Manyworks of art are claimed to have been designed usingthe golden ratio.However, many of these claims are disputed, or refuted by measurement.[1]
Thegolden ratio, anirrational number, is approximately 1.618; it is often denoted by theGreek letter φ (phi).
Various authors have claimed that early monuments havegolden ratio proportions, often on conjectural interpretations, using approximate measurements, and only roughly corresponding to 1.618.[1] For example, claims have been made about golden ratio proportions in Egyptian, Sumerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from the late Bronze Age. These predate by some 1,000 years the Greek mathematicians first known to have studied the golden ratio.[2][3] However, the historical sources are obscure, and the analyses are difficult to compare because they employ differing methods.[2]
It is claimed, for instance, thatStonehenge (3100 BC – 2200 BC) has golden ratio proportions between its concentric circles.[2][4] Kimberly Elam proposes this relation as early evidence of human cognitive preference for the golden ratio.[5] However, others point out that this interpretation of Stonehenge "may be doubtful" and that the geometric construction that generates it can only be surmised.[2] As another example,Carlos Chanfón Olmos states that the sculpture of KingGudea (c. 2350 BC) has golden proportions between all of its secondary elements repeated many times at its base.[3]
TheGreat Pyramid of Giza (constructed c. 2570 BC byHemiunu) exhibits the golden ratio according to variouspyramidologists, including Charles Funck-Hellet.[3][6] John F. Pile, interior design professor and historian, has claimed that Egyptian architects sought the golden proportions without mathematical techniques and that it is common to see the 1.618:1 ratio, along with many other simpler geometrical concepts, in their architectural details, art, and everyday objects found in tombs. In his opinion, "That the Egyptians knew of it and used it seems certain."[7]
From before the beginning of these theories, other historians and mathematicians have proposed alternative theories for the pyramid designs that are not related to any use of the golden ratio, and are instead based on purely rational slopes that only approximate the golden ratio.[8] The Egyptians of those times apparently did not know thePythagorean theorem; the only right triangle whose proportions they knew was the 3:4:5 triangle.[9]
TheAcropolis of Athens (468–430 BC), including theParthenon, according to some studies, has many proportions that approximate the golden ratio.[10] Other scholars question whether the golden ratio was known to or used by Greek artists and architects as a principle of aesthetic proportion.[11] Building the Acropolis is calculated to have been started around 600 BC, but the works said to exhibit the golden ratio proportions were created from 468 BC to 430 BC.
The Parthenon (447–432 BC), was a temple of theGreek goddessAthena. The Parthenon's facade as well as elements of its facade and elsewhere are claimed to be circumscribed by a progression ofgolden rectangles.[12] Some more recent studies dispute the view that the golden ratio was employed in the design.[1][11][13]
Hemenway claims that the Greek sculptorPhidias (c. 480–c. 430 BC) used the divine proportion in some of his sculptures.[14] He createdAthena Parthenos inAthens andStatue of Zeus (one of theSeven Wonders of the Ancient World) in theTemple of Zeus atOlympia. He is believed to have been in charge of other Parthenon sculptures, although they may have been executed by his disciple or peers. In the early 20th century, American mathematicianMark Barr proposed theGreek letterphi (φ), the first letter of Phidias's name, to denote the golden ratio.[15]
Lothar Haselberger claims that theTemple of Apollo inDidyma (c. 334 BC), designed by Daphnis of Mileto and Paionios of Ephesus, has golden proportions.[3]
It is claimed that the upper level of 21 rows and the lower level of 34 rows of theAncient Theatre of Epidaurus form an approximation of the Golden number since 21 and 34 are successiveFibonacci numbers with their ratio at and a careful examination of the theatre's center reveals two back-to-back triangles balanced by the Golden number.[16]
Between 1950 and 1960, Manuel Amabilis applied some of the analysis methods ofFrederik Macody Lund andJay Hambidge in several designs of prehispanic buildings, such asEl Toloc andLa Iglesia de Las Monjas (the Nuns Church), a notable complex ofTerminal Classic buildings constructed in thePuuc architectural style atChichen Itza. According to his studies, their proportions are concretized from a series of polygons, circles and pentagrams inscribed, asLund found in his studies of Gothic churches. Manuel Amabilis published his studies along with several self-explanatory images of otherpre-columbian buildings made with golden ratio proportions inLa Arquitectura Precolombina de Mexico.[17] The work was awarded the gold medal and the title ofAcademico by theReal Academia de Bellas Artes de San Fernando (Spain) in theFiesta de la Raza (Columbus Day) of 1929.
The Castle of Chichen Itza was built by theMaya civilization between the 11th and 13th centuries AD as a temple to the godKukulcan. John Pile claims that its interior layout has golden ratio proportions. He says that the interior walls are placed so that the outer spaces are related to the central chamber by the golden ratio.[18]
TheGreat Mosque of Kairouan (built byUqba ibn Nafi c. 670 C.E.) uses the golden ratio in the design including its plan, the prayer space, court, and minaret,[19] but the ratio does not appear in the original parts of the mosque.[20]
The Stupa ofBorobudur inJava,Indonesia (built eighth to ninth century AD), the largest known Buddhist stupa, has the dimension of the square base related to the diameter of the largest circular terrace as 1.618:1, according to Pile.[21]
TheRomanesque style of architecture prevailed in Europe between 900 and 1200, a period which ends with the transition toGothic architecture. The contrast between Romanesque and Gothic concepts in religious buildings can be understood in the epistolary betweenSt. Bernard,Cistercian, and theAbbot Suger of the order ofCluny, the initiator ofGothic art inSt. Denis.
