- List of works designed with golden ratio
Works designed with the golden ratio are works of human design that are proportioned according to the
golden ratio, an irrational numberthat is approximately 1.618; it is often denoted by the Greek letter φ (phi).
It is claimed that
Stonehenge(3100 BC – 2200 BC) has golden ratio proportions bewteen its concentric circles. [Prash Trivede. "The 27 Celestial Portals: The Real Secret Behind the 12 Star-Signs". Lotus Press. Page 397] Kimberly Elam proposes this relation as early evidence of human cognitive preference for the golden ratio. [Kimberly Elam. "Geometry of Design: Studies in Proportion and Composition By Kimberly Elam". Princeton Architectural Press. p. 6.] However, others point out that this interpretation of Stonehenge "may be doubtful" and that the geometric construction that generates it can only be surmised.cite book | title = Symmetries of Nature: A Handbook for Philosophy of Nature and Science | author = Klaus Mainzer | publisher = Walter de Gruyter | year = 1996 | isbn = 3110129906 | pages = 118 | url = http://books.google.com/books?id=rqzaQo6CaA0C&pg=PA118&dq=stonehenge+golden-section&lr=&as_brr=3&ei=MY_cR9H5HKiwtAP_26WJDg&sig=zTEHmqsZD1uUuqvofxzlSYrIKNY ]
Various authors discern golden ratio proportions in Egyptian, Summerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from the late Bronze age, which predates by about 1,000 years the Greek mathematicians who were first known to have studied the golden ratio. However, the historical sources are obscure, and the analyses are difficult to compare because they employ differing methods.
Great Pyramid of Giza(constructed c. 2570 BC by Hemiunu) exhibits the golden ratio according to various pyramidologists, including Charles Funck-Hellet.Lidwell, William; Holden, Kritina; and Butler, Jill. "Universal Principles of Design". Rockport Publishers. October 1, 2003. Page 96 ] 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 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." [Pile, John F. " [http://books.google.com/books?id=YVQJvcI1XeoC&pg=PA26&dq=intitle:%22interior+design%22+inauthor:pile+%22everyday+objects%22&lr=&as_brr=0&ei=I1LfR63LLY6IswOYn4nvAQ&sig=o55FhLnB7odaYpCVuWP7wA7RfE8 A history of interior design] ". Laurence King Publishing. 2005. Page 29.]
Some historians and mathematicians propose 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.Eli Maor, " [http://books.google.com/books?id=xU0IPNGRXqEC&pg=PA7&dq=intitle:trigonometric+inauthor:maor+seked+rhind&lr=&as_brr=0&ei=GYo5SJHPCJnstAPNk5S8Cw&sig=dCMf0wTbtg6KwHNbxMS-2Ra_yDc Trigonometric Delights] ", Princeton Univ. Press, 2000] The Egyptians of those times apparently did not know the
Pythagorean theorem; the only right triangle whose proportions they knew was the 3:4:5 triangle. [Eric Temple Bell, "The Development of Mathematics", New York: Dover, 1940, p.40]
Carlos Chanfon Olmos says that the Sculpture of King
Gudea(c. 2350 BC) clearly has golden proportions between all of its secondary elements repeated many times at its base.
Acropolis of Athens(468–430 BC), including the Parthenon, according to some studies, has many proportions that approximate the golden ratio.cite journal
url = http://www.questia.com/PM.qst?a=o&se=gglsc&d=5001403053
title = Rhetorical Prototypes in Architecture: Measuring the Acropolis
author = Van Mersbergen, Audrey M.
