Golden ratio
</math> In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities formula_2 and formula_3 with formula_4 where the Greek letter phi ( or formula_5) represents the golden ratio. It is an irrational number that is a solution to the quadratic equation formula_6 with a value of The golden ratio is also called the golden mean or golden section (Latin: "sectio aurea"). Other names include extreme and mean ratio, medial section, divine proportion (Latin: "proportio divina ), divine section (Latin: sectio divina"), golden proportion, golden cut, and golden number. Mathematicians 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 man-made systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement  of leaves and other parts of vegetation. Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing 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. Calculation. Two quantities formula_2 and formula_3 are said to be in the "golden ratio" formula_9 if One method for finding the value of formula_9 is to start with the left fraction. Through simplifying the fraction and substituting in formula_11 Therefore, Multiplying  by formula_9 gives which can be rearranged to Using the quadratic formula, two solutions are obtained: Because formula_9 is the ratio between positive quantities, formula_9 is necessarily the positive one. However, the negative root, formula_15, shares many properties with the golden ratio. History. According to Mario Livio, Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was  neither a whole number nor a fraction (an irrational number), surprising Pythagoreans. Euclid's "Elements" () provides several propositions 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 series of numbers named after him. 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'). 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio. German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about formula_16" 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: 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". 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 letter 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 quasicrystals, some of which exhibit icosahedral symmetry. Applications and observations. Architecture. The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by  an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned." Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers , and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles. Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private  houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house. Art. "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. 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" 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. 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 formula_17 with averages for individual artists ranging from formula_18 (Goya) to formula_19 (Bellini). On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and formula_20 proportions, and others with proportions like formula_21 formula_22 formula_23 and formula_24 Books and design. According to Jan Tschichold, There was a time when deviations from the truly beautiful page proportions formula_25 formula_26 and the Golden Section were rare . Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter. According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions. Flags. The aspect ratio (height to width ratio) of the flag of Togo is in the golden ratio. Music. Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden  ratio in several of his pieces, including "Sonneries de la Rose+Croix". The golden ratio is also apparent in the organization of the sections in the music of Debussy's "Reflets dans l'eau (Reflections in Water)", from "Images" (1st series, 1905), in which "the sequence of keys is marked out by the intervals and and the main climax sits at the phi position". The musicologist Roy Howat has observed that the formal boundaries of Debussy's "La Mer" correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy  consciously sought such proportions. Though Heinz Bohlen proposed the non-octave-repeating cents scale based on combination tones, the tuning features relations based on the golden ratio. As a musical interval the ratio is cents (). Nature. Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio". The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. Zeising wrote in 1854  of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art". However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious. Optimization. The golden ratio is a critical element to golden-section search. Mathematics. Irrationality. The golden ratio is an irrational number. Below are two short proofs of irrationality: Contradiction from an expression in lowest terms. Recall that: If we call the whole formula_27 and the longer part formula_28 then the second statement above becomes To say that the golden  ratio formula_9 is rational means that formula_9 is a fraction formula_31 where formula_27 and formula_33 are integers. We may take formula_31 to be in lowest terms and formula_27 and formula_33 to be positive. But if formula_31 is in lowest terms, then the equally valued formula_38 is in still lower terms. That is a contradiction that follows from the assumption that formula_9 is rational. By irrationality of formula_20. 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 formula_41 is rational, then formula_42 is  also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational. Minimal polynomial. