Note: It is highly recommended that you read So What’s With Jane already? A Primer on Pictorial Composition. (Part I) , “To the makers of music – all worlds, all times.” A Primer on Pictorial Composition. (Part II), Henri Breuil and Alfred Yarbus Walk into a Bar…A Primer on Pictorial Composition. (Part III), and A Spurious Affair. A Primer on Pictorial Composition. (Part IV), before embarking on this installment.
To adequately address the concepts of pictorial composition featured in this installment, I will need to introduce a significant amount of history and math. I encourage each and every reader to research the statements made herein. A list of resources will be included at the close of this installment. I would also ask the readers of this installment to make me aware of any historical or mathematical errors. I spent a great deal of time attempting to make sure that I could find multiple sources for many of the statements and assertions that are shared here. If you notice an error, you may comment below or contact me directly at firstname.lastname@example.org. Thank you in advance!
“Simple mathematics tells us that the population of the Universe must be zero. Why? Well given that the volume of the universe is infinite there must be an infinite number of worlds. But not all of them are populated; therefore only a finite number are. Any finite number divided by infinity is zero, therefore the average population of the Universe is zero, and so the total population must be zero.” -Douglas Adams.
Imagine a math class that began with the assertion that the number ten was the “holiest” of all numbers. This is so because ten is the result of adding the numbers one, two, three and four–numbers that represent (respectively) a point with no dimension, a line with one dimension, a plane with two dimensions, and a solid with three dimensions.
So if it is true that 1+2+3+4=10, does it follow that ten is indeed the holiest of numbers?Before you dismiss the above assertion (and proof) as nonsense, consider that this concept did not come from some new-age pseudoscience guru, but was indeed part of a belief system held by a figure that many consider to be the first “true” or “pure” mathematician in human history. I am of course speaking of none other than the Greek philosopher and mathematician Pythagoras of Samos (c. 580–c. 500 BCE). He and his secretive society of followers (Pythagoreans) are credited with some considerable contributions to the development of mathematics including the theorem which states that the square of the hypotenuse of a right-angle triangle is equal to the sum of the squares of the other two sides as well as the discovery of irrational numbers. (When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable—this is an idea we will revisit in depth shortly.)
As you may suspect, the problem with the above claim lies not with the math, but with the properties and descriptions that are arbitrarily assigned to each number. This arbitrary assigning or association of numbers with people, events, or properties is known as numerology. While often masquerading as a science, numerology is nothing more than a systematic manifestation of superstition—just like the pseudoscientific concepts of astrology and biorhythms.
So what does all of this have to do with devices for pictorial composition? Unfortunately quite a bit. The use of numerology to substantiate myth has given rise to many pseudoscientific devices and heuristics in the visual arts. We begin this installment by examining one of the most viral myths to infiltrate the science of visual imagery–the Golden Ratio.
It was not the Pythagorean’s contributions to modern day numerology that identified them as the ideal starting point for this installment, but rather their connection to irrational numbers. Such numbers are real numbers that cannot be expressed as the ratio of two integers. They have decimal expansions that neither terminate nor become periodic. Common irrational numbers include Pi (3.141592…), √2 (1.414213…), Euler’s Number (2.718…), and Phi (1.618033…) The first proof of the existence of such 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. This discovery was seen as an abomination to the Pythagoreans as they felt that only rational numbers could (or should) exist. Their reaction to this discovery was so severe that it is said that the Pythagoreans threw Hippasus from a ship at sea.Oddly enough, Pythagoras and his followers used the five-pointed star as a symbol or sign of recognition and referred to it as hugieia, or “health”. I state that this is odd as the diagonal and the side of the pentagon are also incommensurable. In his book, The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number, author Mario Livio writes: “It is possible to establish a rigorous proof that the diagonal and the side of the pentagon are “incommensurable,” i.e, that the ratio of their lengths cannot be expressed as a ratio of whole numbers….It has been suggested by several researchers that the Pythagoreans’ discovery of this was the first appearance of incommensurability in history.”