journal = Communication Quarterly
volume = 46
issue = 2
pages = 194–195
year = 1998
publisher = Eastern Communication Association] Other scholars question whether the golden ratio was known to or used by Greek artists and architects as a principle of aesthetic proportion. 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. [Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis", "Philosophical Polemic Communication Quarterly", Vol. 46, 1998.] Some more recent studies dispute the view that the golden ratio was employed in the design.cite journal
doi = 10.2307/2686193
author = Markowsky, George
journal = The College Mathematics Journal
volume = 23
issue = 1
pages = 2–19
year = 1992
month = January
title = Misconceptions About the Golden Ratio
accessdate = 2008-06-30]
The Greek sculptor
Phidias(c. 480–c. 430 BC) used the divine proportion in some of his sculptures, according to Hemenway.cite book
last = Hemenway
first = Priya
title = Divine Proportion: Phi In Art, Nature, and Science
year = 2005
publisher = Sterling
location = New York
isbn = 1-4027-3522-7
pages = p.96
url = http://books.google.com/books?id=akjSa4AdK4AC&pg=PA96&dq=inauthor:Hemenway+phidias+divine&lr=&as_brr=3&ei=EpvcR_TVIabutAO5hMjwAQ&sig=DXfN9iQeiqf8HqeohZ1Oz_5qRRk#PPA96,M1 ] He created "
Athena Parthenos" in Athensand "Statue of Zeus" (one of the Seven Wonders of the Ancient World) in the Temple of Zeusat 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 Barrproposed using the Greek letter phi (φ), the first letter of Phidias's name, to denote the golden radio. [Cook, Theodore Andrea (1979). "The Curves of Life", p. 420. Courier Dover Publications, ISBN 0-486-23701-X.]
Prehispanic Mesoamerican architecture
Carlos Chanfon Olmos, researcher of the
UNAM, exposed in his "Curso de Proporciòn", the presence of the golden ratio in a series of olmecheads, the Aztec calendar stoneand a series of Aztec permission house plans. In the fifties, Manuel Amabilis applied some of the analysis methods of Frederik Macody Lundand Jay Hambidgeto several plans and sections of prehispanic buildings, such as "El Toloc" and "La Iglesia" of "Las Monjas", a notable complex of Terminal Classic buildings constructed in the Puucarchitectural style at Chichen Itza. According to his studies, their proportions derived from a series of successively inscribed pentagons, circles and pentagrams, just as the Gothic churches Lund studied do. Amabilis published his studies along with several self-explanatory images of various other precolumbine buildings with golden proportions in "La Arquitectura Precolombina de Mexico" [ [http://books.google.com.mx/books?um=1&hl=en&q=manuel+amabilis Manue Amabilis] . (1956) "La Arquitectura Precolombina en Mexico". Editorial Orion. P. 200, 202. [http://www.antiqbook.com/boox/dailey/5479.shtml] ] , 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. According to John Pile, The Castle of Chichen Itza, built by the Maya civilizationsometime 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. [PILE, John F. " [http://books.google.com/books?id=YVQJvcI1XeoC&pg=PA23&dq=chichen-itza++intitle:A+intitle:history+intitle:of+intitle:interior+intitle:design&lr=&as_brr=0&ei=-CvfR5vDF5u8swPmxOXzAQ&sig=4FQS1K4hAjcgqmGBNET_BleGQIE#PPA23,M1 A history of interior design] ". Laurence King Publishing. 2005. Page 23.]
A geometrical analysis of the Great Mosque of Kairouan (built by
Uqba ibn Nafic. 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. [Kenza Boussora and Said Mazouz, " [http://www.nexusjournal.com/BouMaz.html 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]
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. [PILE, John F. "A history of interior design ". Laurence King Publishing. 2005. Page 88.]
In his 1919 book "Ad Quadratum",
Frederik Macody Lund, a historian who studied the geometry of several gothic structures, claims that the Cathedral of Chartres(begun in the 12th century), the Notre-Dame of Laon(1157–1205), and the Notre Dame de Paris(1160) are designed according to the golden ratio. Other scholars argue that until Pacioli's 1509 publication (see next section), the golden ratio was unknown to artists and architects.
A 2003 conference on medieval architecture resulted in the book "Ad 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. [cite journal | title = The geometry of Romanesque and Gothic cathedrals. (Ad Quadratum: The Application of Geometry to Medieval Architecture) (Book Review) | journal = Architectural Science Review | volume = 46 | issue = 3 | pages = pp. 337–338 | date = September 1, 2003]
De divina proportione", written by Luca Pacioliin Milan in 1496–1498, published in Venice in 1509,Pacioli, Luca. De Divina Proportione. Venice, 1509. ] 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. Vitruvian Man, created by Leonardo da Vinci around the year 1492, [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] 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. Some authors feel there is no actual evidence that Da Vinci used the golden ratio in "Vitruvian Man";cite web | title = Good stories, pity they're not true | date = June 2004 | work = MAA Online | publisher = Mathematical Association of America | author = Keith Devlin | url = http://www.maa.org/devlin/devlin_06_04.html ] however, Chanfon (1991) observes otherwise through geometrical analysis. He also proposes Leonardo da Vinci's self portrait, Michaelangelo's David (1501–1504), Albrecht Durer's Melencoliaand the classic violindesign by the Masters of Cremona, as having similar regulator lines related to the golden ratio.