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial This quadratic polynomial has two roots, formula_9 and formula_44 The golden ratio is also closely related to the polynomial which has roots formula_45 and formula_46 Golden ratio conjugate. The conjugate root to the minimal polynomial formula_47 is The absolute value of this quantity corresponds to the length ratio taken in reverse order (shorter segment length over longer segment  length, formula_48), and is sometimes referred to as the golden ratio conjugate or silver ratio. It is denoted here by the capital Phi This illustrates the unique property of the golden ratio among positive numbers, that or its inverse: Alternative forms. The formula formula_49 can be expanded recursively to obtain a continued fraction for the golden ratio: and its reciprocal: The convergents of these continued fractions formula_50 formula_50 formula_52 formula_53 formula_54 formula_55 ... or formula_56 formula_57 formula_58 formula_59 formula_60 are ratios of successive Fibonacci numbers. The equation formula_61 likewise produces the continued square root: An infinite series can be derived to  express formula_9: Also: These correspond to the fact that the length of the diagonal of a regular pentagon is formula_9 times the length of its side, and similar relations in a pentagram. Geometry. The number formula_9 turns up often in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is formula_9 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. formula_66 This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3. Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length. Dividing a line segment by exterior division. 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 golden triangle can be characterized as an isosceles triangle formula_100 with the property that bisecting the angle formula_73 produces a new triangle formula_102 which is a similar triangle to the original. If angle formula_103 then formula_104 because of the bisection, and formula_105 because of the similar triangles; formula_106 from the original isosceles symmetry, and formula_107 by similarity. The angles in a triangle add up to formula_108 so formula_109 giving formula_110 So the  angles of the golden triangle are thus formula_111–formula_112–formula_113 The angles of the remaining obtuse isosceles triangle formula_114 (sometimes called the golden gnomon) are formula_111–formula_111–formula_117 Suppose formula_118 has length formula_119 and we call formula_68 length formula_121 Because of the isosceles triangles formula_122 and formula_123 so these are also length formula_121 Length formula_125 therefore equals formula_126 But triangle formula_100 is similar to triangle formula_128 so formula_129 formula_130 and so formula_75 also equals formula_132 Thus formula_133 confirming that formula_9 is indeed the golden ratio. Similarly, the ratio of the area of the larger triangle formula_114 to the smaller formula_102  is equal to formula_137 while the inverse ratio is formula_138 Pentagon. In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. Odom's construction. George Odom has given a remarkably simple construction for formula_9 involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the  intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H. S. M. Coxeter who published it in Odom's name as a diagram in the "American Mathematical Monthly" accompanied by the single word "Behold!" Pentagram. The golden ratio plays an important role in the geometry of 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 two intersecting edges (a side of the pentagon in the pentagram's  center) is formula_137 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 formula_121 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 formula_142 and short edges are formula_143 then Ptolemy's theorem gives  formula_144 which yields 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: In other words, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates formula_145 e.g., formula_146 These approximations are alternately lower and higher than formula_137 and converge to formula_9 as the Fibonacci numbers increase, and: More generally where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when formula_149 Furthermore, the successive powers of formula_9 obey the Fibonacci recurrence This identity allows  any polynomial in formula_9 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 formula_152 is the formula_153th Fibonacci number. However, this is no special property of formula_137 because polynomials in any solution formula_155 to a quadratic equation can be reduced in an analogous manner, by applying: for given coefficients formula_143 formula_3 such that formula_155 satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible formula_27th-degree polynomial over the rationals can be reduced to a  polynomial of degree formula_160 Phrased in terms of field theory, if formula_161 is a root of an irreducible formula_27th-degree polynomial, then formula_163 has degree formula_27 over formula_165 with basis formula_166 Symmetries. The golden ratio and inverse golden ratio formula_167 have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations formula_168 – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps formula_169 – they are reciprocals, symmetric about formula_170 and (projectively) symmetric about formula_171 More deeply, these maps form a subgroup of the  modular group formula_172 isomorphic to the symmetric group on formula_173 letters, formula_174 corresponding to the stabilizer of the set formula_175 of formula_173 standard points on the projective line, and the symmetries correspond to the quotient map formula_177 – the subgroup formula_178 consisting of the identity and the formula_173-cycles, in cycle notation formula_180 fixes the two numbers, while the formula_181-cycles formula_182 interchange these, thus realizing the map. 