List of works designed with golden ratio

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This is a chronological list of works of art apparently proportioned according to the golden ratio, an irrational number that is approximately 1.618; it is often denoted by the Greek letter φ (phi).

The list is long – as Keith Devlin, director of the Stanford Center for the Study of Language and Information, explains, "not only is GR a very special number mathematically – all of its genuine appearances in mathematics and Nature show that – it also has enormous cultural significance as the number that most people have the greatest number of false beliefs about."[1]

Contents

[edit] Early history

It is claimed that Stonehenge (3100 BC – 2200 BC) has its concentric circles proportionally based on the golden proportion.[2][3] Some authors, like Kimberly Elam, propose this relation as a documented early evidence of human cognitive preference for the golden proportion.[4] However, others point out that this interpretation of Stonehenge is "doubtful".[5]

The Great Pyramid of Giza (constructed c. 2570 BC by Hemiunu) exhibits the golden ratio according to various pyramidologists.[6] John F. Pile, interior design professor and historian, has claimed that Egyptian designers sought the golden proportions without mathematical techniques and that it is common to see the 0.618:1 proportion, along with many other simpler geometrical concepts, in their architectural details, art, and everyday objects found in tombs, and states "That the Egyptians knew of it and used it seems certain."[7]

Some historians and mathematicians state that Egyptian mathematics would not have supported the calculations necessary to build pyramids with irrational slopes[8] (for example, they did not know the Pythagorean theorem and the only right triangle shape they were aware of numerically was the 3:4:5 triangle[9]) and the pyramid calculations found in ancient Egyptian documents were based on purely rational slopes close to but different from the golden ratio.[10]

[edit] Greece

The Parthenon west façade
The Parthenon west façade

The Acropolis of Athens (468–430 BC), including the Parthenon, according to some studies, has many proportions that approximate the golden ratio.[11] Other scholars question whether the golden ratio was known to or used by Greek artists and architects as a principle of aesthetic proportion.[12] Building the Acropolis is calculated to have been started around 6000 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 built on the Acropolis in the 5th century BC for the Greek goddess Athena. It is the most important surviving building of Classical Greece. The Parthenon's facade as well as elements of its facade and elsewhere can be circumscribed by a progression of golden rectangles.[13] Some more recent studies dispute the view that the golden ratio was employed in the design.[12][14]

The Greek sculptor Phidias (c. 480–c. 430 BC) used the divine proportion in some of his sculptures, according to Hemenway.[15] He created Athena Parthenos in Athens and Statue of Zeus (one of the Seven Wonders of the Ancient World) in the Temple of Zeus at Olympia. He is believed to have been in charge of other Parthenon sculptures, although they may have been executed by his alumni or peers. Many art historians conclude that Phidias made meticulous use of the golden ratio in proportioning his sculptures. For this reason, in the early 20th century, American mathematician Mark Barr proposed using the Greek letter phi (φ), the first letter of Phidias's name, to denote the golden radio.[16]

[edit] Classic Mayan architecture

According to Manuel Amabilis, who in the fifties applied some of the analysis methods of Frederik Macody Lund and Jay Hambidge to prehispanic buildings such as the plan and section of El Toloc and La Iglesia, buildings of Las Monjas, a notable complex of Terminal Classic buildings constructed in the Puuc architectural style at Chichen Itza, these have proportions derived from a series of successively inscribed pentagons, circles and pentagrams, just as do several Gothic churches Lund studied. Amabilis published his studies along with several self-explanatory images of various other precolumbine buildings with golden proportions in La Arquitectura Precolombina de Mexico[17], which was awarded the gold medal and the title of Academico by the "Real Academia de Bellas Artes de San Fernando" (Spain) in the "Fiesta de la Raza" contest of 1929.

