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Golden ratio

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Golden ratio
Golden ratio line.svg
Line segments in the golden ratio
Algebraic form
Continued fraction
A golden rectangle with long side a and short side b adjacent to a square with sides of length a produces a similar golden rectangle with long side a + b and short side a. This illustrates the relationship

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 and with

where the Greek letter phi ( or ) represents the golden ratio.[a] It is an irrational number that is a solution to the quadratic equation with a value of[2][1]


The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[3][4] Other names include extreme and mean ratio,[5] medial section, divine proportion (Latin: proportio divina),[6] divine section (Latin: sectio divina), golden proportion, golden cut,[7] and golden number.[8][9][10]


Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be aesthetically pleasing. These often appear in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio.


The Greek letter phi symbolizes the golden ratio. Usually, the lowercase form or is used. Sometimes the uppercase form is used for the reciprocal of the golden ratio, [12]

Two quantities and are said to be in the golden ratio if

One method for finding the value of is to start with the left fraction. Through simplifying the fraction and substituting in


Multiplying by gives

which can be rearranged to

Using the quadratic formula, two solutions are obtained:


Because is the ratio between positive quantities, is necessarily the positive one. The negative root is .


According to Mario Livio,


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

Ancient Greek mathematicians first studied what we now call the golden ratio, because of its frequent appearance in geometry;[14] the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.[15] According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising Pythagoreans.[16] Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio,[17][b] and contains its first known definition which proceeds as follows:[18]

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.[19][c]

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 although he never connected it to the series of numbers named after him.[21]

Luca Pacioli named his book Divina proportione (1509) after the ratio, and explored its properties including its appearance in some of the Platonic solids.[10][22] Leonardo da Vinci, who illustrated the aforementioned book, called the ratio the sectio aurea ('golden section').[23] 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.[24]

German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;[25] this was rediscovered by Johannes Kepler in 1608.[26] The first known decimal approximation of the (inverse) golden ratio was stated as "about " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.[27] The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.[6]

18th-century mathematicians Abraham de Moivre, Daniel Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula".[28] Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.[29] James Sully used the equivalent English term in 1875.[30]

By 1910, mathematician Mark Barr began using the Greek letter Phi () as a symbol for the golden ratio.[31][d] It has also been represented by tau (), the first letter of the ancient Greek τομή ('cut' or 'section').[34][35]

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.[36] This led to Dan Shechtman's early 1980s discovery of quasicrystals,[37][38] some of which exhibit icosahedral symmetry.[39][40]

Applications and observations







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

Divina proportione (Divine proportion), a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. Divina proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[45] Pacioli also saw Catholic religious significance in the ratio, which led to his work's title.

Leonardo da Vinci's illustrations of polyhedra in Divina proportione[46] 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.[47] Similarly, although the Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.[48][49]

Salvador Dalí, influenced by the works of Matila Ghyka,[50] 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.[47][51]

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 with averages for individual artists ranging from (Goya) to (Bellini).[52] 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 proportions, and others with proportions like and [53]


Books and design

According to Jan Tschichold,

There was a time when deviations from the truly beautiful page proportions and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.[55]

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.[56][57][58][59]


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

The aspect ratio (height to width ratio) of the flag of Togo is in the golden ratio.


Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[60] though other music scholars reject that analysis.[61] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position".[62]

The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.[63] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[64]



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

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".[66]

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.[67][68] Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".[69]



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




Contradiction from an expression in lowest terms


Recall that:

the whole is the longer part plus the shorter part;

If we call the whole and the longer part then the second statement above becomes

is to as is to

or, algebraically


To say that the golden ratio is rational means that is a fraction where and are integers. We may take to be in lowest terms and and to be positive. But if is in lowest terms, then the identity labeled (☆) above says is in still lower terms. That is a contradiction that follows from the assumption that is rational.

By irrationality of


Minimal polynomial

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


This quadratic polynomial has two roots, and

The golden ratio is also closely related to the polynomial

which has roots and

Golden ratio conjugate

The conjugate root to the minimal polynomial is

The absolute value of this quantity () corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ), and is sometimes referred to as the golden ratio conjugate[12] or silver ratio.[e][71] It is denoted here by the capital Phi ():


or its inverse:

Alternative forms


The formula can be expanded recursively to obtain a continued fraction for the golden ratio:[72]

and its reciprocal:

The convergents of these continued fractions ( ... or ...) are ratios of successive Fibonacci numbers.

