Golden ratio

Template:Short description {{#invoke:other uses|otheruses}} Template:Pp Template:Infobox non-integer number}}</math> | algebraic = <math>\frac{1 + \sqrt5}{2}</math> }}

File:SimilarGoldenRectangles.svg

A golden rectangle with long side Template:Math and short side Template:Mvar can be divided into two pieces: a similar golden rectangle (shaded red, right) with long side Template:Mvar and short side Template:Mvar and a square (shaded blue, left) with sides of length Template:Mvar. This illustrates the relationship Template:Math

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 Template:Tmath and Template:Tmath with Template:Tmath, Template:Tmath is in a golden ratio to Template:Tmath if

<math display=block> \frac{a+b}{a} = \frac{a}{b} = \varphi,</math>

where the Greek letter phi (Template:Tmath or Template:Tmath) denotes the golden ratio.Template:Efn The constant Template:Tmath satisfies the quadratic equation Template:Tmath and is an irrational number with a value of<ref name=a001622 />

The golden ratio was called the extreme and mean ratio by Euclid,<ref name="Elements 6.3" /> and the divine proportion by Luca Pacioli;<ref name=Pacioli /> it also goes by other names.Template:Efn

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.Template:Sfn A golden rectangle—that is, a rectangle with an aspect ratio of Template:Tmath—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.<ref name="strogatz nytimes" /> 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 Template:Tmath and Template:Tmath are in the golden ratio Template:Tmath if<ref name=schielack /> <math display=block> \frac{a+b}{a} = \frac{a}{b} = \varphi. </math>

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

The quadratic formula yields two solutions:

Because Template:Tmath is a ratio between positive quantities, Template:Tmath is necessarily the positive root.<ref name=peters /> The negative root is in fact the negative inverse Template:Tmath, which shares many properties with the golden ratio.

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.Template:Sfn{{#if:The Golden Ratio: The Story of Phi, the World's Most Astonishing Number|{{#if:|}}

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Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry;Template:Sfn the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.Template:Sfn 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.Template:Sfn Euclid's Elements (Template:Nowrap) provides several propositions and their proofs employing the golden ratio,Template:SfnTemplate:Efn and contains its first known definition which proceeds as follows:<ref name=hemenway />

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.Template:SfnTemplate:Efn{{#if:|{{#if:|}}

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File:Michael Maestlin.jpg

Michael Maestlin, the first to write a decimal approximation of the ratio

The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers.Template:Sfn

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.<ref name=mackinnon />Template:Sfn Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section').<ref name=baravalle /> Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.Template:Sfn 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.Template:Sfn

German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;<ref name=schreiber /> this was rediscovered by Johannes Kepler in 1608.Template:Sfn The first known decimal approximation of the (inverse) golden ratio was stated as "about Template:Tmath" in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.<ref name=mactutor /> The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:

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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".<ref name="beutelspacher petri" /> Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.Template:Sfn James Sully used the equivalent English term in 1875.Template:Sfn

By 1910, inventor Mark Barr began using the Greek letter phi (Template:Tmath) as a symbol for the golden ratio.Template:SfnTemplate:Efn It has also been represented by tau (Template:Tmath), the first letter of the ancient Greek τομή ('cut' or 'section').Template:Sfn

File:Dan Shechtman in 1985.jpg

Dan Shechtman demonstrates quasicrystals at the NIST in 1985 using a Zometoy model.

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.<ref name=gardner /> 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.<ref name=quasicrystals />

Mathematics

Irrationality

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

Contradiction from an expression in lowest terms

File:Whirling squares.svg

If Template:Mvar were rational, then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the positive integers have a lower bound, so Template:Mvar cannot be rational.