Many years would pass before another Greek mathematician would formally address this concept. In his book Elements, Euclid (c.300 BCE) wrote, “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.”In other words, as shown in the diagram above, point C divides the line in such a way that the ratio of AC to CB is equal to the ratio of AB to AC. Some elementary algebra shows that in this case, the ratio of AC to CB is equal to the irrational number 1.618 (precisely half the sum of 1 and the square root of 5). However, unlike Pythagoras and his followers, Euclid attached no numerological properties to the number, ultimately giving the ratio the seemingly unromantic moniker, “extreme and mean ratio”.
Centuries would pass before we would see another significant chapter in this tale being written. In 1202, a mathematician named Leonardo Bonacci (also known as Fibonacci) published a text titled Liber Abaci (Book of Calculation). This important text not only introduced the western world to the Hindu-Arabic numeral system but, 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. For example, if we look at the sequence of 0,1,1,2,3,5,8,13,21,..we can see that each number equals the sum of the two numbers before it. So after 1 and 1, the next number is 1+1=2, the next is 1+2=3, the next is 2+3=5 and so on. This is known as a recursion. It is a process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term.
Fibonacci applied this recursion to resolve the following problem contained in Liber Abaci::
“If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth?”
With a few assumptions in place, the solution would follow that we would see only one pair at the end of the first month, two pairs at the end of the second month, three by the end of the third, and five pairs by the end of the fourth (the original female has produced another new pair, the female born two months ago produces her first pair also, making five pairs total.) So starting with one pair in this scenario, the sequence that we find for the solution IS the Fibonacci sequence.
So what does this recursive number series have to do with the extreme and mean ratio defined by Euclid?
While it might seem completely unrelated at first, closer examination reveals that dividing each number in the Fibonacci sequence by the previous number in the sequence give rise to numbers nearing the extreme and mean ratio. For example:
The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
So, 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…
While this relationship would not be proven until many years later by the Scottish mathematician Robert Simpson (1687-1768), the linking of these concepts would eventually do much to expand the mysticism of this interesting number.
Even though Euclid and Fibonacci did not seem to promote the same numerology that permeated the work of Pythagoras and his followers–the mysticism would not disappear into history quietly. In 1509, a three-volume work by Luca Pacioli titled De Divina Proportione (The Divine Proportion) was published. Pacioli, a Franciscan friar, was known mostly known as a mathematician, but he was also trained in, and keenly interested in, art. Leonardo Da Vinci, a long time friend and collaborator of Pacioli’s created a number of illustrations for Divina.
Just as Pythagoras saw divinity in mathematics, Pacioli saw religious significance in the ratio. As such, Pacioli renamed Euclid’s extreme and mean ratio, The Divine Proportion (the same title as the three-volume treatise). So why was this number said to be divine? He offers five reasons:
- “That it is one and only one and not more”. That is, there’s only one value for the divine proportion and only one Christian God.
- The geometric expression of divine proportion involves three lengths and God also contains three component parts (the Father, Son, and Holy Ghost).
- “Just like God cannot be properly defined, nor can be understood through words, likewise our proportion cannot be ever designated by intelligible numbers, nor can be expressed by rational quantity, but always remains concealed and secret, and is called irrational by mathematicians.”
- The omnipresence and invariability of God is like the self-similarity associated with the divine proportion: its value is always the same and does not depend on the length of the line being divided or the size of the pentagon in which ratios of lengths are calculated.
- Just as God has conferred being to the entire cosmos through the fifth essence (the fifth essence being beyond the four simple elements (earth, water, air and fire)), represented by the dodecahedron, so does the divine proportion confer being to the dodecahedron, since one cannot construct the dodecahedron without the divine proportion,
And there you have it. With the publication of De Divina Proportione, Pacioli fuels a reweaving of numerology with mathematics. What’s more, Divina went beyond the mystical flourishing of numbers that defined the Pythagoreans. Rather, it married mysticism with both mathematics and the arts. In fact, the very first page of Divina contains Pacioli’s desire to reveal to artists the “secret of harmonic forms” via the divine proportion. He states that his book is “...a work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.’”
