Many works of art are claimed to have been designed using the golden ratio. However, many of these claims are disputed, or refuted by measurement.

The golden ratio, an irrational number, is approximately 1.618; it is often denoted by the Greek letter φ (phi).

Early history

Various authors have claimed that early monuments have golden ratio proportions, often on conjectural interpretations, using approximate measurements, and only roughly corresponding to 1.618. 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. However, the historical sources are obscure, and the analyses are difficult to compare because they employ differing methods.

It is claimed, for instance, that Stonehenge (3100 BC – 2200 BC) has golden ratio proportions between its concentric circles.Trivede, Prash. 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. As another example, Carlos Chanfón Olmos states that the sculpture of King Gudea (c. 2350 BC) has golden proportions between all of its secondary elements repeated many times at its base.

The 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 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."

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.Maor, Eli. 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.Bell, Eric Temple. The Development of Mathematics, New York: Dover, 1940, p.40

Ancient and medieval architecture

Greece

The Acropolis of Athens (468–430 BC), including the Parthenon, according to some studies, has many proportions that approximate the golden ratio. 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 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 the Greek goddess Athena. The Parthenon's facade as well as elements of its facade and elsewhere are claimed to 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.Markowsky, George

Hemenway claims that the Greek sculptor Phidias (c. 480–c. 430 BC) used the divine proportion in some of his sculptures. 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 disciple or peers. In the early 20th century, American mathematician Mark Barr proposed the Greek letter phi (φ), the first letter of Phidias's name, to denote the golden ratio.Cook, Theodore Andrea (1979). The Curves of Life, p. 420. Courier Dover Publications, .

Lothar Haselberger claims that the Temple of Apollo in Didyma (c. 334 BC), designed by Daphnis of Mileto and Paionios of Ephesus, has golden proportions.

It is claimed that the upper level of 21 rows and the lower level of 34 rows of the Ancient Theatre of Epidaurus form an approximation of the Golden number since 21 and 34 are successive Fibonacci 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.

Prehispanic Mesoamerican architecture

Between 1950 and 1960, Manuel Amabilis applied some of the analysis methods of Frederik Macody Lund and Jay Hambidge in several designs of prehispanic buildings, such as El Toloc and La Iglesia de Las Monjas (the Nuns Church), a notable complex of Terminal Classic buildings constructed in the Puuc architectural style at Chichen Itza. According to his studies, their proportions are concretized from a series of polygons, circles and pentagrams inscribed, as Lund found in his studies of Gothic churches. Manuel Amabilis published his studies along with several self-explanatory images of other pre-columbian buildings made with golden ratio proportions in La Arquitectura Precolombina de Mexico.Manue Amabilis. (1956) La Arquitectura Precolombina en Mexico. Editorial Orion. P. 200, 202. The work 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 (Columbus Day) of 1929.

The Castle of Chichen Itza was built by the Maya civilization between the 11th and 13th centuries AD as a temple to the god Kukulcan. 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.

Islamic architecture

The Great Mosque of Kairouan (built by Uqba ibn Nafi c. 670 C.E.) uses the golden ratio in the design including its plan, the prayer space, court, and minaret, but the ratio does not appear in the original parts of the mosque.

Buddhist architecture The Stupa 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.

Romanesque architecture The Romanesque style of architecture prevailed in Europe between 900 and 1200, a period which ends with the transition to Gothic architecture. The contrast between Romanesque and Gothic concepts in religious buildings can be understood in the epistolary between St. Bernard, Cistercian, and the Abbot Suger of the order of Cluny, the initiator of Gothic art in St. Denis.

One of the most beautiful works of Romanesque Cistercian is the Sénanque Abbey in Provence. The Sénanque abbatial was founded in 1148 and consecrated in 1178. It was initiated in life of St Bernard of Clairvaux. "La Lumière à Sénanque" (The Light in Sénanque), a chapter of Cîteaux : commentarii cistercienses, a publication of the Cistercian 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.

Gothic architecture

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 Luca Pacioli's 1509 De 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.

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.

The Australian architectural historian John James made a detailed study of the Cathedral of Chartres. In his work The 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.James, John, The Master Masons of Chartres. Edition 1990. 273 the Mall, Leura NSW 2780 Australia: West Grinstead Publishing. p. 157.

Art

Renaissance

De divina proportione, written by Luca Pacioli in 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.