This is a proof by infinite descent. Recall that: Template:Bi

If we call the whole Template:Tmath and the longer part Template:Tmath, then the second statement above becomes

To say that the golden ratio Template:Tmath is rational means that Template:Tmath is a fraction Template:Tmath where Template:Tmath and Template:Tmath are integers. We may take Template:Tmath to be in lowest terms and Template:Tmath and Template:Tmath to be positive. But if Template:Tmath is in lowest terms, then the equally valued Template:Tmath is in still lower terms. That is a contradiction that follows from the assumption that Template:Tmath 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 Template:Tmath is assumed to be rational, then Template:Tmath, the square root of Template:Tmath, must also be rational. This is a contradiction, as the square roots of all non-square natural numbers are irrational.Template:Efn

Minimal polynomial

File:Golden ratio parabolas.png

The golden ratio Template:Mvar and its negative reciprocal Template:Math are the two roots of the quadratic polynomial Template:Math. The golden ratio's negative Template:Math and reciprocal Template:Math are the two roots of the quadratic polynomial Template:Math.

The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial <math display=block>x^2 - x - 1.</math>

This quadratic polynomial has two roots, Template:Tmath and Template:Tmath.

The golden ratio is also closely related to the polynomial Template:Tmath, which has roots Template:Tmath and Template:Tmath. As the root of a quadratic polynomial, the golden ratio is a constructible number.<ref name=constructions />

Golden ratio conjugate and powers

The conjugate root to the minimal polynomial Template:Tmath is

<math display=block>-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt5}{2} = -0.618033\dots.</math>

The absolute value of this quantity (Template:Tmath) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, Template:Tmath).

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 Template:Tmath:

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

The sequence of powers of Template:Tmath contains these values Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath; more generally, any power of Template:Tmath 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 Template:Tmath into a multiple of Template:Tmath and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of Template:Tmath:

If Template:Tmath, then: <math display=block>\begin{align} \varphi^n &= \varphi^{n-1} + \varphi^{n-3} + \cdots + \varphi^{n-1-2m} + \varphi^{n-2-2m} \\[5mu] \varphi^n - \varphi^{n-1} &= \varphi^{n-2}. \end{align}</math>

Continued fraction and square root

File:Golden mean.png

Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers

The formula Template:Tmath can be expanded recursively to obtain a simple continued fraction for the golden ratio:<ref name="Concrete Abstractions" /> <math display=block> \varphi = [1; 1, 1, 1, \dots] = 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; 1, 1, 1, \dots] = 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}} </math>

The convergents of these continued fractions, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Nowrap or Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Nowrap 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 Template:Tmath, there are infinitely many distinct fractions Template:Tmath such that, <math display=block>\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}.</math>

This means that the constant Template:Tmath cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.<ref name=hardy />

A continued square root form for Template:Tmath can be obtained from Template:Tmath, yielding:<ref name=sizer/> <math display=block> \varphi = \sqrt{1 + \sqrt{\textstyle 1 + \sqrt{ 1 + \cdots \vphantom)}}}. </math>

Relationship to Fibonacci and Lucas numbers

Fibonacci numbers and Lucas numbers 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 Template:Tmath as the 0th and 1st terms <math>F_0</math> and <math>F_1</math>:

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 <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 Template:Tmath as the 0th and 1st terms <math>L_0</math> and <math>L_1</math>:

Exceptionally, the golden ratio is equal to the limit 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 Template:Tmath. For example,

Template:Bi{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 Template:Tmath, and converge to Template:Tmath as the Fibonacci and Lucas numbers increase.

Closed-form expressions 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[\left({ 1+ \sqrt{5} \over 2}\right)^n - \left({ 1- \sqrt{5} \over 2}\right)^n\right],
</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 Template:Tmath 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 Template:Tmath, which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five: <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, \\[5mu] & \bigl(\tfrac12 L_{3n}\bigr)^2 = 5 \bigl(\tfrac12 F_{3n}\bigr)^2 + (-1)^n. \end{align}</math>

These values describe Template:Tmath as a fundamental unit of the algebraic number field Template:Tmath.