The equation likewise produces the continued square root:

An infinite series can be derived to express :[73]


These correspond to the fact that the length of the diagonal of a regular pentagon is times the length of its side, and similar relations in a pentagram.



The number turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is 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 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. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[74]

Dividing a line segment by interior division

Dividing a line segment by interior division according to the golden ratio
  1. Having a line segment construct a perpendicular at point with half the length of Draw the hypotenuse
  2. Draw an arc with center and radius This arc intersects the hypotenuse at point
  3. Draw an arc with center and radius This arc intersects the original line segment at point Point divides the original line segment into line segments and with lengths in the golden ratio.

Dividing a line segment by exterior division

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


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 triangle, pentagon and pentagram

Golden triangle. The double-red-arched angle is or radians.
Golden triangle

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

If angle then because of the bisection, and because of the similar triangles; from the original isosceles symmetry, and by similarity. The angles in a triangle add up to so giving So the angles of the golden triangle are thus The angles of the remaining obtuse isosceles triangle (sometimes called the golden gnomon) are

Suppose has length and we call length Because of the isosceles triangles and so these are also length Length therefore equals But triangle is similar to triangle so and so also equals Thus confirming that is indeed the golden ratio.

Similarly, the ratio of the area of the larger triangle to the smaller is equal to while the inverse ratio is



Odom's construction
Let and be midpoints of the sides and of an equilateral triangle Extend to meet the circumcircle of at




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 The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.

Ptolemy's theorem

The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are and short edges are then Ptolemy's theorem gives which yields

Scalenity of triangles

Consider a triangle with sides of lengths and in decreasing order. Define the "scalenity" of the triangle to be the smaller of the two ratios and The scalenity is always less than and can be made as close as desired to [76]

Triangle whose sides form a geometric progression

If the side lengths of a triangle form a geometric progression and are in the ratio where is the common ratio, then must lie in the range as a consequence of the triangle inequality (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If then the shorter two sides are and but their sum is thus A similar calculation shows that A triangle whose sides are in the ratio is a right triangle (because ) known as a Kepler triangle.[77]

Golden triangle, rhombus, and rhombic triacontahedron

One of the rhombic triacontahedron's rhombi
All of the faces of the rhombic triacontahedron are golden rhombi

A golden rhombus is a rhombus whose diagonals are in the golden ratio. The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is which is twice the isosceles angle of a golden triangle and four times its most acute angle.[78]

Relationship to Fibonacci sequence



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

A Fibonacci spiral which approximates the golden spiral, using Fibonacci sequence square sizes up to The spiral is drawn starting from the inner square and continues outwards to successively larger squares.

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as shown by Kepler:[79]

In other words, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates e.g., These approximations are alternately lower and higher than and converge to as the Fibonacci numbers increase, and:

More generally


Furthermore, the successive powers of obey the Fibonacci recurrence

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


where is the th Fibonacci number.

However, this is no special property of because polynomials in any solution to a quadratic equation can be reduced in an analogous manner, by applying:

for given coefficients such that satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible th-degree polynomial over the rationals can be reduced to a polynomial of degree Phrased in terms of field theory, if is a root of an irreducible th-degree polynomial, then has degree over with basis



More deeply, these maps form a subgroup of the modular group isomorphic to the symmetric group on letters, corresponding to the stabilizer of the set of standard points on the projective line, and the symmetries correspond to the quotient map – the subgroup consisting of the identity and the -cycles, in cycle notation fixes the two numbers, while the -cycles interchange these, thus realizing the map.

Other properties



The sequence of powers of contains these values more generally, any power of is equal to the sum of the two immediately preceding powers:

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

If then:

The golden ratio is a fundamental unit of the algebraic number field and is a Pisot–Vijayaraghavan number.[81] In the field we have where is the -th Lucas number.

