Editing Golden ratio (section)

==Disputed observations==
Examples of disputed observations of the golden ratio include the following:
[[File:NautilusCutawayLogarithmicSpiral.jpg|thumb|[[Nautilus]] shells are often erroneously claimed to be golden-proportioned.]]
* Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive [[Phalanx bone|phalangeal]] and [[metacarpal bones]] (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.<ref name=pheasant /><ref name=vanLaack />
* The shells of mollusks such as the [[nautilus]] are often claimed to be in the golden ratio.<ref name=dunlap /> The growth of nautilus shells follows a [[logarithmic spiral]], and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio,<ref name=falbo /> or sometimes claimed that each new chamber is golden-proportioned relative to the previous one.<ref name=moscovich /> However, measurements of nautilus shells do not support this claim.<ref name=shellspirals />
* Historian [[John Man (author)|John Man]] states that both the pages and text area of the [[Gutenberg Bible]] were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is {{tmath|1.45}}.<ref name=gutenberg />
* Studies by psychologists, starting with [[Gustav Fechner]] {{Circa|1876}},<ref name=Fechner /> have been devised to test the idea that the golden ratio plays a role in human perception of [[beauty]]. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/7 7]}}<ref name="livio plus" />
* In investing, some practitioners of [[technical analysis]] use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.<ref name=osler/> The use of the golden ratio in investing is also related to more complicated patterns described by [[Fibonacci numbers]] (e.g. [[Elliott wave principle]] and [[Fibonacci retracement]]). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.<ref name=magicdow />

===Egyptian pyramids===
[[File:Egypt. Pyramids. Pyramids and palm grove reflections LOC matpc.23063.jpg|thumb|The [[Great Pyramid of Giza]]]]
The [[Great Pyramid of Giza]] (also known as the Pyramid of Cheops or Khufu) has been analyzed by [[pyramidology|pyramidologists]] as having a doubled [[Kepler triangle]] as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on [[pi]] or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.<ref name=greatpyramid />

===The Parthenon===
[[File:The_Parthenon_in_Athens.jpg|thumb|280px|Many of the proportions of the [[Parthenon]] are alleged to exhibit the golden ratio, but this has largely been discredited.{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/74 74–75]}}]]

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.<ref name=Polemic /> 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."<ref name=mathinstinct /> [[Midhat J. Gazalé]] affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."<ref name=gazalé />

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.<ref name=foutakis />
Later sources like Vitruvius (first century&nbsp;BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

===Modern art===
[[File:Albert Gleizes, 1912, Les Baigneuses, oil on canvas, 105 x 171 cm, Paris, Musée d'Art Moderne de la Ville de Paris.jpg|thumb|[[Albert Gleizes]], ''[[Les Baigneuses (Gleizes)|Les Baigneuses]]'' (1912)]]
The [[Section d'Or]] ('Golden Section') was a collective of [[Painting|painters]], sculptors, poets and critics associated with [[Cubism]] and [[Orphism (art)|Orphism]].<ref name=centrepompidou1 /> 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]].<ref name=centrepompidou2 /> (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat's writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.)<ref name=seuratclaims /> The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception".<ref name=herbert /> However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 [[Section d'Or#Salon de la Section d'Or, 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,{{sfn|Livio|2002|p=[https://archive.org/details/goldenratiostory00livi/page/169 169]}} and [[Marcel Duchamp]] said as much in an interview.<ref name=camfield /> 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.<ref name=camfield /><ref name=juangris /> Art historian [[Daniel Robbins (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.<ref name=allard />

[[Piet Mondrian]] has been said to have used the golden section extensively in his geometrical paintings,<ref name=bouleau /> though other experts (including critic [[Yve-Alain Bois]]) have discredited these claims.<ref name="livio plus" />{{sfn|Livio|2002|pp=[https://archive.org/details/goldenratiostory00livi/page/177 177–178]}}