Successive powers of the golden ratio obey the Fibonacci recurrence, Template:Tmath.

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 Template:Tmath 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 \\[5mu] &= 3\bigl((\varphi + 1) + \varphi\bigr) - 5(\varphi + 1) + 4 \\[5mu] &= \varphi + 2 \approx 3.618033. \end{align}</math>

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation: <math display=block>\sum_{n=1}^{\infty}\bigl|F_n\varphi-F_{n+1}\bigr| = \varphi.</math>

In particular, the powers of Template:Tmath themselves round to Lucas numbers (in order, except for the first two powers, Template:Tmath and Template:Tmath, are in reverse order):

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

and so forth.<ref name=parker4d /> The Lucas numbers also directly generate powers of the golden ratio; for Template:Tmath: <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 Template:Tmath; and, importantly, that Template:Tmath.

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.<ref name=BurgerStarbird /> It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.

Construction

Dividing by interior division

1. Having a line segment Template:Tmath, construct a perpendicular Template:Tmath at point Template:Tmath, with Template:Tmath half the length of Template:Tmath. Draw the hypotenuse Template:Tmath.
2. Draw an arc with center Template:Tmath and radius Template:Tmath. This arc intersects the hypotenuse Template:Tmath at point Template:Tmath.
3. Draw an arc with center Template:Tmath and radius Template:Tmath. This arc intersects the original line segment Template:Tmath at point Template:Tmath. Point Template:Tmath divides the original line segment Template:Tmath into line segments Template:Tmath and Template:Tmath with lengths in the golden ratio.

Dividing by exterior division

1. Draw a line segment Template:Tmath and construct off the point Template:Tmath a segment Template:Tmath perpendicular to Template:Tmath and with the same length as Template:Tmath.
2. Do bisect the line segment Template:Tmath with Template:Tmath.
3. A circular arc around Template:Tmath with radius Template:Tmath intersects in point Template:Tmath the straight line through points Template:Tmath and Template:Tmath (also known as the extension of Template:Tmath). The ratio of Template:Tmath to the constructed segment Template:Tmath 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

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File:Golden Angle.svg

style }}

When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure Template:Tmath:

<math display=block>\begin{align} \frac{2\pi - g}{g} &= \frac{2\pi}{2\pi - g} = \varphi, \\[8mu] 2\pi - g &= \frac{2\pi}{\varphi} \approx 222.5^\circ\!, \\[8mu] g &= \frac{2\pi}{\varphi^2} \approx 137.5^\circ\!. \end{align}</math>

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

Pentagonal symmetry system

Pentagon and pentagram

File:Pentagram-phi.svg

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

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 Template:Tmath, and short edges are Template:Tmath, then Ptolemy's theorem gives Template:Tmath. Dividing both sides by Template:Tmath yields (see Template:Slink above), <math display=block> \frac ab = \frac{a + b}{a} = \varphi. </math>

The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by Template:Tmath. 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 Template:Tmath, as the four-color illustration shows.

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

Golden triangle and golden gnomon

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File:Golden triangle (math).svg

A golden triangle Template:Mvar can be subdivided by an angle bisector into a smaller golden triangle Template:Mvar and a golden gnomon Template:Mvar.

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 Template:Tmath and base angles Template:Tmath.<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 Template:Tmath and base angle Template:Tmath. 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 regular pentagram are golden triangles,<ref name=loeb /> as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.<ref name=miller />

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

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

Penrose tilings

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File:Kite Dart.svg

The kite and dart tiles of the Penrose tiling. The colored arcs divide each edge in the golden ratio; when two tiles share an edge, their arcs must match.

