Golden ratio

Un article de Vev.

Modèle:Distinguish2

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887.

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter <math>\varphi</math> (phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

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

This equation has as its unique positive solution the algebraic irrational number

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

Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and the Greek letter phi (<math>\varphi</math>).<ref name="livio">Modèle:Cite book</ref><ref>Piotr Sadowski, The Knight on His Quest: Symbolic Patterns of Transition in Sir Gawain and the Green Knight, Cranbury NJ: Associated University Presses, 1996</ref><ref name="dunlap">Richard A Dunlap, The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997</ref> Other terms encountered include extreme and mean ratio, medial section, divine proportion (Italian: proporzione divina), 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) pp.37 . "And the same applies in architecture, to the rectangles 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 mean of Phidias.<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>Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.</ref>

Calculation

Two quantities (positive numbers) a and b are said to be in the golden ratio <math>\varphi</math> if

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

This equation unambiguously defines <math> \varphi\, </math>.

The right equation shows that <math>a=b\varphi</math>, which can be substituted in the left part, giving

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

Cancelling b yields

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

Multiplying both sides by <math>\varphi</math> and rearranging terms leads to:

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

The only positive solution to this quadratic equation is

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

History

The golden ratio has fascinated intellectuals of diverse interests for at least 2,400 years:

Modèle:Cquote2

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry. The ratio is important in the geometry of regular pentagrams and pentagons. The Greeks usually attributed discovery of the ratio to Pythagoras or his followers. The regular pentagram, which has a regular pentagon inscribed within it, was the Pythagoreans' symbol.

Euclid's Elements (Greek: Modèle:Polytonic) 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 less."<ref name=Elements 6.3/> Euclid explains a construction for cutting (sectioning) a line "in extreme and mean ratio", i.e. the golden ratio.<ref> Euclid, Elements, Book 6, Proposition 30.</ref> Throughout the Elements, several propositions (theorems in modern terminology) and their proofs employ the golden ratio.<ref> Euclid, Elements, Book 2, Proposition 11; Book 4, Propositions 10–11; Book 13, Propositions 1–6, 8–11, 16–18.</ref> Some of these propositions show that the golden ratio is an irrational number.

The name "extreme and mean ratio" was the principal term used from the 3rd century BC<ref name=Elements 6.3>Euclid, Elements, Book 6, Definition 3.</ref> until about the 18th century.

The modern history of the golden ratio starts with Luca Pacioli's Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.

The first known decimal calculation of the golden ratio as a decimal of "about 0.6180340" was written in 1597 by Prof. Michael Maestlin of the University of Tübingen to his former student Johannes Kepler.<ref> The Golden Ratio

. The MacTutor History of Mathematics archive

 

. Retrieved on 2007-09-18. </ref>

Since the twentieth century, the golden ratio has been represented by the Greek letter <math>\varphi</math> (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by <math>\tau</math> (tau, the first letter of the ancient Greek root τομή—meaning cut).

Timeline

Timeline according to Priya Hemenway<ref name=Hemenway,P>Modèle:Cite book</ref>.

• Phidias (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.
• Plato (427–347 BC), in his Timaeus, describes five possible regular solids (the Platonic solids, the tetrahedron, cube, octahedron, dodecahedron and icosahedron), some of which are related to the golden ratio.
• Euclid (c. 325–c. 265 BC), in his Elements, gave the first recorded definition of the golden ratio, which he called, as translated into English, "extreme and mean ratio" (Greek: ακρος και μεσος λογος).<ref name=Elements 6.3/>
• Fibonacci (1170–1250) mentioned the numerical series now named after him in his Liber Abaci; the Fibonacci sequence is closely related to the golden ratio.
• Luca Pacioli (1445–1517) defines the golden ratio as the "divine proportion" in his Divina Proportione.
• Johannes Kepler (1571–1630) describes the golden ratio as a "precious jewel": "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel."
• Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
• Martin Ohm (1792–1872) is believed to be the first to use the term goldene Schnitt (golden section) to describe this ratio.
• Edouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
• Mark Barr (20th century) uses the Greek letter phi (φ), the initial letter of Greek sculptor Phidias's name, as a symbol for the golden ratio.
• Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.

Aesthetics

Beginning in the Renaissance, a body of literature on the aesthetics of the golden ratio has developed. As a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works.

The first and most influential of these was De Divina Proportione by Luca Pacioli, a three-volume work published in 1509. Pacioli, a Franciscan friar, 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 that interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.<ref name="livio"/> Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. Containing illustrations of regular solids by Leonardo Da Vinci, Pacioli's longtime friend and collaborator, De Divina Proportione was a major influence on generations of artists and architects.

