}} algebraic=$\backslash frac\{1\; +\; \backslash sqrt5\}\{2\}$
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation to the larger of the two quantities. Expressed algebraically, for quantities $a$ and $b$ with $a\; >\; b\; >\; 0,$
where the Greek letter phi or $\backslash phi$) represents the golden ratio. It is an irrational number that is a solution to the quadratic equation $x^2\; \; x\; \; 1\; =\; 0,$ with a value of
The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).Dunlap, Richard A., The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997 Other names include extreme and mean ratio,Euclid, Elements, Book 6, Definition 3. medial section, divine proportion (Latin: proportio divina), divine section (Latin: sectio divina), golden proportion, golden cut,Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 37. "And the same applies in architecture, to the 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." and golden number.Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A CrossDisciplinary Reference, Gloucester MA: Rockport Publishers, 2003Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.
since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as manmade systems such as , in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the phyllotaxis and other plant parts.
Some twentiethcentury and , including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be aesthetics 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.
One method for finding the value of $\backslash varphi$ is to start with the left fraction. Through simplifying the fraction and substituting in $b/a\; =\; 1/\backslash varphi,$
Therefore,
Multiplying by $\backslash varphi$ gives
which can be rearranged to
Using the quadratic formula, two solutions are obtained:
Because $\backslash varphi$ is the ratio between positive quantities, $\backslash varphi$ is necessarily the positive one. The negative root is $\backslash frac\{1\}\{\backslash varphi\}$.
Ancient Greece mathematicians first studied what we now call the golden ratio, because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and . According to one story, 5thcentury BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising Pythagoreanism. Euclid's Elements () provides several theorem and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:
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 Fibonacci number.
Luca Pacioli named his book Divina proportione (1509) after the ratio, and explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated the aforementioned book, called the ratio the sectio aurea ('golden section'). 16thcentury mathematicians such as Rafael Bombelli solved geometric problems using the ratio.
German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about $0.6180340$" in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:
18thcentury mathematicians Abraham de Moivre, Daniel Bernoulli, and Leonhard Euler used a golden ratiobased formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.
James Sully used the equivalent English term in 1875.By 1910, mathematician Mark Barr began using the Greek alphabet Phi as a symbol for the golden ratio.
It has also been represented by tau the first letter of the ancient Greek τομή ('cut' or 'section').The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.
This led to Dan Shechtman's early 1980s discovery of , some of which exhibit icosahedral symmetry.
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 anthropometry, 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.Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. : "Both the paintings and the architectural designs make use of the golden section".
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.Urwin, Simon. Analysing Architecture (2003) pp. 154–155,
Leonardo da Vinci's illustrations of polyhedra in Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although 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.
Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47,
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).Olariu, Agata, Golden Section and the Art of Painting Available online On the other hand, Pablo Tosto listed over 350 works by wellknown artists, including more than 100 which have canvasses with golden rectangle and $\backslash sqrt5$ proportions, and others with proportions like $\backslash sqrt2,$ $3,$ $4,$ and $6.$Tosto, Pablo, La composición áurea en las artes plásticas – El número de oro, Librería Hachette, 1969, pp. 134–144
There was a time when deviations from the truly beautiful page proportions $2\backslash mathbin\{:\}3,$ $1\backslash mathbin\{:\}\backslash sqrt3,$ and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.
According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.
The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.
Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.Though Heinz Bohlen proposed the nonoctaverepeating cents scale based on , the tuning features relations based on the golden ratio. As a musical interval the ratio is cents ()." An 833 Cents Scale: An experiment on harmony", HuygensFokker.org. Accessed December 1, 2012.
The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law.
Zeising wrote in 1854 of a universal orthogenesis law of "striving for beauty and completeness in the realms of both nature and art".However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.Pommersheim, James E., Tim K. Marks, and Erica Flapan, eds. 2010. "Number Theory: A Lively Introduction with Proofs, Applications, and Stories". John Wiley and Sons: 82.