The golden ratio appears prominently in the Penrose tiling, a family of aperiodic 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.<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 kites with three interior angles of Template:Tmath and one interior angle of Template:Tmath, and darts, concave quadrilaterals with two interior angles of Template:Tmath, one of Template:Tmath, and one non-convex angle of Template:Tmath. 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 Template:Tmath and Template:Tmath, and a thick rhombus with angles of Template:Tmath and Template:Tmath. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals Template:Tmath, 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" />

In triangles and quadrilaterals

Odom's construction

File:Odom.svg

Odom's construction: Template:Math

George Odom found a construction for Template:Tmath 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.<ref name=triangleconstruction />

Kepler triangle

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:Multiple image The Kepler triangle, named after Johannes Kepler, is the unique right triangle with sides in geometric progression: <math display=block> 1\mathbin:\sqrt{\varphi\vphantom+}\mathbin:\varphi.</math> These side lengths are the three Pythagorean means of the two numbers Template:Tmath. The three squares on its sides have areas in the golden geometric progression Template:Tmath.

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.<ref name="Liber mensurationum" /> The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter.<ref name=bruce />

For a Kepler triangle with smallest side length Template:Tmath, the area and acute internal angles are: <math display=block>\begin{align} A &= \tfrac12 s^2\sqrt{\varphi\vphantom+}, \\[5mu] \theta &= \sin^{-1}\frac{1}{\varphi}\approx 38.1727^\circ\!, \\[5mu] \theta &= \cos^{-1}\frac{1}{\varphi}\approx 51.8273^\circ\!. \end{align}</math>

Golden rectangle

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File:Golden Rectangle Construction.svg

To construct a golden rectangle with only a straightedge and compass in four simple steps:

Draw a square.

Draw a line from the midpoint of one side of the square to an opposite corner.

Use that line as the radius to draw an arc that defines the height of the rectangle.

Complete the golden rectangle.

The golden ratio proportions the adjacent side lengths of a golden rectangle in Template:Tmath ratio.Template:Sfn Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in Template:Tmath 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).<ref name=BurgerStarbird />

Golden rhombus

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly Template:Tmath.<ref name=hexecontahedron /> For a rhombus of such proportions, its acute angle and obtuse angles are:

<math display=block>\begin{align} \alpha &= 2\arctan{1\over\varphi}\approx63.43495^\circ\!, \\[5mu] \beta &= 2\arctan\varphi=\pi-\arctan2 = \arctan1+\arctan3 \approx 116.56505^\circ\!. \end{align}</math>

The lengths of its short and long diagonals Template:Tmath and Template:Tmath, in terms of side length Template:Tmath are:

<math display=block>\begin{align} d &= \frac{2a}{\sqrt{2+\varphi}}

  = 2\sqrt{\frac{3-\varphi}{5}}a \approx 1.05146a, \\[5mu]

D &= 2\sqrt{\frac{2+\varphi}{5}}a \approx 1.70130a. \end{align}</math>

Its area, in terms of Template:Tmath and Template:Tmath:

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

Its inradius, in terms of side Template:Tmath:

<math display=block> r = \frac{a}{\sqrt{5}}. </math>

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

Golden spiral

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File:FakeRealLogSpiral.svg

The golden spiral (red) and its approximation by quarter-circles (green), with overlaps shown in yellow

File:Golden triangle and Fibonacci spiral.svg

A logarithmic spiral whose radius grows by the golden ratio per Template:Math of turn, surrounding nested golden isosceles triangles. This is a different spiral from the golden spiral, which grows by the golden ratio per Template:Math of turn.<ref name=loeb-varney />

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,<ref name=quarter-circles /> or their approximations generated from Fibonacci numbers,<ref name=diedrichs /> often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the polar equation with Template:Tmath: <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 Template:Tmath that it turns, instead of the Template:Tmath 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 />

Dodecahedron and icosahedron

File:Dodecahedron vertices.svg

Cartesian coordinates of the dodecahedron :
Template:Math
Template:Math
Template:Math
Template:Math
A nested cube inside the dodecahedron is represented with dotted lines.

