A (VERY) Brief History of the Golden Ratio

When faced with the question of actual function, Golden Ratio proponents are quick to throw out many claims of historical use and tradition. Unfortunately, many that point to the history of this marriage of the GR and visual art are demonstrably unfamiliar with it. So, for those interested, here’s a brief look at the history of this ratio and how it was shoehorned (mostly via error and misrepresentation) into the arts: (based on my article “Fool’s Gold.”)

Hippasus of Metapontum (late 5th Century BC): The first proof of the existence of incommensurable numbers is usually attributed not to Pythagoras, but to one of his followers, Hippasus of Metapontum. As the story goes, Hippasus realized that the sides of a square were incommensurable with its diagonal and that this incommensurability could not be expressed as the ratio of two integers.

Euclid of Alexandria (c.300 BCE): In his book Elements, Euclid states that “a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”

Leonardo Bonacci (also known as Fibonacci) (c. 1170 – c. 1250): published a text in 1202 titled Liber Abaci (Book of Calculation) which contained, among a list of challenging brain-teasers, a fascinating number sequence that would be used to model or describe an amazing variety of mathematical concepts as well as natural phenomena. The sequence (which would eventually be named the “Fibonacci sequence” by French mathematician Édouard Lucas in the 19th century.) starts with a one or a zero, followed by a one, and proceeds based on the rule that each number is equal to the sum of the preceding two numbers. (e.g., 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …)

How does Fibonacci relate to our focus here? If we divide each number by the previous number gives: 1 / 1 = 1, 2 / 1 = 2, 3 / 2 = 1.5, and so on up to 144 / 89 = 1.6179…. The resulting sequence is: 1, 2, 1.5, 1.666…, 1.6, 1.625, 1.615…, 1.619…, 1.6176…, 1.6181…, 1.6179…, or a series of numbers that seems to oscillate very near the numerical value of phi, 1.618 (attributed to Scottish mathematician Robert Simpson (1687-1768.))

Luca Pacioli (c. 1447 – 1517): published a three-volume work in 1509 titled De Divina Proportione (The Divine Proportion.) Leonardo Da Vinci, a longtime friend and collaborator of Pacioli’s created a number of illustrations for Divina. Just as the Pythagoreans saw divinity in mathematics, Pacioli saw religious significance in the ratio. As such, Pacioli renamed Euclid’s extreme and mean ratio, The Divine Proportion.

French Mathematicians Jean Etienne Montucla and Jérôme de Lalande’s 1799 edition of Histoire des Mathématiques (History of mathematics) mistakenly confuses the Divine Proportion with the Vitruvian system (a system of proportions based on the work of Roman architect Marcus Vitruvius Pollio who in fact advocated a system of measurement based on rational numbers–not irrational ones.)

Johannes Kepler (1571 – 1630) mentions the connection between Fibonacci and the divine proportion in a 1611 essay titled, “In De nive sexangula” (On the Six-Cornered Snowflake.)

Martin Ohm (1792 – 1872): In 1835, the German mathematician (and younger brother of physicist Georg Ohm) would be the first to refer to the extreme and mean ratio as “Golden”.

James Sulley (1842 –1923): writes an article on aesthetics for the 9th edition of the Encyclopedia Britannica which contains the first instance of the term “Golden Ratio.” In 1909, an American mathematician would use the Greek letter phi (Φ) to designate this proportion. Barr took the letter from the name of the Greek sculptor Phidias whom he believed applied the ratio in his work.

Adolf Zeising (1810 –1876): One of the largest contributors to the marriage of the Golden Ratio and art. Zeising’s work in this area began with a series of publications (described by mathematician Mario Livio as “crankish”) including an 1854 work titled A New Theory of the proportions of the human body, developed from a basic morphological law which stayed hitherto unknown, and which permeates the whole nature and art, accompanied by a complete summary of the prevailing systems. (yes, that is all one title). After Zeising’s death, this and other publications would be combined into a large book titled Der Goldne Schnitt (The Golden Section).

Charles-Édouard Jeanneret-Gris (1887- 1965), (better known as Le Corbusier): While It has been stated that Le Corbusier was originally skeptical of the aesthetic claims associated with the Golden Ratio and Fibonacci—however, this did not stop him from developing a proportion system based on both that he dubbed The Modulor. (His figure illustrations for this system are laughably awful.)

Matila Ghyka (1881-1965): Publishes a number of texts containing blatant misstatements in his texts about the use of the Golden Ratio among artists. Of the two most influential books by Ghyka, author Mario Livio states “Both books are composed of semimystical interpretations of mathematics. Alongside correct descriptions of the mathematical properties of the Golden Ratio, the books contain a collection of inaccurate anecdotal materials on the occurrence of the Golden Ratio in arts.

Gustav Fechner (1801 – 1887): An early pioneer of experimental psychology, Fechner was among the first to set up controlled experiments to test the aesthetic claims surrounding the golden ratio. His first major treatise on the topic was Zur experimentalen Aesthetik (1871). While Fechner did report a demonstrable aesthetic preference for the ratio, his testing methods have faced significant criticism. Furthermore, while some have claimed to have replicated Fechner’s results (e.g., Lightner Witmer), many attempts to replicate Fechner’s original study as closely as possible found that the golden ratio was indeed not a “preferred proportion.” Professor of Psychology Holger Höge writes of his own study, “Thus, as there are so many results on the golden section hypothesis showing contradictory outcomes it seemed necessary to replicate Fechner’s original study as far as possible: giving the same proportions, using white cards on black ground. Other specifics could not be kept constant because Fechner’s report on the experiment is not very precise (cf. Fechner, 1876/1925/1997). As a complete replication is not possible, three experiments were carried out, each of them being slightly different in methodology. However, regardless of the conditions under which the choices were made, the golden section did not turn out to be the preferred proportion. The comparison with Fechner’s results makes this research only quasi-experimental in character and, hence, inevitably there are some restrictions with respect to the strength of the conclusions to be drawn. But, nevertheless, the nice peak of preference Fechner reported for the golden section seems to be either an artifact or it is an effect of still unknown factors. Two possible hypotheses (change-of-taste and color-of-paper) are discussed. It is concluded that the golden section hypothesis is a myth.”

Jay Hambidge (1867-1924): With a series of articles and books, Hambidge defined two types of symmetry in classical and modern art. One, which he called “static symmetry,” was based on regular figures like the square and equilateral triangle, and was supposed to produce lifeless art. The other, which he dubbed “dynamic symmetry,” had the Golden Ratio and the logarithmic spiral in leading roles. Hambidge’s basic thesis was that the use of “dynamic symmetry” in design leads to vibrant and moving art. Unfortunately, in all of his writings, Hambidge never bothers to offer any valid substantiation for his claims leaving them locked in the realm of bald assertions.

Hope you find this informative. Happy Painting!

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