ⁠${\displaystyle 72^{\circ }}$⁠ and one interior angle of ⁠${\displaystyle 144^{\circ }\!}$⁠, and darts, concave quadrilaterals with two interior angles of ⁠${\displaystyle 36^{\circ }\!}$⁠, one of ⁠${\displaystyle 72^{\circ }\!}$⁠, and one non-convex angle of ⁠${\displaystyle 216^{\circ }\!}$⁠. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.
• The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context Robinson triangles, can be used as the prototiles for a form of the Penrose tiling.
• The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of ⁠${\displaystyle 36^{\circ }}$⁠ and ⁠${\displaystyle 144^{\circ }\!}$⁠, and a thick rhombus with angles of ⁠${\displaystyle 72^{\circ }}$⁠ and ⁠${\displaystyle 108^{\circ }\!}$⁠. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals ⁠${\displaystyle 1\mathbin {:} \varphi }$⁠, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.

Original four-tile Penrose tilingRhombic Penrose tiling

In triangles and quadrilaterals

Odom's construction

George Odom found a construction for ⁠${\displaystyle \varphi }$⁠ involving an equilateral triangle: if the line segment joining the midpoints of two sides is extended to intersect the circumcircle, then the two midpoints and the point of intersection with the circle are in golden proportion.

Kepler triangle

Main article: Kepler triangle Geometric progression of areas of squares on the sides of a Kepler triangleAn isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length

The Kepler triangle, named after Johannes Kepler, is the unique right triangle with sides in geometric progression: $${\displaystyle 1\mathbin {:} {\sqrt {\varphi {\vphantom {+}}}}\mathbin {:} \varphi .}$$ These side lengths are the three Pythagorean means of the two numbers ⁠${\displaystyle \varphi \pm 1}$⁠. The three squares on its sides have areas in the golden geometric progression ⁠${\displaystyle \textstyle 1\mathbin {:} \varphi \mathbin {:} \varphi ^{2}}$⁠.

Among isosceles triangles, the ratio of inradius to side length is maximized for the triangle formed by two reflected copies of the Kepler triangle, sharing the longer of their two legs. The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter.

For a Kepler triangle with smallest side length ⁠${\displaystyle s}$⁠, the area and acute internal angles are: $${\displaystyle {\begin{aligned}A&={\tfrac {1}{2}}s^{2}{\sqrt {\varphi {\vphantom {+}}}},\\\theta &=\sin ^{-1}{\frac {1}{\varphi }}\approx 38.1727^{\circ }\!,\\\theta &=\cos ^{-1}{\frac {1}{\varphi }}\approx 51.8273^{\circ }\!.\end{aligned}}}$$

Golden rectangle

Main article: Golden rectangle

The golden ratio proportions the adjacent side lengths of a golden rectangle in ⁠${\displaystyle 1\mathbin {:} \varphi }$⁠ ratio. Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in ⁠${\displaystyle \varphi }$⁠ ratio. They can be generated by golden spirals, through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail).

Golden rhombus

Main article: Golden rhombus

A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly ⁠${\displaystyle 1\mathbin {:} \varphi }$⁠. For a rhombus of such proportions, its acute angle and obtuse angles are:

$${\displaystyle {\begin{aligned}\alpha &=2\arctan {1 \over \varphi }\approx 63.43495^{\circ }\!,\\\beta &=2\arctan \varphi =\pi -\arctan 2=\arctan 1+\arctan 3\approx 116.56505^{\circ }\!.\end{aligned}}}$$

The lengths of its short and long diagonals ⁠${\displaystyle d}$⁠ and ⁠${\displaystyle D}$⁠, in terms of side length ⁠${\displaystyle a}$⁠ are:

$${\displaystyle {\begin{aligned}d&={\frac {2a}{\sqrt {2+\varphi }}}=2{\sqrt {\frac {3-\varphi }{5}}}a\approx 1.05146a,\\D&=2{\sqrt {\frac {2+\varphi }{5}}}a\approx 1.70130a.\end{aligned}}}$$

Its area, in terms of ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle d}$⁠:

$${\displaystyle {\begin{aligned}A&=\sin(\arctan 2)\cdot a^{2}={2 \over {\sqrt {5}}}~a^{2}\approx 0.89443a^{2},\\A&={{\varphi } \over 2}d^{2}\approx 0.80902d^{2}.\end{aligned}}}$$

Its inradius, in terms of side ⁠${\displaystyle a}$⁠:

$${\displaystyle r={\frac {a}{\sqrt {5}}}.}$$

Golden rhombi form the faces of the rhombic triacontahedron, the two golden rhombohedra, the Bilinski dodecahedron, and the rhombic hexecontahedron.