One of the most beautiful works of Romanesque Cistercian is theSénanque Abbey in Provence. TheSénanque abbatial was founded in 1148 and consecrated in 1178. It was initiated in life ofSt Bernard of Clairvaux."La Lumière à Sénanque" (The Light in Sénanque),[22]a chapter ofCîteaux : commentarii cistercienses, a publication of theCistercian Order. Its author, Kim Lloveras i Montserrat, made in 1992 a complete study of the abbatial, and argues that the abbatial church was designed using a system of measures founded in the golden ratio, and that the instruments used for its construction were the "Vescica" and the medieval squares used by the constructors, both designed with the golden ratio. The "Vescica" of Sénanque is located in the cloister of the monastery, in front of the Chapter, the site of the workshop.
In his 1919 bookAd Quadratum,Frederik Macody Lund, a historian who studied the geometry of several Gothic structures, claims that theCathedral of Chartres (begun in the 12th century), theNotre-Dame of Laon (1157–1205), and theNotre-Dame de Paris (1160) are designed according to the golden ratio.[3] Other scholars argue that until Luca Pacioli's 1509De Divina Proportione (see next section), the golden ratio was unknown to artists and architects, although this is not likely the case since the ratio was explicitly defined by Euclid.[11]
A 2003 conference on medieval architecture resulted in the bookAd Quadratum: The Application of Geometry to Medieval Architecture. According to a summary by one reviewer:
Most of the contributors consider that the setting out was done ad quadratum, using the sides of a square and its diagonal. This gave an incommensurate ratio of [square root of (2)] by striking a circular arc (which could easily be done with a rope rotating around a peg). Most also argued that setting out was done geometrically rather than arithmetically (with a measuring rod). Some considered that setting out also involved the use of equilateral or Pythagorean triangles, pentagons, and octagons. Two authors believe the Golden Section (or at least its approximation) was used, but its use in medieval times is not supported by most architectural historians.[23]
The Australian architectural historian John James made a detailed study of the Cathedral of Chartres. In his workThe Master Masons of Chartres he says that Bronze, one of the master masons, used the golden ratio. It was the same relation as between the arms of their metal square:
Bronze by comparison was an innovator, in practical rather than in philosophic things. Amongst other things Bronze was one of the few masters to use the fascinating ratio of the golden mean. For the builder, the most important function Fi, as we write the golden mean, is that if the uses is consistently he will find that every subdivision, no matter how accidentally it may have been derived, will fit somewhere into the series. Is not too difficult a ratio to reproduce, and Bronze could have had the two arms of his metal square cut to represent it. All he would than have had to do was to place the square on the stone and, using the string draw between the corners, relate any two lengths by Phi. Nothing like making life easy.[24]
De divina proportione, written byLuca Pacioli in Milan in 1496–1498, published in Venice in 1509,[25] features 60 drawings byLeonardo da Vinci, some of which illustrate the appearance of the golden ratio in geometric figures. Starting with part of the work of Leonardo da Vinci, this architectural treatise was a major influence on generations of artists and architects.
Vitruvian Man, created by Leonardo da Vinci around the year 1492,[26] is based on the theories of the man after which the drawing takes its name,Vitruvius, who inDe Architectura: The Planning of Temples (c. I BC) pointed that the planning of temples depends on symmetry, which must be based on the perfect proportions of the human body. Some authors feel there is no actual evidence that Da Vinci used the golden ratio inVitruvian Man;[27] however, Olmos[3] (1991) observes otherwise through geometrical analysis. He also proposesLeonardo da Vinci'sself portrait,Michelangelo'sDavid (1501–1504),Albrecht Dürer'sMelencolia I and the classicviolin design by the masters of Cremona (Guarneri,Stradivari and several members of theAmati family) as having similar regulator lines related to the golden ratio.