With this strong emphasis on the importance of the divine proportion, it might be surprising to find that the second text of Pacioli’s three-volume work was based on the work of Roman architect Marcus Vitruvius Pollio (born c. 80–70 BC, died after c. 15 BC) who advocated a system of measurement based on rational numbers–not irrational ones. Further compounding the aforementioned influences of numerology, this favoring of the Vitruvian system would be later misrepresented in a book published in 1799. In Mario Livio’s book, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number, he writes, “Author Roger Herz-Fischler traced the fallacy of the Golden Ratio [Divine Proportion] as Pacioli’s canon for proportion to a false statement made in the 1799 edition of Histoire des mathématiques (History of mathematics) by the French mathematicians Jean Etienne Montucla and Jérôme de Lalande” (Livio, p135).
Both the divine proportion and the Fibonacci sequence would get another boost in the 17th century with a German mathematician, astronomer, and astrologer named Johannes Kepler. It can be demonstrated that Kepler did understand the divine proportion’s relationship to Fibonacci’s sequence via a 1608 letter he penned to a professor. He revisited this connection in 1611, in a 24-page essay titled, “In De nive sexangula” (On the Six-Cornered Snowflake):
“Of the two regular solids, the dodecahedron and the isohedron…both of these solids, and indeed the structure of the pentagon itself, cannot be formed without the divine proportion as the geometers of today call it. It is so arranged that the two lesser terms of a progressive series together constitute the third and the two last, when added, make the immediately subsequent term and so on to infinity, as the proportion continues unbroken…the further we advance from the number one, the more perfect the example becomes.”
Over the years, mysticism would indeed continue to swirl around these mathematical concepts, and in 1835, the German mathematician Martin Ohm (younger brother of physicist Georg Ohm) would be the first to refer to the extreme and mean ratio as “Golden”. In the second edition of Die Reine Elementar-Mathematik, Ohm writes:
“One also customarily calls this division of an arbitrary line in two such parts the golden section [Goldene Schnitt].“
40 years later, James Sulley’s 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica would be the first instance of the term in an English textbook. And so the “Golden Ratio” had finally received a moniker that reflected its mystic gilding. In 1909, an American mathematician would use the Greek letter phi (Φ) to designate this proportion. This may sound a tad less “sacred”, but Barr felt the letter, taken from the name of the Greek sculptor Phidias whom he believed applied the ratio in his work (c. 480 – 430 BC), was apt.
The nineteenth and twentieth centuries brought additional contributions to the merging of the Golden Ratio (GR), the Fibonacci sequence, mysticism, and art. Pareidolia and GR gave birth to elaborate geometrical armatures for pictorial composition that persist to this day. Like Pythagoras, many believed that the numbers related to these geometries held special powers, or at the very least–some aesthetic advantage for the artist.
Several artists published books that aimed to demonstrate that the irrational number that may have once led a bunch of cranky mathematicians to drown a man at sea, held a “secret” formula for beauty. One such book, Dynamic Symmetry, was written by a Canadian-born American artist named Jay Hambidge in 1920. We will explore this book in more detail a little later on.
One of the largest contributors to the marriage of the Golden Ratio and art was German psychologist, Adolf Zeising (24 September 1810 – 27 April 1876). 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). In his writings, Zeising claimed that in the Golden Section “is contained the fundamental principle of all formation striving to beauty and totality in the realm of nature and in the field of the pictorial arts, and that it, from the very first beginning was the highest aim and ideal of all figurations and formal relations, whether cosmic or individualizing, organic or inorganic, acoustic or optical, which had found its most perfect realization however only in the human figure.” Of Zeising’s work, Mario Livio writes, “In these works, Zeising combined his own interpretation of Pythagorean and Vitruvian ideas to argue that “the partition of the human body, the structure of many animals which are characterized by well-developed building, the fundamental types of many forms of plants,…the harmonics of the most satisfying musical accords, and the proportionality of the most beautiful works in architecture and sculpture” are all based on the Golden Ratio. To him, therefore, the Golden Ratio offered the key to the understanding of all proportions in “the most refined forms of nature and art”.