If we call the whole $n$ and the longer part $m,$ then the second statement above becomes
or, algebraically
To say that the golden ratio $\backslash varphi$ is rational means that $\backslash varphi$ is a fraction $n/m$ where $n$ and $m$ are integers. We may take $n/m$ to be in lowest terms and $n$ and $m$ to be positive. But if $n/m$ is in lowest terms, then the identity labeled (☆) above says $m/(nm)$ is in still lower terms. That is a contradiction that follows from the assumption that $\backslash varphi$ is rational.
This quadratic polynomial has two roots, $\backslash varphi$ and $\backslash varphi^\{1\}.$
The golden ratio is also closely related to the polynomial
which has roots $\backslash varphi$ and $\backslash varphi^\{1\}.$
The absolute value of this quantity 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 or silver ratio.Weisstein, Eric W. (2002). "Golden Ratio Conjugate". CRC Concise Encyclopedia of Mathematics, Second Edition, pp. 1207–1208. CRC Press. . It is denoted here by the capital Phi
This illustrates the unique property of the golden ratio among positive numbers, that
or its inverse:
and its reciprocal:
The convergents of these continued fractions $2/1,$ $2/1,$ $3/2,$ $5/3,$ $8/5,$ $13/8,$ ... or $1/1,$ $1/2,$ $2/3,$ $3/5,$ $5/8,$ are ratios of successive Fibonacci numbers.
The equation $\backslash varphi^2\; =\; 1\; +\; \backslash varphi$ likewise produces the Nested radical:
An infinite series can be derived to express $\backslash varphi$:Brian Roselle, "Golden Mean Series"
Also:
These correspond to the fact that the length of the diagonal of a regular pentagon is $\backslash varphi$ times the length of its side, and similar relations in a pentagram.
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. $360^\backslash circ/\backslash varphi\; \backslash approx\; 222.5^\backslash circ.$ This method was used to arrange the 1500 mirrors of the studentparticipatory satellite STARSHINE.
Both of the above displayed different produce geometric constructions that determine two aligned where the ratio of the longer one to the shorter one is the golden ratio.
If angle $BCX\; =\; \backslash alpha,$ then $XCA\; =\; \backslash alpha$ because of the bisection, and $CAB\; =\; \backslash alpha$ because of the similar triangles; $ABC\; =\; 2\backslash alpha$ from the original isosceles symmetry, and $BXC\; =\; 2\backslash alpha$ by similarity. The angles in a triangle add up to $180^\backslash circ,$ so $5\backslash alpha\; =\; 180^\backslash circ,$ giving $\backslash alpha\; =\; 36^\backslash circ.$ So the angles of the golden triangle are thus $36^\backslash circ$–$72^\backslash circ$–$72^\backslash circ.$ The angles of the remaining obtuse isosceles triangle $AXC$ (sometimes called the golden gnomon) are $36^\backslash circ$–$36^\backslash circ$–$108^\backslash circ.$
Suppose $XB$ has length $1,$ and we call $BC$ length $\backslash varphi.$ Because of the isosceles triangles $XC\; =\; XA$ and $BC\; =\; XC,$ so these are also length $\backslash varphi.$ Length $AC\; =\; AB,$ therefore equals $\backslash varphi\; +\; 1.$ But triangle $ABC$ is similar to triangle $CXB,$ so $AC/BC\; =\; BC/BX,$ $AC/\backslash varphi\; =\; \backslash varphi/1,$ and so $AC$ also equals $\backslash varphi^2.$ Thus $\backslash varphi^2\; =\; \backslash varphi\; +\; 1,$ confirming that $\backslash varphi$ is indeed the golden ratio.
Similarly, the ratio of the area of the larger triangle $AXC$ to the smaller $CXB$ is equal to $\backslash varphi,$ while the inverse ratio is $\backslash varphi\; \; 1.$
The pentagram includes ten isosceles triangles: five acute triangle and five obtuse triangle isosceles triangles. In all of them, the ratio of the longer side to the shorter side is $\backslash varphi.$ The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.