Golden spiral

Main article: Golden spiral

Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral. These spirals can be approximated by quarter-circles that grow by the golden ratio, or their approximations generated from Fibonacci numbers, often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the polar equation with ⁠${\displaystyle (r,\theta )}$⁠: $${\displaystyle r=\varphi ^{2\theta /\pi }.}$$

Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each ⁠${\displaystyle 108^{\circ }}$⁠ that it turns, instead of the ⁠${\displaystyle 90^{\circ }}$⁠ turning angle of the golden spiral. Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.

Dodecahedron and icosahedron

The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the golden ratio. A dodecahedron has ⁠${\displaystyle 12}$⁠ regular pentagonal faces, whereas an icosahedron has ⁠${\displaystyle 20}$⁠ equilateral triangles; both have ⁠${\displaystyle 30}$⁠ edges.

For a dodecahedron of side ⁠${\displaystyle a}$⁠, the radius of a circumscribed and inscribed sphere, and midradius are (⁠${\displaystyle r_{u}}$⁠, ⁠${\displaystyle r_{i}}$⁠, and ⁠${\displaystyle r_{m}}$⁠, respectively):

${\displaystyle r_{u}=a\,{\frac {{\sqrt {3}}\varphi }{2}},}$ ${\displaystyle r_{i}=a\,{\frac {\varphi ^{2}}{2{\sqrt {3-\varphi }}}},}$ and ${\displaystyle r_{m}=a\,{\frac {\varphi ^{2}}{2}}.}$

While for an icosahedron of side ⁠${\displaystyle a}$⁠, the radius of a circumscribed and inscribed sphere, and midradius are:

${\displaystyle r_{u}=a{\frac {\sqrt {\varphi {\sqrt {5}}}}{2}},}$ ${\displaystyle r_{i}=a{\frac {\varphi ^{2}}{2{\sqrt {3}}}},}$ and ${\displaystyle r_{m}=a{\frac {\varphi }{2}}.}$

The volume and surface area of the dodecahedron can be expressed in terms of ⁠${\displaystyle \varphi }$⁠:

${\displaystyle A_{d}={\frac {15\varphi }{\sqrt {3-\varphi }}}}$ and ${\displaystyle V_{d}={\frac {5\varphi ^{3}}{6-2\varphi }}.}$

As well as for the icosahedron:

${\displaystyle A_{i}=20{\frac {\varphi ^{2}}{2}}}$ and ${\displaystyle V_{i}={\frac {5}{6}}(1+\varphi ).}$

These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving ⁠${\displaystyle \varphi }$⁠. The coordinates of the dodecahedron are displayed on the figure to the right, while those of the icosahedron are:

$${\displaystyle (0,\pm 1,\pm \varphi ),\ (\pm 1,\pm \varphi ,0),\ (\pm \varphi ,0,\pm 1).}$$

Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings. In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. The three golden rectangles together contain all ⁠${\displaystyle 12}$⁠ vertices of the icosahedron, or equivalently, intersect the centers of all ⁠${\displaystyle 12}$⁠ of the dodecahedron's faces.

A cube can be inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is ⁠${\displaystyle 2/(2+\varphi )}$⁠ times that of the dodecahedron's. In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in ⁠${\displaystyle \textstyle \varphi \mathbin {:} \varphi ^{2}}$⁠ ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's ⁠${\displaystyle 12}$⁠ vertices touch the ⁠${\displaystyle 12}$⁠ edges of an octahedron at points that divide its edges in golden ratio.