[edit] Islamic architecture

A geometrical analysis of the Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 A.D.) reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz, who say it is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret.[18]

Panorama of the minaret and the courtyard (on the right)
Panorama of the minaret and the courtyard (on the right)

[edit] Buddhist architecture

The Stuppa of Borobudur in Java, 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.[19]

[edit] Classic Mayan architecture

The Castle of Chichen Itza, built by the Maya civilization sometime between the 11th and 13th centuries AD to serve as a temple to the god Kukulcan, has golden proportions in its interior layout with walls placed so that the outer spaces relate to the center chamber as 0.618:1, according to Pile.[20]

[edit] Gothic era

The regulator lines that, according to Macody Lund, give the Cathedral of Laon golden proportions
The regulator lines that, according to Macody Lund, give the Cathedral of Laon golden proportions
The regulator lines that give, according to Macody Lund, Notre Dame of Paris golden proportions
The regulator lines that give, according to Macody Lund, Notre Dame of Paris golden proportions

In his 1919 book Ad Quadratum, Frederik Macody Lund, a historian who studied the geometry of several gothic structures by their geometry, claims that the Cathedral of Chartres (begun in the 12th century), the Cathedral of Notre-Dame in Laon (1157–1205), and the Notre Dame de Paris (1160) are designed according to the golden ratio.[21] Other scholars argue that until Pacioli's 1509 publication (see next section), the golden ratio was unknown to artists and architects.[12]

A 2003 conference on medieval architecture resulted in the book Ad Quadratum: The Application of Geometry to Medieval Architecture, a review of which summarizes:[22]

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.

[edit] Renaissance

Leonardo Da Vinci's illustration of a human head from De Divina Proportione
Leonardo Da Vinci's illustration of a human head from De Divina Proportione[23]

De divina proportione, written by Luca Pacioli in Milan in 1496–1498, published in Venice in 1509,[23] features 60 drawings by Leonardo 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.

Besides his illustrations in De Divina Proportione, golden proportions have been observed in several other works of Leonardo Da Vinci.[24] Vitruvian Man, created by Leonardo da Vinci around the year 1492,[25] is based on the theories of the man after which the drawing takes its name, Vitruvius, who in De 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. There is no actual evidence that Da Vinci used the golden ratio in Vitruvian Man, or in faces, however.[1]

Da Vinci's Mona 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.[12]

The Tempietto chapel at the Monastery of Saint Peter in Montorio, Rome, built by Bramante, has relations to the golden ratio in its elevation and interior lines.[26]

[edit] The baroque and the Spanish empire

Cristo Crucificado de Diego Velazquez (1639)
Cristo Crucificado de Diego Velazquez (1639)

Jose Villagran Garcia has claimed[21] that the golden ratio is an important part of the design of the Mexico City Metropolitan Cathedral (circa 1667–1813) and Carlos Chaflon Olmos claims the same for the Cristo Crucificado de Diego Velazquez (1639).

[edit] Romanticism

Symphony No. 5 in C minor, Op. 67 (c. 1804–08): Derek Haylock[27] claims that the opening motto of Ludwig van Beethoven's piece, occurs exactly at the golden mean point 0.618 in bar 372 of 601 and again at bar 228 which is the other golden section point (0.618034 from the end of the piece) but he has to use 601 bars to get these figures. This he does by ignoring the final 20 bars that occur after the final appearance of the motto and also ignoring bar 387.

Leonid Sabaneev 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 and that the greatest musical pieces based on the golden section meets in the works of Beethoven (97%), Gaidn (97%), Arensky (95%), Shopen (92%), Schubert (91%) and Mozart (91%).[28] However the author attributes this incidence to the instinct of the mucisians: "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."

[edit] Impressionism

Matila Ghyka[29] and others[30][24] point out that Georges-Pierre Seurat used golden proportions in paintings like La Parade, Le Pont de Courbevoie and Une Baignade, Asnières, but there is no direct evidence for these claims.[1]

[edit] Neogothic

According to the official tourism page of Buenos Aires, Argentina, the ground floor of the Palacio Barolo (1923), located at Avenida de Mayo 1370, and built by the Italian Architect Mario Palanti, is built according to the golden section.[31]

[edit] Surrealism

The Sacrament of the Last Supper‎ (1955): The dimensions of the canvas of this surrealist work of Spanish painter and intellectual, Salvador Dali, are a golden rectangle. A huge dodecahedron, with edges in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[32][12]

[edit] Neomayan Architecture

Among the many historicist and eclectic architecture movements in Mexico and Latin America, the Neomayan designs of Manuel Amabilis in Yucatan features several elements of his studies of Golden Proportion in Classic Mayan and Toltec architecture.