The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonaccilike sequence), as shown by Kepler:
In other words, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates $\backslash varphi;$ e.g., $987/610\; \backslash approx\; 1.6180327868852.$ These approximations are alternately lower and higher than $\backslash varphi,$ and converge to $\backslash varphi$ as the Fibonacci numbers increase, and:
More generally
where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when $a\; =\; 1.$
Furthermore, the successive powers of $\backslash varphi$ obey the Fibonacci recurrence
This identity allows any polynomial in $\backslash varphi$ to be reduced to a linear expression. For example:
The reduction to a linear expression can be accomplished in one step by using the relationship
where $F\_k$ is the $k$th Fibonacci number.
However, this is no special property of $\backslash varphi,$ because polynomials in any solution $x$ to a quadratic equation can be reduced in an analogous manner, by applying:
for given coefficients $a,$ $b$ such that $x$ satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible $n$thdegree polynomial over the rationals can be reduced to a polynomial of degree $n1.$ Phrased in terms of field theory, if $\backslash alpha$ is a root of an irreducible $n$thdegree polynomial, then $\backslash mathbb\{Q\}(\backslash alpha)$ has degree $n$ over $\backslash mathbb\{Q\},$ with basis $\backslash \{\; 1,\; \backslash alpha,\; \backslash ldots,\; \backslash alpha^\{n1\}\; \backslash \}.$
More deeply, these maps form a subgroup of the modular group $\backslash operatorname\{PSL\}(2,\; \backslash mathbb\{Z\})$ isomorphic to the symmetric group on $3$ letters, $S\_3,$ corresponding to the stabilizer of the set $\backslash \{0,\; 1,\; \backslash infty\backslash \}$ of $3$ standard points on the projective line, and the symmetries correspond to the quotient map $S\_3\; \backslash to\; S\_2$ – the subgroup $C\_3\; <\; S\_3$ consisting of the identity and the $3$cycles, in cycle notation $\backslash \{(1),\; (0\backslash ,1\backslash ,\backslash infty),\; (0\backslash ,\backslash infty\backslash ,\; 1)\backslash \},$ fixes the two numbers, while the $2$cycles $\backslash \{(0\backslash ,1),\; (0\backslash ,\backslash infty),\; (1\backslash ,\backslash infty)\backslash \}$ interchange these, thus realizing the map.
The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with $\backslash varphi$:
The sequence of powers of $\backslash varphi$ contains these values $0.618033\backslash ldots,$ $1.0,$ $1.618033\backslash ldots,$ $2.618033\backslash ldots;$ more generally, any power of $\backslash varphi$ is equal to the sum of the two immediately preceding powers:
As a result, one can easily decompose any power of $\backslash varphi$ into a multiple of $\backslash varphi$ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of $\backslash varphi$:
If $\backslash lfloor\; n/2\; \; 1\; \backslash rfloor\; =\; m,$ then:
The golden ratio is a fundamental unit of the algebraic number field $\backslash mathbb\{Q\}(\backslash sqrt5)$ and is a Pisot–Vijayaraghavan number. In the field $\backslash mathbb\{Q\}(\backslash sqrt5)$ we have $\backslash varphi^n\; =\; \backslash tfrac12\backslash bigl(L\_n\; +\; F\_n\backslash sqrt5\backslash bigr),$ where $L\_n$ is the $n$th Lucas number.
When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or $\backslash varphi$ nary), quadratic integers in the ring $\backslash mathbb\{Z\}\backslash varphi$ – that is, numbers of the form $a\; +\; b\backslash varphi$ for $a,\; b\; \backslash in\; \backslash mathbb\{Z\}$ – have terminating representations, but rational fractions have nonterminating 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 $4\backslash mathbin\{:\}\backslash log(\backslash varphi).$ Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved 20090721.