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.
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. [PILE, John F. "A history of interior design ". Laurence King Publishing. 2005. Page 130.]
The Baroque and the Spanish empire
Jose Villagran Garciahas claimedVILLAGRAN GARCIA, Jose. "Los Trazos Reguladores de la Proporcion Arquitectonica". Memoria de el Colegio Nacional, Volume VI, No. 4, Editorial de El Colegio Nacional, Mexico, 1969] CHANFÓN OLMOS, Carlos. "Curso sobre Proporción. Procedimientos reguladors en construcción". Convenio de intercambio UNAM– UADY. México - Mérica, 1991 ] that the golden ratio is an important element in the design of the Mexico City Metropolitan Cathedral(circa 1667–1813). Carlos Chaflon Olmosclaims the same for the design of the cities of Coatepec(1579), Chicoaloapa (1579) and Huejutla (1580), as well as the Merida Cathedral, the Acolman Temple, "Cristo Crucificado" by Diego Velazquez(1639) and La Madona de Media Luna of Bartolomé Esteban Murillo.
Neoclassicism and romanticism
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%). [SABANEEV, Leonid and JOFFE, Judah A. "Modern Russian Composers". 1927.] 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." In Surrey's Internet site, Ron Knott [KNOTT, Ron, " [Ron Knott's web pages on Mathematic] ", " [http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html Fibonacci Numbers and The Golden Section in Art, Architecture and Music] ", Surrey University] exposes how the golden ratio is unintentionally present in several pieces of classicel music:
*An article of "American Scientist" [MAY, Mike, "Did Mozart use the Golden mean?", "American Scientist", March/April 1996] ("Did Mozart use the Golden mean?", March/April 1996), reports that John Putz found that there was considerable deviation from ratio section division in many of Mozart's sonatas and claimed that any proximity to this number can be explained by constraints of the sonata form itself.
*Derek Haylock [HEYLOCK, Derek. "Mathematics Teaching, Volume 84", p. 56-57. 1978] claims that the opening motto of
Ludwig van Beethoven's Symphony No. 5in C minor, Op. 67 (c. 1804–08), 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.
Matila Ghyka[GHYKA, Matila. "The Geometry of Art and Life". 1946. Page 162] and othersSTASZKOW, Ronald and BRADSHAW, Robert. "The Mathematical Palette". Thomson Brooks/Cole. P. 372] contend that Georges-Pierre Seuratused golden ratio proportions in paintings like "La Parade", "Le Pont de Courbevoie" and " Une Baignade, Asnières". However, there is no direct evidence to support these claims.
According to the official tourism page of
Buenos Aires, Argentina, the ground floor of the Palacio Barolo(1923), designed by Italian architect Mario Palanti, is built according to the golden section. [http://www.bue.gov.ar/recorridos/index.php?menu_id=52&info=auto_contenido&lang=en Official tourism page of the city of Buenos Aires] ]
Henri Poincaré, taught the properties of the golden ratio to Juan Gris, who developed Cubismfeaturing them. [BALMORI, Santos, "Aurea mesura", UNAM, 1978, 189 p. P. 23-24.]
The Sacrament of the Last Supper" (1955): The canvas of this surrealist masterpiece by Salvador Daliis a golden rectangle. A huge dodecahedron, with edges 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, ISBN 1-883001-51-X] cite book|last=Livio|first=Mario|year=2002|title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number|publisher=Broadway Books|location=New York|id=ISBN 0-7679-0815-5]
Some works in the Dutch artistic movement called
De Stijl, or neoplasticism, exhibit golden ratio proportions. Piet Mondrianused the golden section extensively in his neoplasticist, geometrical paintings, created circa 1918–38. [Bouleau, Charles, "The Painter's Secret Geometry: A Study of Composition in Art" (1963) pp. 247-48, Harcourt, Brace & World, ISBN 0-87817-259-9] Mondrian sought proportion in his paintings by observation, knowledge and intuition, rather than geometrical or mathematical methods. [PADOVAN, Richard. "Proportion: Science, Philosophy, Architecture". Taylor & Francis. Page 26.] Juan Grisalso used golden ratio proportions.