Architecture

Some studies of the Acropolis, including the Parthenon, conclude that many of its proportions approximate the golden ratio. The Parthenon's facade as well as elements of its facade and elsewhere can be circumscribed by golden rectangles.<ref>Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis", Philosophical Polemic Communication Quarterly, Vol. 46, 1998.</ref> To the extent that classical buildings or their elements are proportioned according to the golden ratio, this might indicate that their architects were aware of the golden ratio and consciously employed it in their designs. Alternatively, it is possible that the architects used their own sense of good proportion, and that this led to some proportions that closely approximate the golden ratio. On the other hand, such retrospective analyses can always be questioned on the ground that the investigator chooses the points from which measurements are made or where to superimpose golden rectangles, and that these choices affect the proportions observed.

Some 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 B.C.) 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 decagons was duly observed, as well as in the dodecahedron (a regular polyhedron 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 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. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 B.C., 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 Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

A geometrical analysis of the Great Mosque of Kairouan 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), Available online</ref> It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. Boussora and Mazouz also examined earlier archaeological theories about the mosque, and demonstrate the geometric constructions based on the golden ratio by applying these constructions to the plan of the mosque to test their hypothesis.

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> 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>

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 Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.<ref>Le Corbusier, The Modulor, p. 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>

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.<ref> Urwin, Simon. Analysing Architecture (2003) pp. 154-5, ISBN 0-415-30685-X</ref>

Art

Painting

Leonardo da Vinci's illustrations in De Divina Proportione (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 own paintings. Some suggest that his Mona Lisa, for example, employs the golden ratio in its geometric equivalents. Whether Leonardo proportioned his paintings according to the golden ratio has been the subject of intense debate. The secretive Leonardo seldom disclosed the bases of his art, and retrospective analysis of the proportions in his paintings can never be conclusive.

Salvador Dalí 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, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.<ref> Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X</ref><ref name=livio/>

Mondrian 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>

Interestingly, 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 Available online</ref>

Sculpture

Australian sculptor Andrew Rogers's 50-ton stone and gold sculpture entitled Golden Ratio, installed outdoors in Jerusalem. The height of each stack of stones, beginning from either end and moving toward the center, is the beginning of the Fibonacci sequence: 1, 1, 2, 3, 5, 8.

Book design

See Canons of page construction.

According to Jan Tschichold,<ref>Jan Tschichold, The Form of the Book, Hartley & Marks (1991), ISBN 0-88179-116-4.</ref> "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 millimetre."

Perceptual studies

Studies by psychologists, starting with 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.<ref name="livio" /><ref>The golden ratio and aesthetics, by Mario Livio</ref>

Music

James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,<ref>Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.</ref> though other music scholars reject that analysis.<ref name="livio"/> In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.<ref name=Smith>Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X</ref> French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.

The golden ratio is also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, 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 />

This Binary Universe, an experimental album by Brian Transeau (aka BT), includes a track entitled "1.618" in homage to the golden ratio. The track features musical versions of the ratio and the accompanying video displays various animated versions of the golden mean.

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.<ref>

  Pearl Masters Premium 
. Pearl Corporation  
 

 

. Retrieved on Dec. 2, 2007. </ref>

According to author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio."<ref>Modèle:Cite book</ref>

Nature

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors. In these phenomena he saw the golden ratio operating as a universal law.<ref>Ibid. Padovan, R. Proportion: Science, Philosophy, Architecture , pp. 305-06</ref> Zeising wrote in 1854:

[The Golden Ratio is 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 structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.<ref>Zeising, Adolf, Neue Lehre van den Proportionen des meschlischen Körpers, Leipzig, 1854, preface.</ref>

See also History of aesthetics (pre-20th-century)

Mathematics

Golden ratio conjugate

The negative root of the quadratic equation for φ (the "conjugate root") is <math>1 - \varphi \approx -0.618</math>. The absolute value of this quantity (≈ 0.61Image:Cool.gif 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>Modèle:MathWorld</ref> It is denoted here by the capital Phi (<math> \Phi </math>):

<math>\Phi = {1 \over \varphi} \approx 0.61803\,39887\,.</math>

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

<math>\Phi = \varphi -1\,.</math>

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

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

or its inverse:

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

Short proofs of irrationality

Recall that we denoted the "larger part" by a and the "smaller part" by b. If the golden ratio is a positive rational number, then it must be expressible as a fraction in lowest terms in the form a / b where a and b are coprime positive integers. The algebraic definition of the golden ratio then indicates that if a / b = φ, then