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or -nary), quadratic integers in the ring – that is, numbers of the form for – have terminating representations, but rational fractions have non-terminating representations.


The golden ratio appears in the theory of modular functions as well. Let


Also for positive real numbers and then[83]


For the gamma function, the only solutions to the equation Γ(z − 1) = Γ(z + 1) are z = φ and z = −1/φ.

Decimal expansion

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

with 2.236067977.... OEISA002163. The square root of can be calculated via the Babylonian method, starting with an initial estimate such as and iterating

for until the difference between and becomes zero to the desired number of digits. Then

The Babylonian algorithm for is equivalent to Newton's method for solving the equation and it converges quadratically, meaning that the number of correct digits is roughly doubled each iteration.

To avoid the computationally expensive division operation, Newton's method can instead be used to solve the equation for the root Then and the update step is

Alternately Newton's method can be applied directly to any equation that has the golden ratio as a solution, such as In this case, and the update step is

Halley's method has cubic convergence (roughly tripling the number of correct digits with each iteration), but may be slower for practical computation because each step takes more work. To solve the update step is

The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute digits of the golden ratio is proportional to the time needed to divide two -digit numbers. This is considerably faster than known algorithms for the transcendental numbers and .

An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers and each over digits, yields over significant digits of the golden ratio.

The decimal expansion of the golden ratio [1] has been calculated to an accuracy of ten trillion () digits.[84]


A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid's apothem (), semi-base (), and height (); the face inclination angle is also marked. Mathematical proportions of and and are of particular interest in relation to Egyptian pyramids.


Mathematical pyramids

A pyramid in which the apothem (slant height along the bisector of a face) is equal to 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 times the semi-base (that is, the slope of the face is ); the square of the height is equal to the area of a face, times the square of the semi-base. The medial right triangle of this "golden" pyramid (see diagram), with sides is interesting in its own right, demonstrating via the Pythagorean theorem the relationship or This Kepler triangle[85] is the only right triangle proportion with edge lengths in geometric progression,[86][77] just as the triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent corresponds to the angle that the side of the pyramid makes with respect to the ground, ().[87]

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 triangle;[88] the face slope corresponding to the angle with tangent is, to two decimal places, (). The slant height or apothem is 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,[89] and the rational inverse slope (run/rise, multiplied by a factor of to convert to their conventional units of palms per cubit) was used in the building of pyramids.[88]

Egyptian pyramids very close in proportion to these mathematical pyramids are known.[90][77]

Egyptian pyramids

One Egyptian pyramid that is close to a "golden pyramid" is the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of is close to the "golden" pyramid inclination of – and even closer to the -based pyramid inclination of However, several other mathematical theories of the shape of the great pyramid, based on rational slopes, have been found to be both more accurate and more plausible explanations for the slope.[77]

In the mid-nineteenth century, Friedrich Röber studied various Egyptian pyramids including those of Khafre, Menkaure, and some of the Giza, Saqqara, and Abusir groups. He did not apply the golden ratio to the Great Pyramid of Giza, but instead agreed with John Shae Perring that its side-to-height ratio is For all the other pyramids he applied measurements related to the Kepler triangle, and claimed that either their whole or half-side lengths are related to their heights by the golden ratio.[91]

In 1859, the pyramidologist John Taylor misinterpreted Herodotus (c. 440 BC) as indicating that the Great Pyramid's height squared equals the area of one of its face triangles.[f] This led Taylor to claim that, in the Great Pyramid, the golden ratio is represented by the ratio of the length of the face (the slope height, inclined at an angle to the ground) to half the length of the side of the square base (equivalent to the secant of the angle ).[93] The above two lengths are about 186.4 metres (612 ft) and 115.2 metres (378 ft), respectively.[92] The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, Howard Vyse reported the great pyramid height 148.2 metres (486 ft), and half-base 116.4 metres (382 ft), yielding for the ratio of slant height to half-base, again more accurate than the data variability.[86]

Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the pyramid, since the 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 or [94] Example geometric problems of pyramid design in the Rhind papyrus correspond to various rational slopes.[77]