Mies Van der Rohe
Farnsworth Househas been described as "the proportions, within the glass walls, approach 1:2" [cite book | title = The Modern Steel House | author = Neil Jackson | publisher = Taylor & Francis | year = 1996 | ibsn = 0419217207 | url = http://books.google.com/books?id=3pyDEb9RRPwC&pg=PA71&dq=%22Farnsworth+House%22+golden&ei=3KTdR4OgLIrysgP6yMXqAQ&sig=SRQygYWI50dtLCaOXRf7KOwOyQw#PPA71,M1 ] and "with a width to length ratio of 1:1.75 (nearly the golden section)" [cite book | title = American Architecture: A History | author = Leland M. Roth | publisher = Westview Press | year = 2001 | ibsn = 0813336619 | url = http://books.google.com/books?id=pH7rd6EFImgC&pg=PA433&dq=%22Farnsworth+House%22+golden&ei=3KTdR4OgLIrysgP6yMXqAQ&sig=IkeDUUCZ6ruL_SMAEkQNEhdqTaE ] and has been studied with his other works in relation to the golden ratio.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 / ," ]
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, "The Modulor" p. 25, as cited in Padovan, Richard, "Proportion: Science, Philosophy, Architecture" (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6] 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 Modulorsystem.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".]
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. 154-5, ISBN 0-415-30685-X]
James Tenneyreconceived 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 temperedminor 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 Lendvaianalyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, [Lendvai, Ernő (1971). "Béla Bartók: An Analysis of His Music". London: Kahn and Averill.] though other music scholars reject that analysis. 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.Smith, Peter F. " [http://books.google.com/books?id=ZgftUKoMnpkC&pg=PA83&dq=bartok+intitle:The+intitle:Dynamics+intitle:of+intitle:Delight+intitle:Architecture+intitle:and+intitle:Aesthetics&as_brr=0&ei=WkkSR5L6OI--ogLpmoyzBg&sig=Ijw4YifrLhkcdQSMVAjSL5g4zVk The Dynamics of Delight: Architecture and Aesthetics] " (New York: Routledge, 2003) pp 83, ISBN 0-415-30010-X] French composer Erik Satieused 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."
The musicologist Roy Howat has observed that the formal boundaries of "La Mer" correspond exactly to the golden section. [cite book | title = Debussy in Proportion: A Musical Analysis | author = Roy Howat | url = http://books.google.com/books?id=4bwKykNp24wC&pg=PA169&dq=intitle:Debussy+intitle:in+intitle:Proportion+golden+la-mer&lr=&as_brr=0&ei=KFKlR5b5O4bOiQGQt82pCg&sig=oBWbHkWkhTG11w_BNdx89SWjQTY#PPA169,M1 | publisher = Cambridge University Press | year = 1983 | isbn = 0521311454 ] Trezise finds the intrinsic evidence "remarkable," but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions. [cite book | title = Debussy: La Mer | author = Simon Trezise | publisher = Cambridge University Press | year = 1994 | isbn = 0521446562 | pages = p.53 | url = http://books.google.com/books?id=THD1nge_UzcC&pg=PA53&dq=inauthor:Trezise+golden+evidence&lr=&as_brr=0&ei=DlSlR-7HJJXEigGR2sS5Cg&sig=5auw0tRu24Jq0aFKOjLyZ2u7BGo ]
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 [http://stage6.divx.com/BT/show_video/1051714 accompanying video] displays various animated versions of the golden mean. Pearl Drumspositions 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 patenton this innovation. [cite web
url = http://www.pearldrum.com/premium-birch.asp
title = Pearl Masters Premium
accessmonthday =Dec. 2
publisher = Pearl Corporation
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." [cite book | title = The Math Behind the Music | author = Leon Harkleroad | publisher = Cambridge University Press | year = 2006 | isbn = 0521810957 | url = http://books.google.com/books?id=C3dsb7Qysh4C&pg=RA4-PA120&dq=misguided+music+mathematics+%22golden+ratio%22&lr=&as_brr=0&ei=9GJwR7m-HIbktAO_-ayeBw&sig=h_YrTJ6LYBsfmhjexiQvmmPZFFM ]
* [http://www.emis.de/journals/NNJ/Frings.html Nexux Network Journal – Architecture and Mathematics Online.] Kim Williams Books
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