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

Multiplying both sides by ab leads to:

<math>a^2 = ab+b^2\,.</math>

Subtracting ab from both sides and factoring out a gives:

<math>a(a-b) = b^2\,.</math>

Finally, dividing both sides by b(a − b) yields:

<math>\frac{a}{b} = \frac{b}{a-b}\,.</math>

This last equation indicates that a / b could be further reduced to b / (a − b), where a − b is still positive, which is an equivalent fraction with smaller numerator and denominator. But since a / b was already given in lowest terms, this is a contradiction. Thus this number cannot be so written, and is therefore irrational.

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

Alternate forms

The formula <math>\varphi = 1 + 1/\varphi</math> can be expanded recursively to obtain a continued fraction for the golden ratio:<ref>Modèle:Cite book</ref>

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

and its reciprocal:

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

The convergents of these continued fractions (1, 2, 3/2, 5/3, 8/5, 13/8, ..., or 1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers.

The equation <math>\varphi^2 = 1 + \varphi</math> likewise produces the continued square root form:

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

Also:

<math>\varphi = 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ</math>

<math>\varphi = {1 \over 2}\csc(\pi/10) = {1 \over 2}\csc 18^\circ</math>

<math>\varphi = 2\cos(\pi/5)=2\cos 36^\circ\,</math>

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 pentagram.

If x agrees with <math>\varphi</math> to n decimal places, then <math>\frac{x^2+2x}{x^2+1}</math> agrees with it to 2n decimal places.

An equation derived in 1994 connects the golden ratio to the Number of the Beast (666):<ref name="livio">x</ref>

<math>-\frac{\varphi}{2}=\sin666^\circ=\cos(6\cdot 6 \cdot 6^\circ).</math>

Which can be combined into the expression:

<math>-\varphi=\sin666^\circ+\cos(6\cdot 6 \cdot 6^\circ).</math>

Geometry

The number φ turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles.

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ Modèle:Unicode 222.5°. This approach was used to arrange mirrors on the Starshine 3 satellite<ref> A Disco Ball in Space

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Golden triangle, pentagon and pentagram

The golden triangle can be characterised as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original.

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

Suppose XB has length 1, and we call BC length φ. Because of the isosceles traingles BC=XC and XC=XA, so these are also length φ. Length AC = AB, therefore equals φ+1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, and so AC also equals φ². Thus φ² = φ+1, confirming that φ is indeed the golden ratio.

Pentagram

Modèle:Details The golden ratio plays an important role in regular pentagons and pentagrams. 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 2 intersecting edges (a side of the pentagon in the pentagram's center) is φ, as the four-color illustration shows.

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomon.

Ptolemy's theorem

The golden ratio can also be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one vertex from a regular pentagon. If the quadrilateral's long edge and diagonals are b, and short edges are a, then Ptolemy's theorem gives b² = a² + ab which yields

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

Scalenity of triangles

Consider a triangle with sides of lengths a, b, and c in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than φ and can be made as close as desired to φ.<ref>American Mathematical Monthly, pp. 49-50, 1954.</ref>

Relationship to Fibonacci sequence

Modèle:Details

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

The explicit expression for the Fibonacci sequence involves the golden ratio:

<math>F\left(n\right)

= {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}} = {{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}}\,.</math>

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence):

<math>\lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi.</math>

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:

<math>\sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)|

= \varphi\,.</math>

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

<math>\varphi^{n+1}

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

This identity allows any polynomial in φ to be reduced to a linear expression. For example:

<math>3\varphi^3 - 5\varphi^2 + 4

= 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 = 3[(\varphi + 1) + \varphi] - 5(\varphi + 1) + 4 = \varphi + 2 \approx 3.618\,.</math>

Other properties

The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases of the Lagrange's approximation theorem. This may be the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants).

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

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

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

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

<math>\varphi^n = \varphi^{n-1} + \varphi^{n-2} = \varphi \cdot \operatorname{F}_n + \operatorname{F}_{n-1}\,.</math>

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

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

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

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

The golden ratio is the fundamental unit of the algebraic number field <math>\mathbb{Q}(\sqrt{5})</math> and is a Pisot-Vijayaraghavan number.