Other properties

The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation ⁠${\displaystyle \textstyle x^{2}-x-1=0}$⁠ or on ⁠${\displaystyle \textstyle x^{2}-5=0}$⁠ (to compute ⁠${\displaystyle {\sqrt {5}}}$⁠ first). The time needed to compute ⁠${\displaystyle n}$⁠ digits of the golden ratio using Newton's method is essentially ⁠${\displaystyle O(M(n))}$⁠, where ⁠${\displaystyle M(n)}$⁠ is the time complexity of multiplying two ⁠${\displaystyle n}$⁠-digit numbers. This is considerably faster than known algorithms for π and 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 ⁠${\displaystyle F_{25001}}$⁠ and ⁠${\displaystyle F_{25000}}$⁠, each over ⁠${\displaystyle 5000}$⁠ digits, yields over ⁠${\displaystyle 10{,}000}$⁠ significant digits of the golden ratio. The decimal expansion of the golden ratio ⁠${\displaystyle \varphi }$⁠ has been calculated to an accuracy of ten trillion (⁠${\displaystyle \textstyle 1\times 10^{13}=10{,}000{,}000{,}000{,}000}$⁠) digits.

In the complex plane, the fifth roots of unity ⁠${\displaystyle \textstyle z=e^{2\pi ki/5}}$⁠ (for an integer ⁠${\displaystyle k}$⁠) satisfying ⁠${\displaystyle \textstyle z^{5}=1}$⁠ are the vertices of a pentagon. They do not form a ring of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, ⁠${\displaystyle z+{\bar {z}}}$⁠, is a quadratic integer, an element of ⁠${\displaystyle \mathbb {Z} }$⁠. Specifically,

$${\displaystyle {\begin{aligned}e^{0}+e^{-0}&=2,\\e^{2\pi i/5}+e^{-2\pi i/5}&=\varphi ^{-1}=-1+\varphi ,\\e^{4\pi i/5}+e^{-4\pi i/5}&=-\varphi .\end{aligned}}}$$

This also holds for the remaining tenth roots of unity satisfying ⁠${\displaystyle \textstyle z^{10}=1}$⁠,

$${\displaystyle {\begin{aligned}e^{\pi i}+e^{-\pi i}&=-2,\\e^{\pi i/5}+e^{-\pi i/5}&=\varphi ,\\e^{3\pi i/5}+e^{-3\pi i/5}&=-\varphi ^{-1}=1-\varphi .\end{aligned}}}$$

For the gamma function ⁠${\displaystyle \Gamma }$⁠, the only solutions to the equation ⁠${\displaystyle \Gamma (z-1)=\Gamma (z+1)}$⁠ are ⁠${\displaystyle z=\varphi }$⁠ and ⁠${\displaystyle \textstyle z=-\varphi ^{-1}}$⁠.

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or ⁠${\displaystyle \varphi }$⁠-nary), quadratic integers in the ring ⁠${\displaystyle \mathbb {Z} }$⁠ – that is, numbers of the form ⁠${\displaystyle a+b\varphi }$⁠ for ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ in ⁠${\displaystyle \mathbb {Z} }$⁠ – have terminating representations, but rational fractions have non-terminating representations.

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 ⁠${\displaystyle 4\log(\varphi )}$⁠.

The golden ratio appears in the theory of modular functions as well. For ${\displaystyle |q|<1,}$ let $${\displaystyle R(q)={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+{{\vphantom {1}} \atop \ddots }}}}}}}}}.}$$ Then $${\displaystyle R(e^{-2\pi })={\sqrt {\varphi {\sqrt {5}}}}-\varphi ,\quad R(-e^{-\pi })=\varphi ^{-1}-{\sqrt {2-\varphi ^{-1}}}}$$ and $${\displaystyle R(e^{-2\pi i/\tau })={\frac {1-\varphi R(e^{2\pi i\tau })}{\varphi +R(e^{2\pi i\tau })}}}$$ where ⁠${\displaystyle \operatorname {Im} \tau >0}$⁠ and ⁠${\displaystyle \textstyle (e^{z})^{1/5}}$⁠ in the continued fraction should be evaluated as ⁠${\displaystyle \textstyle e^{z/5}}$⁠. The function ⁠${\displaystyle \textstyle \tau \mapsto R(e^{2\pi i\tau })}$⁠ is invariant under ⁠${\displaystyle \Gamma (5)}$⁠, a congruence subgroup of the modular group. Also for positive real numbers ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ such that ⁠${\displaystyle \textstyle ab=\pi ^{2},}$⁠