The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the golden ratio. A dodecahedron has Template:Tmath regular pentagonal faces, whereas an icosahedron has Template:Tmath equilateral triangles; both have Template:Tmath edges.<ref name ="Regular dodecahedron">Template:Harvtxt</ref>

For a dodecahedron of side Template:Tmath, the radius of a circumscribed and inscribed sphere, and midradius are (Template:Tmath, Template:Tmath, and Template:Tmath, respectively):

Template:Bi,</math> and <math>r_m = a\, \frac{\varphi^2}{2}.</math>}}

While for an icosahedron of side Template:Tmath, the radius of a circumscribed and inscribed sphere, and midradius are:

Template:Bi}{2},</math> <math>r_i = a\frac{\varphi^2}{2 \sqrt{3}},</math> and <math>r_m = a\frac{\varphi}{2}.</math>}}

The volume and surface area of the dodecahedron can be expressed in terms of Template:Tmath:

Template:Bi</math> and <math>V_d = \frac{5\varphi^3}{6-2\varphi}.</math>}}

As well as for the icosahedron:

Template:Bi{2}</math> and <math>V_i = \frac{5}{6}(1 + \varphi).</math>}}

File:Icosahedron-golden-rectangles.svg

Three golden rectangles touch all of the Template:Math vertices of a regular icosahedron.

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

<math display=block> (0,\pm1,\pm\varphi),\ (\pm1,\pm\varphi,0),\ (\pm\varphi,0,\pm1). </math>

Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings.<ref name=borromean /><ref name=BurgerStarbird /> In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. The three golden rectangles together contain all Template:Tmath vertices of the icosahedron, or equivalently, intersect the centers of all Template:Tmath of the dodecahedron's faces.<ref name="Regular dodecahedron" />

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 Template:Tmath 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 Template:Tmath ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's Template:Tmath vertices touch the Template:Tmath edges of an octahedron at points that divide its edges in golden ratio.<ref name="59 Icosahedra" />

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 Template:Tmath or on Template:Tmath (to compute Template:Tmath first). The time needed to compute Template:Tmath digits of the golden ratio using Newton's method is essentially Template:Tmath, where Template:Tmath is the time complexity of multiplying two Template:Tmath-digit numbers.<ref name=muller /> This is considerably faster than known algorithms for [[pi|Template:Mvar]] and [[e (mathematical constant)|Template:Mvar]]. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers Template:Tmath and Template:Tmath, each over Template:Tmath digits, yields over Template:Tmath significant digits of the golden ratio. The decimal expansion of the golden ratio Template:Tmath<ref name=a001622 /> has been calculated to an accuracy of twenty trillion (Template:Tmath) digits.<ref name=ycruncher />

In the complex plane, the fifth roots of unity Template:Tmath (for an integer Template:Tmath) satisfying Template:Tmath 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, Template:Tmath, is a quadratic integer, an element of Template:Tmath. Specifically,

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

This also holds for the remaining tenth roots of unity satisfying Template:Tmath,

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

For the gamma function Template:Tmath, the only solutions to the equation Template:Tmath are Template:Tmath and Template:Tmath.

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or Template:Tmath-nary), quadratic integers in the ring Template:Tmath – that is, numbers of the form Template:Tmath for Template:Tmath and Template:Tmath in Template:Tmath – 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 Template:Tmath.<ref name=horocycle />

The golden ratio appears in the theory of modular functions as well. For <math>|q|<1,</math> let <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 Template:Tmath and Template:Tmath in the continued fraction should be evaluated as Template:Tmath. The function Template:Tmath is invariant under Template:Tmath, a congruence subgroup of the modular group. Also for positive real numbers Template:Tmath and Template:Tmath such that Template:Tmath<ref name=rrcf />

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

Template:Tmath is a Pisot–Vijayaraghavan number.<ref name=duffin />

Applications and observations

File:GoldenSquare 6.png

Rhythms apparent to the eye: rectangles in aspect ratios Template:Math (left, middle) and Template:Math (right side) tile the square.