Decimal expansion

Calculation methods

The golden ratio's decimal expansion can be calculated directly from the expression

<math>\varphi = {1+\sqrt{5} \over 2},</math>

with √5 ≈ 2.2360679774997896964. The square root of 5 can be calculated with the Babylonian method, starting with an initial estimate such as x₁ = 2 and iterating

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

for n = 1, 2, 3, ..., until the difference between x_n and x_n−1 becomes zero, to the desired number of digits.

The Babylonian algorithm for √5 is equivalent to Newton's method for solving the equation x² − 5 = 0. In its more general form, Newton's method can be applied directly to any algebraic equation, including the equation x² − x − 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself,

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

for an appropriate initial estimate x₁ such as x₁ = 1. A slightly faster method is to rewrite the equation as x − 1 − 1/x = 0, in which case the Newton iteration becomes

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

These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental numbers 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 F₂₅₀₀₁ and F₂₅₀₀₀, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.

Decimal expansion to 1,050 places

Modèle:OEIS

1.6180339887 4989484820 4586834365 6381177203 0917980576
  2862135448 6227052604 6281890244 9707207204 1893911374
  8475408807 5386891752 1266338622 2353693179 3180060766
  7263544333 8908659593 9582905638 3226613199 2829026788
  0675208766 8925017116 9620703222 1043216269 5486262963
  1361443814 9758701220 3408058879 5445474924 6185695364
  8644492410 4432077134 4947049565 8467885098 7433944221
  2544877066 4780915884 6074998871 2400765217 0575179788
  3416625624 9407589069 7040002812 1042762177 1117778053
  1531714101 1704666599 1466979873 1761356006 7087480710
  1317952368 9427521948 4353056783 0022878569 9782977834
  7845878228 9110976250 0302696156 1700250464 3382437764
  8610283831 2683303724 2926752631 1653392473 1671112115
  8818638513 3162038400 5222165791 2866752946 5490681131
  7159934323 5973494985 0904094762 1322298101 7261070596
  1164562990 9816290555 2085247903 5240602017 2799747175
  3427775927 7862561943 2082750513 1218156285 5122248093
  9471234145 1702237358 0577278616 0086883829 5230459264
  7878017889 9219902707 7690389532 1968198615 1437803149
  9741106926 0886742962 2675756052 3172777520 3536139362
  1076738937 6455606060 5921658946 6759551900 4005559089
  ...

Pyramids

Both Egyptian pyramids and those mathematical regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.

Mathematical pyramids and triangles

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

The medial right triangle of this "golden" pyramid (see diagram), with sides <math>1:\sqrt{\varphi}:\varphi</math> is interesting in its own right, demonstrating via the Pythagorean theorem the relationship <math>\sqrt{\varphi} = \sqrt{\varphi^2 - 1}</math> or <math>\varphi = \sqrt{1 + \varphi}</math>. This "Kepler triangle"<ref>Modèle:Cite book</ref> is the only right triangle proportion with edge lengths in geometric progression,<ref>Modèle:Cite book</ref> just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. 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>

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle;<ref name="maor">Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000</ref> the face slope corresponding to the angle with tangent 4/3 is 53.13 degrees (53 degrees and 8 minutes).<ref name=Herkommer> The Great Pyramid, The Great Discovery, and The Great Coincidence

. Retrieved on 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, mutliplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.<ref name="maor">Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000</ref>

Another mathematical pyramid with proportions almost identical to the "golden" one is the one with perimeter equal to 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 golden triangle. This pyramid relationship corresponds to the coincidental relationship <math>\sqrt{\varphi} \approx 4/\pi</math>.

Egyptian pyramids very close in proportion to these mathematical pyramids are known.<ref name=Herkommer/>

Egyptian pyramids

The shapes of Egyptian pyramids include one that is remarkably close to a "golden pyramid". This is the Great Pyramid of Giza (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">Eli 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 a topic of controversy. Several other Egyptian pyramids are very close to the rational 3:4:5 shape.<ref name=Herkommer/>

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 Egyptian architecture 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> He also asserts that some recent historians of science have denied that the Egyptians had any such knowledge, contending rather that its appearance in an Egyptian building is the result of chance.

In 1859, the Pyramidologist John Taylor (1781-1864) asserted that in the Great Pyramid of Giza built around 2600 BC, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant 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.

Howard Vyse, 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.

Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, asserts that Egyptian mathematics as understood in modern times, 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.<ref>Eric Temple Bell, The Development of Mathematics, New York: Dover, 1940, p.40</ref>

Disputed sightings of the golden ratio

Empirical sightings of the golden ratio in numerous natural proportions and artistic proportions are necessarily just approximations, to a wide range of accuracies. For example, historian John Man suggests that Gutenberg's Bible page was "based on the golden section shape," even though its page size is 44.5 cm × 30.7 cm, which is a ratio of 1.45.<ref>Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166-67, Wiley, ISBN 0-471-21823-5. "The half-folio page (30.7 x 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 (footnote: The ratio is 0.618.... ad inf commonly rounded to 0.625)"</ref>

Examples of disputed observations of the golden ratio include:

• A connection has been proposed between the Fibonacci numbers (Golden Ratio) and Chargaff's second rule concerning the proportions of nucleobases in the human genome.<ref name=Yamagishi2006> Yamagishi M. E. B. and Shimabukuro A. I. (2007) Nucleotide Frequencies in Human Genome and Fibonacci Numbers. Bulletin of Mathematical Biology </ref>

• It is sometimes claimed that the number of bees in a beehive divided by the number of drones yields the golden ratio.<ref name=moscovich>Ivan Moscovich, Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004</ref> In reality, the proportion of drones in a beehive varies greatly by beehive, by bee race, by season, and by beehive health status; the ratio is normally much greater than the golden ratio (usually close to 20:1 in healthy colonies).^{[citation needed]} This misunderstanding may arise because in theory bees have approximately this ratio of male to female ancestors (See The Bee Ancestry Code) - the caveat being that ancestry can trace back to the same drone by more than one route, so the actual numbers of bees do not need to match the formula.
• Some specific proportions in the bodies of many animals (including humans<ref name=pheasant>Stephen Pheasant, Bodyspace, London: Taylor & Francis, 1998</ref><ref name=vanLaack>Walter van Laack, A Better History Of Our World: Volume 1 The Universe, Aachen: van Laach GmbH, 2001.</ref>) and parts of the shells of mollusks<ref name="dunlap">x</ref> and cephalopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in a specific individual and the proportion in question is often significantly different from the golden ratio.<ref name=pheasant/> The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.<ref name=vanLaack/> The Nautilus shell, whose construction proceeds in a logarithmic spiral, is often cited, usually under 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/>
• The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species.<ref>Derek Thomas, Architecture and the Urban Environment: A Vision for the New Age, Oxford: Elsevier, 2002</ref> In practice, there are significant variations between individuals, seasonal variations, and age variations in these species. While the golden ratio may be found in some proportions in some individuals at particular times in their life cycles, there is no consistent ratio in their proportions.^{[citation needed]}
• In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.<ref>For instance, Osler writes that "38.2 percent and 61.8 percent

retracements of recent rises or declines are common," in Modèle:Cite journal</ref> The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers; see, e.g. Elliott wave principle. However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.<ref>Roy Batchelor and Richard Ramyar, "Magic numbers in the Dow," 25th International Symposium on Forecasting, 2005, p. 13, 31. "Not since the 'big is beautiful' days have giants looked better", Tom Stevenson, The Daily Telegraph, Apr. 10, 2006, and "Technical failure", The Economist, Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.</ref>

• ISO 7810 cards such as Visa or MasterCard have an aspect ratio of 1.586, which is only 2% smaller than the golden ratio.
• A rectangle that is one mile long by one kilometer wide is within 1% of a golden rectangle, with a mile being exactly 1.609344 km.

See also

References and footnotes

Further reading

• Modèle:Cite book

• Modèle:Cite book

• Modèle:Cite book Citations in the text are to this online edition.

• Modèle:Cite book

• Modèle:Cite book
• Modèle:Cite book

• Modèle:Cite journal

• Plato

       (360 BC) (Benjamin Jowett trans.)
     
   
 
.    Timaeus 
. The Internet Classics Archive 
   

. Retrieved on May 30, 2006.

• Modèle:Cite book

• Modèle:Cite book

External links

• Green , Thomas M.

     (updated June 20, 2005)
   
.    The Pentagram & The Golden Ratio 

. Retrieved on Dec. 1, 2007.

Geometry instruction with problems to solve.

• Khan , Amore

     (revised Feb. 2, 2007)
   
.    Khan Amore’s Commentary on the Divine Proportion 
 (HTML; PDF available)

. Retrieved on Dec. 1, 2007.

• Knott , Ron

.    The Golden section ratio: Phi 

. Information and activities by a mathematics professor.

• Modèle:MathWorld

• Yarrow , David

     (updated Oct. 21, 2005)
   
.    PHI: The Divine Ratio 

. Retrieved on Dec. 1, 2007.

• "Golden Section" by Michael Schreiber, The Wolfram Demonstrations Project, 2007.Modèle:Link FA

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