[edit] De Stijl

Various authors[30][24] point that Piet Mondrian used the golden section extensively in his neoplasticist, geometrical paintings. The related works were produced circa 1918–1938.[33] Mondrian, aware of such concepts of golden ratio, sought proportion in his paintings by observation, knowledge and intuition rather than geometrical or mathematical methods.[34]

Juan Gris also used golden proportions.[30]

[edit] Modern Architecture

[edit] Mies Van der Rohe

The Farnsworth House has been described as "the proportions, within the glass walls, approach 1:2"[35] and "with a width to length ratio of 1:1.75 (nearly the golden section)"[36] and has been studied with his other works in relation to the golden ratio.[37]

[edit] Le Corbusier

United Nations Headquarters (New York City, viewed from the East River), one of the clearest examples of the use golden ratio in architecture.
United Nations Headquarters (New York City, viewed from the East River), one of the clearest examples of the use golden ratio in 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."[38]

Modulor: Le Corbusier explicitly used the golden ratio in his system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system.[39]

He helped design the United Nations Headquarters in New York City, possibly using such proportions,[40] or perhaps not.[1]

[edit] Post-modern architecture

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.[41]

[edit] Contemporary music

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

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[42] though other music scholars reject that analysis.[12] In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.[43] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.

The golden ratio is also apparent in the organisation of the sections in the music of Debussy's Image, Reflections in Water, in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position."[43]

The musicologist Roy Howat has observed that the formal boundaries of La Mer correspond exactly to the golden section.[44] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[45]

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

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.[46]

According to author Leon Harkleroad, "Some of the most misguided attempts to link music and mathematics have involved Fibonacci numbers and the related golden ratio."[47]