$${\displaystyle {\begin{aligned}{\Bigl (}\varphi +R{{\bigl (}e^{-2a}{\bigr )}}{\Bigr )}{\Bigl (}\varphi +R{{\bigl (}e^{-2b}{\bigr )}}{\Bigr )}&=\varphi {\sqrt {5}},\\{\Bigl (}\varphi ^{-1}-R{{\bigl (}{-e^{-a}}{\bigr )}}{\Bigr )}{\Bigl (}\varphi ^{-1}-R{{\bigl (}{-e^{-b}}{\bigr )}}{\Bigr )}&=\varphi ^{-1}{\sqrt {5}}.\end{aligned}}}$$

⁠${\displaystyle \varphi }$⁠ is a Pisot–Vijayaraghavan number.

Applications and observations

Architecture

Further information: Mathematics and architecture

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

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of 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 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.

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.

Art

Further information: Mathematics and art and History of aesthetics

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. 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.

Salvador Dalí, influenced by the works of Matila Ghyka, 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.

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 ⁠${\displaystyle 1.34}$⁠, with averages for individual artists ranging from ⁠${\displaystyle 1.04}$⁠ (Goya) to ⁠${\displaystyle 1.46}$⁠ (Bellini). 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 ⁠${\displaystyle {\sqrt {5}}}$⁠ proportions, and others with proportions like ⁠${\displaystyle {\sqrt {2}}}$⁠, ⁠${\displaystyle 3}$⁠, ⁠${\displaystyle 4}$⁠, and ⁠${\displaystyle 6}$⁠.

Books and design

Main article: Canons of page construction

According to Jan Tschichold,

There was a time when deviations from the truly beautiful page proportions ⁠${\displaystyle 2\mathbin {:} 3}$⁠, ⁠${\displaystyle 1\mathbin {:} {\sqrt {3}}}$⁠, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.

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.

Flags

The aspect ratio (width to height ratio) of the flag of Togo was intended to be the golden ratio, according to its designer.

Music

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. 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".

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

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 cents, a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.

Nature

Main article: 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".

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. Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.

Physics

The quasi-one-dimensional Ising ferromagnet ${\textstyle {\ce {CoNb2O6}}}$ (cobalt niobate) has ⁠${\displaystyle 8}$⁠ predicted excitation states (with ⁠${\displaystyle E_{8}}$⁠ 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 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.

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. ⁠${\displaystyle 360^{\circ }~\!/\varphi \approx 222.5^{\circ }\!}$⁠. This method was used to arrange the ⁠${\displaystyle 1500}$⁠ mirrors of the student-participatory satellite Starshine-3.

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

Disputed observations

Examples of disputed observations of the golden ratio include the following:

• Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive phalangeal and metacarpal bones (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.
• The shells of mollusks such as the nautilus are often claimed to be in the golden ratio. The growth of nautilus shells follows a logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio, or sometimes claimed that each new chamber is golden-proportioned relative to the previous one. However, measurements of nautilus shells do not support this claim.
• Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is ⁠${\displaystyle 1.45}$⁠.
• Studies by psychologists, starting with Gustav Fechner c. 1876, 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.
• 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. 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.

Egyptian pyramids

The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.

The Parthenon

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. 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." Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."

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. Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

Modern art

The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism. 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. (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat's writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.) The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception". However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not, and Marcel Duchamp said as much in an interview. 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. 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.

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings, though other experts (including critic Yve-Alain Bois) have discredited these claims.

See also

• List of works designed with the golden ratio
• Metallic mean
• Plastic ratio
• Sacred geometry
• Supergolden ratio
• Silver ratio

References

Explanatory footnotes

1. If the constraint on ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. ⁠${\displaystyle \varphi }$⁠ is defined as the positive solution. The negative solution is ⁠${\displaystyle \textstyle -\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}$⁠. The sum of the two solutions is ⁠${\displaystyle 1}$⁠, and the product of the two solutions is ⁠${\displaystyle -1}$⁠.
2. Other names include the golden mean, golden section, golden cut, golden proportion, golden number, medial section, and divine section.
3. Euclid, Elements, Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.
4. "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."
5. After Classical Greek sculptor Phidias (c. 490–430 BC); Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.
6. The theorem that non-square natural numbers have irrational square roots can be found in Euclid's Elements, Book X, Proposition 9.