Da Vinci'sMona Lisa (c. 1503–1506) "has been the subject of so many volumes of contradicting scholarly and popular speculations that it virtually impossible to reach any unambiguous conclusions" with respect to the golden ratio, according to Livio.[11]
TheTempietto chapel at theMonastery of Saint Peter inMontorio, Rome, built byBramante, has relations to the golden ratio in its elevation and interior lines.[28]
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José Villagrán García has claimed[29] that the golden ratio is an important element in the design of theMexico City Metropolitan Cathedral (circa 1667–1813). Olmos claims the same for the design of the cities ofCoatepec (1579), Chicoaloapa (1579) and Huejutla (1580), as well as theMérida Cathedral, the Acolman Temple,Christ Crucified byDiego Velázquez (1639) andThe Immaculate Conception byBartolomé Esteban Murillo.[3]
Matila Ghyka[30] and others[31] contend thatGeorges Seurat used golden ratio proportions in paintings likeParade de cirque,Le Pont de Courbevoie, andBathers at Asnières. However, there is no direct evidence to support these claims.[27]
While the golden ratio appears to govern the geometric structure of Seurat'sParade de cirque (Circus Sideshow),[32][33] modern consensus among art historians is that Seurat never used this "divine proportion" in his work.[34][35][36]
The final study ofParade, executed prior to the oil on canvas, is divided horizontally into fourths and vertically into sixths (4 : 6 ratio) corresponding to the dimensions of the canvas, which is one and one-half times wider than its vertical dimension. These axes do not correspond precisely to the golden section, 1 : 1.6, as might have been expected. Rather, they correspond to basic mathematical divisions (simple ratios that appear to approximate the golden section), as noted by Seurat with citations from the mathematician, inventor, estheticianCharles Henry.[34]
The idea of theSection d'Or (or Groupe de Puteaux) originated in the course of conversations betweenAlbert Gleizes,Jean Metzinger andJacques Villon. The group's title was suggested by Villon, after reading a 1910 translation ofLeonardo da Vinci'sA Treatise on Painting byJoséphin Péladan. Péladan attached greatmystical significance to thegolden section (French:nombre d'or), and other similar geometric configurations. For Villon, this symbolized his belief in order and the significance of mathematical proportions, because it reflected patterns and relationships occurring in nature. Jean Metzinger and the Duchamp brothers were passionately interested in mathematics. Jean Metzinger,Juan Gris and possiblyMarcel Duchamp at this time were associates ofMaurice Princet, an amateur mathematician credited for introducing profound and rational scientific arguments into Cubist discussions.[37] The name 'Section d'Or' represented simultaneously a continuity with past traditions and current trends in related fields, while leaving open future developments in the arts.[38][39]
The Sacrament of the Last Supper (1955): The canvas of thissurrealist masterpiece bySalvador Dalí is a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[11][40]
Some works in the Dutch artistic movement calledDe Stijl, or neoplasticism, exhibit golden ratio proportions.Piet Mondrian used the golden section extensively in his neoplasticist, geometrical paintings, created circa 1918–38.[31][41] Mondrian sought proportion in his paintings by observation, knowledge and intuition, rather than geometrical or mathematical methods.[42]
TheFarnsworth House, designed byLudwig Mies van der Rohe, has been described as "the proportions, within the glass walls, approach 1:2"[43] and "with a width to length ratio of 1:1.75 (nearly the golden section)"[44] and has been studied with his other works in relation to the golden ratio.[45]
The Swiss architectLe Corbusier, famous for his contributions to themoderninternational 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 theFibonacci number, 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."[46]
Le Corbusier explicitly used the golden ratio in his system for thescale ofarchitectural proportion. He saw this system as a continuation of the long tradition ofVitruvius, Leonardo da Vinci'sVitruvian Man, the work ofLeon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function ofarchitecture. In addition to the golden ratio, Le Corbusier based the system onhuman measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in theModulor system.[47]
InThe Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and Mechanics Le Corbusier reveals he used his system in the MarseillesUnité d'habitation (in the general plan and section, the front elevation, plan and section of the apartment, in the woodwork, the wall, the roof and some prefabricated furniture), a small office in 35 rue de Sèvres, a factory in Saint-Die and theUnited Nations Headquarters building in New York City.[48] Many authors claim that the shape of the facade of the second is the result of three golden rectangles;[49] however, each of the three rectangles that can actually be appreciated have different heights.
Catalan architectJosep Lluis Sert, a disciple ofLe Corbusier, applied the measures of theModulor in all his particular works, including the Sert's House in Cambridge[50] and theJoan Miró Foundation in Barcelona.[51]
According to the official tourism page ofBuenos Aires,Argentina, the ground floor of thePalacio Barolo (1923), designed by Italian architectMario Palanti, is built according to the golden ratio.[52]
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 inOriglio, the golden ratio is the proportion between the central section and the side sections of the house.[53]
Ernő Lendvai analyzesBéla Bartók's works as being based on two opposing systems, that of the golden ratio and theacoustic scale,[54] though other music scholars reject that analysis.[11]
The musicologistRoy Howat has observed that the formal boundaries of Debussy'sLa mer correspond exactly to the golden section.[55] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[56]
Leonid Sabaneyev hypothesizes that the separate time intervals of the musical pieces connected by the "culmination event", as a rule, are in the ratio of the golden section.[57] However, the author attributes this incidence to the instinct of the musicians: "All such events are timed by author's instinct to such points of the whole length that they divide temporary durations into separate parts being in the ratio of the golden section."
Ron Knott[58] exposes how the golden ratio is unintentionally present in several pieces of classical music:
According to author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involvedFibonacci numbers and the related golden ratio."[61]
With few exceptions, numerators for the meter signatures (over 100) inKarlheinz Stockhausen'sKlavierstück IX are either Fibonacci or Lucas numbers.[62]
{{cite web}}: CS1 maint: archived copy as title (link)The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organisation. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period.
Farnsworth House golden.
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