Michael Rice[95] asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of the mathematics of the pyramids, citing Giedon (1957).[96] Historians of science have long debated whether the Egyptians had any such knowledge, contending that its appearance in the Great Pyramid is the result of chance.[97]

Disputed observations

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

Nautilus shells are often erroneously claimed to be golden-proportioned.
  • Some specific proportions in the bodies of many animals (including humans)[98][99] and parts of the shells of mollusks[4] are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.[98] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[99] The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is golden-proportioned relative to the previous one.[100] However, measurements of nautilus shells do not support this claim.[101]
  • 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 [102]
  • Studies by psychologists, starting with Gustav Fechner c. 1876,[103] 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.[104][47]

The Parthenon

Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.[107]

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.[108] 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."[109] Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."[110]

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

Modern art

The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism.[112] 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.[113] The Cubists observed in its harmonies, geometric structuring of motion and form, the primacy of idea over nature, an absolute scientific clarity of conception.[114] 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,[115] and Marcel Duchamp said as much in an interview.[116] 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.[116][117][118] 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.[119]

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,[120] though other experts (including critic Yve-Alain Bois) have discredited these claims.[47][121]

See also


Explanatory footnotes

  1. ^ If the constraint on and each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. is defined as the positive solution. The negative solution is The sum of the two solutions is and the product of the two solutions is
  2. ^ Euclid, Elements, Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.
  3. ^ "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."[20]
  4. ^ After Classical Greek sculptor Phidias (c. 490–430 BC);[32] Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.[33]
  5. ^ Not to be confused with the silver mean, also known as the silver ratio.
  6. ^ Taylor translated Herodotus: "this Pyramid, which is four-sided, each face is, on every side 8 plethra, and the height equal." He interpreted this imaginatively, and in 1860, John Herschel was the first of many authors to repeat his false claim. In 2000, Roger Herz-Fischler traced the error back to Taylor.[92]


  1. ^ a b c OEISA001622
  2. ^ Weisstein, Eric W. "Golden Ratio". Retrieved 2020-08-10.
  3. ^ Livio 2003, pp. 3, 81.
  4. ^ a b Dunlap, Richard A., The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
  5. ^ Euclid, Elements, Book 6, Definition 3.
  6. ^ a b Fink, Karl; Beman, Wooster Woodruff; Smith, David Eugene (1903). A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik (2nd ed.). Chicago: Open Court Publishing Co. p. 223.
  11. ^ Strogatz, Steven (September 24, 2012). "Me, Myself, and Math: Proportion Control". The New York Times.
  13. ^ Livio 2003, p. 6.
  14. ^ Livio 2003, p. 4: "... line division, which Euclid defined for ... purely geometrical purposes ..."
  15. ^ Livio 2003, pp. 7–8.
  16. ^ Livio 2003, pp. 4–5.
  17. ^ Livio 2003, p. 78.
  18. ^ Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN 978-1-4027-3522-6.
  19. ^ Livio 2003, p. 3.
  20. ^ Richard Fitzpatrick (translator) (2007). Euclid's Elements of Geometry. p. 156. ISBN 978-0615179841. {{cite book}}: |author= has generic name (help)
  21. ^ Livio 2003, pp. 88–96.
  22. ^ Livio 2003, pp. 131–132.
  23. ^ Baravalle, H. V. (1948). "The geometry of the pentagon and the golden section". Mathematics Teacher. 41: 22–31. doi:10.5951/MT.41.1.0022.
  24. ^ Livio 2003, p. 141.
  25. ^ Schreiber, Peter (1995). "A Supplement to J. Shallit's Paper "Origins of the Analysis of the Euclidean Algorithm"". Historia Mathematica. 22 (4): 422–424. doi:10.1006/hmat.1995.1033.
  26. ^ Livio 2003, pp. 151–152.
  28. ^ Weisstein, Eric W. "Binet's Fibonacci Number Formula". MathWorld.
  29. ^ Herz-Fischler, Roger (1987). A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier University Press. ISBN 978-0889201521.
  30. ^ Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. p. 8. ISBN 9-781-61614-424-1.
  31. ^ Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. p. 285. ISBN 9-781-61614-424-1.
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Works cited

Further reading

External links