[edit] References

  1. ^ a b c d Keith Devlin (June 2004). Good stories, pity they're not true. MAA Online. Mathematical Association of America.
  2. ^ TRIVEDE, Prash. The 27 Celestial Portals: The Real Secret Behind the 12 Star-Signs. Lotus Press. Page 397
  3. ^ MAINZER, Klaus. Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter. Page 118.
  4. ^ ELAM, Kimberly. Geometry of Design: Studies in Proportion and Composition By Kimberly Elam. Princeton Architectural Press. Page 6
  5. ^ Klaus Mainzer (1996). Symmetries of Nature: A Handbook for Philosophy of Nature and Science. Walter de Gruyter, 118. ISBN 3110129906. 
  6. ^ Lidwell, William; Holden, Kritina; and Butler, Jill. Universal Principles of Design. Rockport Publishers. October 1, 2003. Page 96
  7. ^ PILE, John F. A history of interior design. Laurence King Publishing. 2005. Page 29.
  8. ^ 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
  9. ^ Eric Temple Bell, The Development of Mathematics, New York: Dover, 1940, p.40
  10. ^ Eli Maor, Trigonometric Delights, Princeton Univ. Press, 2000
  11. ^ Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis", Philosophical Polemic Communication Quarterly, Vol. 46, 1998.
  12. ^ a b c d e f Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books. ISBN 0-7679-0815-5. 
  13. ^ Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis", Philosophical Polemic Communication Quarterly, Vol. 46, 1998.
  14. ^ George Markowsky, [http://www.math.nus.edu.sg/aslaksen/teaching/maa/markowsky.pdf "Misconceptions about the Golden Ratio." The College Mathematics Journal. Volume 23, No 1, January 1992.
  15. ^ Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling, p.96. ISBN 1-4027-3522-7. 
  16. ^ Cook, Theodore Andrea (1979). The Curves of Life, p. 420. Courier Dover Publications, ISBN 0-486-23701-X.
  17. ^ Manue Amabilis. (1956) La Arquitectura Precolombina en Mexico. Editorial Orion. P. 200, 202.[1]
  18. ^ Kenza Boussora and Said Mazouz, "The Use of the Golden Section in the Great Mosque of Kairouan", Nexus Network Journal, vol. 6 no. 1 (Spring 2004), pp. 7-16. DOI 10.1007/s00004-004-0002-y
  19. ^ PILE, John F. A history of interior design . Laurence King Publishing. 2005. Page 88.
  20. ^ PILE, John F. A history of interior design. Laurence King Publishing. 2005. Page 23.
  21. ^ a b CHAFLÓN OLMOS, Carlos. Curso sobre Proporción. Procedimientos reguladors en construcción. Convenio de intercambio UNAM–UADY. México - Mérica, 1991
  22. ^ "The geometry of Romanesque and Gothic cathedrals. (Ad Quadratum: The Application of Geometry to Medieval Architecture) (Book Review)" (September 1, 2003). Architectural Science Review 46 (3): pp.337–338. 
  23. ^ a b Pacioli, Luca. De Divina Proportione. Venice, 1509.
  24. ^ a b c WILLIAMS, Gareth. Linear Algebra With Applications. Jones & Bartlett Publishers. Page 309
  25. ^ TUBERVILLE, Joseph. A Glimmer of Light from the Eye of a Giant: Tabular Evidence of a Monument in Harmony with the Universe. 2001. Page 1
  26. ^ PILE, John F. A history of interior design . Laurence King Publishing. 2005. Page 130.
  27. ^ HEYLOCK, Derek. Mathematics Teaching, Volume 84, p. 56-57. 1978
  28. ^ SABANEEV, Leonid and JOFFE, Judah A. Modern Russian Composers. 1927.
  29. ^ GHYKA, Matila. The Geometry of Art and Life. 1946. Page 162
  30. ^ a b c STASZKOW, Ronald and BRADSHAW, Robert. The Mathematical Palette. Thomson Brooks/Cole. P. 372
  31. ^ Official tourism page of the city of Buenos Aires
  32. ^ Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X
  33. ^ Bouleau, Charles, The Painter's Secret Geometry: A Study of Composition in Art (1963) pp.247-8, Harcourt, Brace & World, ISBN 0-87817-259-9
  34. ^ PADOVAN, Richard. Proportion: Science, Philosophy, Architecture. Taylor & Francis. Page 26.
  35. ^ Neil Jackson (1996). The Modern Steel House. Taylor & Francis. 
  36. ^ Leland M. Roth (2001). American Architecture: A History. Westview Press. 
  37. ^ SANO, Junichi. Study on the Golden Ratio in the works of Mies van der Rolle : On the Golden Ratio in the plans of House with three Courts and IIT Chapel. Journal of Arch tecture, Planning and Environmental Engineering （Academic Journal ,1993 ） 453,153-158 / ,
  38. ^ Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6
  39. ^ Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: "Both the paintings and the architectural designs make use of the golden section".
  40. ^ Daniel Pedoe (1983). Geometry and the Visual Arts. Courier Dover Publications, p.121. ISBN 048624458X. 
  41. ^ Urwin, Simon. Analysing Architecture (2003) pp. 154-5, ISBN 0-415-30685-X
  42. ^ Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.
  43. ^ a b Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X
  44. ^ Roy Howat (1983). Debussy in Proportion: A Musical Analysis. Cambridge University Press. ISBN 0521311454. 
  45. ^ Simon Trezise (1994). Debussy: La Mer. Cambridge University Press, p.53. ISBN 0521446562. 
  46. ^ Pearl Masters Premium. Pearl Corporation. Retrieved on Dec. 2, 2007.
  47. ^ Leon Harkleroad (2006). The Math Behind the Music. Cambridge University Press. ISBN 0521810957. 

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