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Works cited

• Herz-Fischler, Roger (1998) . A Mathematical History of the Golden Number. Dover. ISBN 9780486400075. (Originally titled A Mathematical History of Division in Extreme and Mean Ratio.)
• Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
• Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. ISBN 9-781-61614-424-1.

Further reading

• Doczi, György (1981). The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala.
• Hargittai, István, ed. (1992). Fivefold Symmetry. World Scientific. ISBN 9789810206000.
• Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover. ISBN 978-0-486-22254-7.
• Schaaf, William L., ed. (1967). The Golden Measure (PDF). California School Mathematics Study Group Reprint Series. Stanford University. Archived (PDF) from the original on 2015-04-25.
• Scimone, Aldo (1997). La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. ISBN 978-88-7231-025-0.
• Walser, Hans (2001) . The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. ISBN 978-0-88385-534-8.

External links

• Weisstein, Eric W. "Golden Ratio". MathWorld.
• Bogomolny, Alexander (2018). "Golden Ratio in Geometry". Cut-the-Knot.
• Knott, Ron. "The Golden section ratio: Phi". Information and activities by a mathematics professor.
• The Myth That Will Not Go Away Archived 2020-11-12 at the Wayback Machine, by Keith Devlin, addressing multiple allegations about the use of the golden ratio in culture.
• Spurious golden spirals collected by Randall Munroe
• YouTube lecture on Zeno's mice problem and logarithmic spirals

v t e Algebraic numbers
Algebraic integer Chebyshev nodes Constructible number Conway's constant Cyclotomic field Doubling the cube Eisenstein integer Gaussian integer Golden ratio (φ) Perron number Pisot–Vijayaraghavan number Plastic ratio (ρ) Quadratic irrational number Rational number Root of unity Salem number Silver ratio (σ) Square root of 2 Square root of 3 Square root of 5 Square root of 6 Square root of 7 Supergolden ratio (ψ) Supersilver ratio (ς) Twelfth root of 2
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v t e Irrational numbers
Chaitin's (Ω) Liouville Prime (ρ) Omega Cahen Logarithm of 2 Dottie Lemniscate (ϖ) Twelfth root of 2 Apéry's (ζ(3)) Cube root of 2 Plastic ratio (ρ) Square root of 2 Supergolden ratio (ψ) Erdős–Borwein (E) Golden ratio (φ) Square root of 3 Supersilver ratio (ς) Square root of 5 Silver ratio (σ) Square root of 6 Square root of 7 Euler's (e) Pi (π)
Schizophrenic Transcendental Trigonometric

v t e Metallic means
Pisot number Gold Angle Base Fibonacci sequence Kepler triangle Rectangle Rhombus Section search Spiral Triangle Supergolden ratio Silver Pell number Supersilver ratio Bronze Copper Nickel etc...