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 Lagrange's approximation theorem and it is an extremal case of the Hurwitz inequality for Diophantine approximations. This may be why angles close to the golden ratio often appear in phyllotaxis. The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with formula_9: The sequence of powers of formula_9 contains these values formula_185 formula_186 formula_187 formula_188 more generally, any power of formula_9 is equal to the sum of the two immediately preceding powers: As a result, one can easily decompose any power of formula_9 into a multiple of  formula_9 and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of formula_9: If formula_193 then: The golden ratio is a fundamental unit of the algebraic number field formula_194 and is a Pisot–Vijayaraghavan number. In the field formula_194 we have formula_196 where formula_197 is the formula_27-th Lucas number. When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed "phinary" or formula_9"-nary"), quadratic integers in the ring formula_200 – that is, numbers of the form formula_201 for formula_202 – have  terminating representations, but rational fractions have non-terminating representations. The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is formula_203 The golden ratio appears in the theory of modular functions as well. For formula_204, let Then and where formula_205 and formula_206 in the continued fraction should be evaluated as formula_207. The function formula_208 is invariant under  formula_209, a congruence subgroup of the modular group. Also for positive real numbers formula_210 and formula_211 then and For the gamma function, the only solutions to the equation are and . Decimal expansion. The golden ratio's decimal expansion can be calculated from the expression with formula_212 . The square root of formula_213 can be calculated via the Babylonian method, starting with an initial estimate such as formula_214 and iterating for formula_215 until the difference between formula_216 and formula_217 becomes zero to the desired number of digits. Then formula_218 The Babylonian algorithm for formula_20 is equivalent to Newton's method for solving  the equation formula_220 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 formula_221 for the root formula_222 Then formula_223 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 formula_224 In this case, formula_225 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 formula_6 the update step is The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute formula_27 digits of the golden ratio is proportional to the time needed to divide two formula_27-digit numbers. This is considerably faster than known algorithms for the transcendental numbers formula_229 and formula_230. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers formula_231 and formula_232 each over formula_233 digits, yields over formula_234 significant  digits of the golden ratio. The decimal expansion of the golden ratio formula_9 has been calculated to an accuracy of ten trillion digits. Pyramids. Both Egyptian pyramids and the regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios. Mathematical pyramids. A pyramid in which the apothem (slant height along the bisector of a face) is equal to formula_9 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 formula_237 times the semi-base (that is, the slope of the face is formula_237); the square of the height is equal to the area of a face, formula_9 times the square of the semi-base. The medial right triangle of this "golden" pyramid (see diagram), with sides formula_240 is interesting in its own right, demonstrating via the Pythagorean theorem the relationship formula_241 or formula_242 This Kepler triangle is the only right triangle proportion with  edge lengths in geometric progression, just as the formula_243 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent formula_237 corresponds to the angle that the side of the pyramid makes with respect to the ground, formula_245 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 formula_243 triangle; the face slope corresponding to the angle with tangent formula_247 is, to two decimal places, formula_248 The slant height or apothem is  formula_249 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, and the rational inverse slope (run/rise, multiplied by a factor of formula_250 to convert to their conventional units of palms per cubit) was used in the building of pyramids. Egyptian pyramids very close in proportion to these mathematical pyramids are known. 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 formula_251 is close to the "golden" pyramid inclination of formula_252 – and even closer to the formula_229-based pyramid inclination of formula_254 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 formula_251 slope. 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 formula_256 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. In 1859, the pyramidologist 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  formula_257 to the ground) to half the length of the side of the square base (equivalent to the secant of the angle formula_257). 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 half-base , yielding formula_259 for the ratio of slant height to half-base, 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 formula_243 pyramid, since the formula_243 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 formula_229 or formula_121 Example geometric problems of pyramid design in the Rhind papyrus correspond to various rational slopes. Michael Rice 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). 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. Disputed observations. Examples of disputed observations of the golden ratio include the following: The Parthenon. The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens  is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation." Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied." From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries. Later sources  like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions. Modern art. The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism. Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat. The Cubists observed in its harmonies, geometric structuring of motion and form, the primacy of idea over nature , an absolute scientific clarity of conception. However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 "Salon de la Section d'Or" exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not, and Marcel Duchamp said as much in an interview. On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition. Art historian Daniel Robbins has argued that in addition to referencing the mathematical term , the exhibition's name also refers to the earlier "Bandeaux d'Or" group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved. Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings, though other experts (including critic Yve-Alain Bois) have discredited these claims.