Architecture

Template:Further 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."<ref name=modulor /><ref name=Frings />

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of 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.<ref name=modulor2 />

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

Art

File:Divina proportione - Illustration 13, crop & monochrome.jpg

Da Vinci's illustration of a dodecahedron from Pacioli's Divina proportione (1509)

Leonardo da Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings.<ref name="livio plus"/> Similarly, although Leonardo's Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.<ref name=devlin /><ref name=simanek />

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

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

File:Medieval manuscript framework.svg

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

Books and design

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

According to Jan Tschichold,

There was a time when deviations from the truly beautiful page proportions Template:Tmath, Template:Tmath, 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 />

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

Flags

File:Flag of Togo.svg

The flag of Togo, whose aspect ratio uses the golden ratio

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

Music

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,<ref name=lendvai /> though other music scholars reject that analysis.Template:Sfn 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 Template:Math Template:Math Template:Math and Template:Math and the main climax sits at the phi position".<ref name=Smith />

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

Music theorists including Hans Zender and Heinz Bohlen have experimented with the 833 cents scale, 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.<ref name=833cents />

Nature

File:Aeonium tabuliforme.jpg

Detail of the saucer plant, Aeonium tabuliforme, showing the multiple spiral arrangement (parastichy)

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also

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".Template:Sfn

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

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

Physics

The quasi-one-dimensional Ising ferromagnet <chem display=inline>CoNb2O6</chem> (cobalt niobate) has Template:Tmath predicted excitation states (with [[E8 (mathematics)|Template:Tmath 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.<ref name=ising />

Optimization

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem or Tammes problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. Template:Tmath. This method was used to arrange the Template:Tmath mirrors of the student-participatory satellite Starshine-3.<ref name=disco />

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:

File:NautilusCutawayLogarithmicSpiral.jpg

Nautilus shells are often erroneously claimed to be golden-proportioned.

• Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive 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.<ref name=pheasant /><ref name=vanLaack />
• The shells of mollusks such as the nautilus are often claimed to be in the golden ratio.<ref name=dunlap /> The growth of nautilus shells follows a logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio,<ref name=falbo /> or sometimes claimed that each new chamber is golden-proportioned relative to the previous one.<ref name=moscovich /> However, measurements of nautilus shells do not support this claim.<ref name=shellspirals />
• Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is Template:Tmath.<ref name=gutenberg />
• Studies by psychologists, starting with Gustav Fechner Template:Circa,<ref name=Fechner /> 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.Template:Sfn<ref name="livio plus" />
• In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.<ref name=osler/> The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.<ref name=magicdow />

Egyptian pyramids

File:Egypt. Pyramids. Pyramids and palm grove reflections LOC matpc.23063.jpg

The Great Pyramid of Giza

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.<ref name=greatpyramid />

The Parthenon

File:The Parthenon in Athens.jpg

Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.Template:Sfn

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

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

Modern art

File:Albert Gleizes, 1912, Les Baigneuses, oil on canvas, 105 x 171 cm, Paris, Musée d'Art Moderne de la Ville de Paris.jpg

Albert Gleizes, Les Baigneuses (1912)

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

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,<ref name=bouleau /> though other experts (including critic Yve-Alain Bois) have discredited these claims.<ref name="livio plus" />Template:Sfn

See also

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

References

Works cited

• Template:Cite book (Originally titled A Mathematical History of Division in Extreme and Mean Ratio.)
• Template:Cite book
• Template:Cite book

Further reading

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:GoldenRatio%7CGoldenRatio.html}} |title = Golden Ratio |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• {{#invoke:citation/CS1|citation

|CitationClass=web }}

• {{#invoke:citation/CS1|citation

|CitationClass=web }} Information and activities by a mathematics professor.

• The Myth That Will Not Go Away Template:Webarchive, 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

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