v t e Ancient Greek mathematics
Mathematicians (timeline)	Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarchus Aristotle Apollonius Archimedes Autolycus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euclid Eudoxus Eutocius Geminus Heliodorus Heron Hipparchus Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Philonides Plato Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus
Treatises	Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement of a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) On the Moving Sphere (Autolycus) Optics (Euclid) On Spirals On the Sphere and Cylinder Ostomachion Planisphaerium Spherics (Theodosius) Spherics (Menelaus) The Quadrature of the Parabola The Sand Reckoner
Problems	Constructible numbers Angle trisection Doubling the cube Squaring the circle Problem of Apollonius
Concepts and definitions	Angle Central Inscribed Axiomatic system Axiom Chord Circles of Apollonius Apollonian circles Apollonian gasket Circumscribed circle Commensurability Diophantine equation Doctrine of proportionality Euclidean geometry Golden ratio Greek numerals Incircle and excircles of a triangle Method of exhaustion Parallel postulate Platonic solid Lune of Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center
Results	In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon Apollonius Apollonius's theorem Other Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Law of sines Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus
Centers	Cyrene Mouseion of Alexandria Platonic Academy
Related	Ancient Greek astronomy Attic numerals Greek numerals Latin translations of the 12th century Non-Euclidean geometry Philosophy of mathematics Neusis construction History of A History of Greek Mathematics by Thomas Heath algebra timeline arithmetic timeline calculus timeline geometry timeline logic timeline mathematics timeline numbers prehistoric counting numeral systems list Other cultures Arabian/Islamic Babylonian Chinese Egyptian Incan Indian Japanese
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v t e Mathematics and art
Concepts	Algorithm Catenary Fractal Golden ratio Hyperboloid structure Minimal surface Paraboloid Perspective Camera lucida Camera obscura Plastic ratio Projective geometry Proportion Architecture Human Symmetry Tessellation Wallpaper group
Forms	Algorithmic art Anamorphic art Architecture Geodesic dome Pyramid Vastu shastra Computer art Fiber arts 4D art Fractal art Islamic geometric patterns Girih Jali Muqarnas Zellij Knotting Celtic knot Croatian interlace Interlace Music Origami Mathematics Sculpture String art String figure Tiling
Artworks	List of works designed with the golden ratio Continuum Mathemalchemy Mathematica: A World of Numbers... and Beyond Octacube Pi Pi in the Sky
Buildings	Cathedral of Saint Mary of the Assumption Hagia Sophia Kresge Auditorium Pantheon Parthenon Pyramid of Khufu Sagrada Família Sydney Opera House Taj Mahal
Artists	Renaissance Paolo Uccello Piero della Francesca Leonardo da Vinci Vitruvian Man Albrecht Dürer Parmigianino Self-portrait in a Convex Mirror 19th–20th Century William Blake The Ancient of Days Newton Jean Metzinger Danseuse au café L'Oiseau bleu Giorgio de Chirico Man Ray M. C. Escher Circle Limit III Print Gallery Relativity Reptiles Waterfall René Magritte La condition humaine Salvador Dalí Crucifixion The Swallow's Tail Crockett Johnson Contemporary Max Bill Martin and Erik Demaine Scott Draves Jan Dibbets John Ernest Helaman Ferguson Peter Forakis Susan Goldstine Bathsheba Grossman George W. Hart Desmond Paul Henry Anthony Hill Charles Jencks Garden of Cosmic Speculation Andy Lomas Robert Longhurst Jeanette McLeod Hamid Naderi Yeganeh István Orosz Hinke Osinga Antoine Pevsner Tony Robbin Alba Rojo Cama Reza Sarhangi Oliver Sin Hiroshi Sugimoto Daina Taimiņa Roman Verostko Margaret Wertheim
Theorists	Ancient Polykleitos Canon Vitruvius De architectura Renaissance Filippo Brunelleschi Leon Battista Alberti De pictura De re aedificatoria Piero della Francesca De prospectiva pingendi Luca Pacioli De divina proportione Leonardo da Vinci A Treatise on Painting Albrecht Dürer Vier Bücher von Menschlicher Proportion Sebastiano Serlio Regole generali d'architettura Andrea Palladio I quattro libri dell'architettura Romantic Samuel Colman Nature's Harmonic Unity Frederik Macody Lund Ad Quadratum Jay Hambidge The Greek Vase Modern Owen Jones The Grammar of Ornament Ernest Hanbury Hankin The Drawing of Geometric Patterns in Saracenic Art G. H. Hardy A Mathematician's Apology George David Birkhoff Aesthetic Measure Douglas Hofstadter Gödel, Escher, Bach Nikos Salingaros The 'Life' of a Carpet
Publications	Journal of Mathematics and the Arts Lumen Naturae Making Mathematics with Needlework Rhythm of Structure Viewpoints: Mathematical Perspective and Fractal Geometry in Art
Organizations	Ars Mathematica The Bridges Organization European Society for Mathematics and the Arts Goudreau Museum of Mathematics in Art and Science Institute For Figuring Mathemalchemy National Museum of Mathematics
Related	Droste effect Mathematical beauty Patterns in nature Sacred geometry
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