The golden ratio appears in the theory of Modular form as well. Let Then Also for positive real numbers $a,\; b\; \backslash in\; \backslash mathbb\{R\}^+$ and $ab\; =\; \backslash pi^2,$ thenBrendt, B. et al. "The Rogers–Ramanujan Continued Fraction"
and
For the gamma function, the only solutions to the equation are and .
with $\backslash sqrt5\; =$ . The square root of $5$ can be calculated via the Babylonian method, starting with an initial estimate such as $x\_0\; =\; 2$ and iterative method
for $n\; =\; 0,\; 1,\; 2,\; 3,\; \backslash ldots,$ until the difference between $x\_n$ and $x\_\{n1\}$ becomes zero to the desired number of digits. Then $\backslash varphi\; \backslash approx\; \backslash tfrac12(1\; +\; x\_n).$
The Babylonian algorithm for $\backslash sqrt5$ is equivalent to Newton's method for solving the equation $x^2\; \; 5\; =\; 0,$ 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 $5x^\{2\}\; \; 4\; =\; 0$ for the root $\backslash tfrac12\backslash sqrt5.$ Then $\backslash varphi\; \backslash approx\; \backslash tfrac12\; +\; x\_n$ 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 $x^2\; \; x\; \; 1\; =\; 0.$ In this case, $\backslash varphi\; \backslash approx\; x\_n$ 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 $x^2\; \; x\; \; 1\; =\; 0,$ the update step is
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 $\backslash pi$ 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 $F\_\{25001\}$ and $F\_\{25000\},$ each over $5000$ digits, yields over $10\{,\}000$ significant digits of the golden ratio.
The decimal expansion of the golden ratio $\backslash varphi$ has been calculated to an accuracy of ten trillion digits. Two independent computations done by Clifford Spielman.
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 $3\backslash mathbin\{:\}4\backslash mathbin\{:\}5$ triangle; the face slope corresponding to the angle with tangent $4/3$ is, to two decimal places, $53.13^\backslash circ$ The slant height or apothem is $5/3\; =\; 1.666\backslash ldots$ times the semibase. 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,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 and the rational inverse slope (run/rise, multiplied by a factor of $7$ to convert to their conventional units of palms per cubit) was used in the building of pyramids.Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000
Egyptian pyramids very close in proportion to these mathematical pyramids are known.
In the midnineteenth 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 sidetoheight ratio is $8\backslash mathbin\{:\}5.$ For all the other pyramids he applied measurements related to the Kepler triangle, and claimed that either their whole or halfside lengths are related to their heights by the golden ratio.
In 1859, the Pyramidology John Taylor misinterpreted Herodotus () as indicating that the Great Pyramid's height squared equals the area of one of its face triangles. 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 $\backslash theta$ to the ground) to half the length of the side of the square base (equivalent to the Cosecant of the angle $\backslash theta$).Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859 The above two lengths are about and , respectively. 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 , and halfbase , yielding $1.6189$ for the ratio of slant height to halfbase, again more accurate than the data variability.
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 $3\backslash mathbin\{:\}4\backslash mathbin\{:\}5$ pyramid, since the $3\backslash mathbin\{:\}4\backslash mathbin\{:\}5$ 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 $\backslash pi$ or $\backslash varphi.$
Example geometric problems of pyramid design in the Rhind papyrus correspond to various rational slopes.Michael RiceRice, Michael, Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C. p. 24 Routledge, 2003, 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).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. p. 24 Routledge, 2003 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.
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.Patrice Foutakis, "Did the Greeks Build According to the Golden Ratio?", Cambridge Archaeological Journal, vol. 24, n° 1, February 2014, pp. 71–86. 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.
Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,Bouleau, Charles, The Painter's Secret Geometry: A Study of Composition in Art (1963) pp. 247–248, Harcourt, Brace & World, though other experts (including critic YveAlain Bois) have discredited these claims.