This idea that the human body exhibited the proportions of the Golden Ratio would next be picked up by Swiss-French architect and artist Charles-Édouard Jeanneret-Gris (1887- 1965), (better known as Le Corbusier). 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. Titled The Modulor, his system was supposed to provide “a harmonic measure to the human scale, universally applicable to architecture and mechanics.” The Modulor presented a six-foot (about 183-centimeter) man, with his arm upraised (to a height of 226 cm; 7’5”). The total height (from the feet to the raised arm) was also divided in a Golden ratio (into 140cm and 86 cm) at the level of the wrist of a downward-hanging arm. The two ratios (113/70) and (140/86) were further subdivided into smaller dimensions according to the Fibonacci series.
Books by Romanian author and mathematician Matila Ghyka (1881-1965) and American Author David Bergamini (1928-1983) made many misstatements about the use of the Golden Ratio among artists. Of the two most influential book 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.” Livio’s assessment here is very important as it describes not only the content of Ghyka’s books—but the typical proponent strategy used to reinforce the idea that the GR holds aesthetic influence via claims of celebrated artist’s “historical use”.
Arguably the two most commonly-referenced historical applications of the Golden Ratio are the Parthenon and the Great Pyramids. Unfortunately, these claims are not supported by any evidence aside from superimposed graphics that are often comically ‘fudged’ into place. Furthermore, many proponents often claim ballpark measurements as evidence, but as mathematician Roger Herz-Fischler points out in his 1981 paper, How to find the golden number without really trying., this is an extremely problematic approach. He writes:
“However measurements, no matter how accurate, cannot be used to reconstruct the original system of proportions used to design an object, for many systems may give rise to approximately the same set of numbers; see [6,7] for an example of this. The only valid way of determining the system of proportions used by an artist is by means of documentation. A detailed investigation of three cases [8, 9, 10, 11] for which it had been claimed in the literature that the artist in question had used the “golden number” showed that these assertions were without any foundation whatsoever.”
A number of authors attempted to substantiate the claims that the GR was present in the Great Pyramid with a host of documented measurements. Martin Gardner, Herbert Turnbull, and David Burton essentially repeat the same story (referencing the 5th-century Greek historian Herodotus as the source):
“Herodotus related in one passage that the Egyptian priests told him that the dimensions of the Great Pyramid were so chosen that the area of a square whose side was the height of the great pyramid equaled the area of the face triangle. ([Bur; p. 62]”
A translation of Herodotus’ History Book II states:
“The Pyramid itself was twenty years in building. It is a square, eight hundred feet each way, and the height the same, built entirely of polished stone fitted together with the utmost care. The stones of which it is composed are none of them less than thirty feet in length”
Not only would these measurements not qualify as proof of design via a specific system of proportions (see Fischler above), but an often cited paper on this topic by mathematician and computer scientist George Markowski demonstrates that the Herodotus’s measurements are not even remotely accurate. .
“A variety of people have looked for phi in the dimensions of the Great Pyramid of Khufu (Cheops), which was built before 2500 BC. According to [Tas; p12] the length of the sides of the base of the Great Pyramid range from 755.43 to 756.08 feet, so it is not a perfect square. The average length is 755.79 feet. The height of the Great Pyramid is given at 481.4 feet. Every source that I have checked for dimensions gives values within 1% of these (e.g., [Gil; p. 185]). Throughout this section, I will use 755.79 feet as the length of the base and 481.4 feet as the height.”
Professor Markowsky goes on to state:
“Furthermore, Herodotus’s figures about the dimensions of the Great Pyramid are wildly off. The Great Pyramid neither is nor ever was (it has lost some height over the years) anywhere near 800 feet tall nor 800 feet square at the base. Finally, we should note that Herodotus wrote roughly two millennia after the Great Pyramid was constructed.
The distorted version of Herodotus’s story makes little sense. Even the authors who quote it do not give a reason why the Egyptians would want to build a pyramid so that its height was the side of a square whose area is exactly the area of one of the faces. This idea sounds like something dreamt up to justify a coincidence rather than a realistic description of how the dimensions of the Great Pyramid were chosen. It does not appear that the Egyptians even knew about the existence of phi much less incorporated it in their buildings (see [Gil; pp238-9]).” –Markowsky, George. “Misconceptions about the Golden Ratio.” The College Mathematics Journal 23.1(1992): 2-19. Web. 17 April 2010.
More often than recorded measurements, evidence for GR’s use in the design of the Parthenon is presented via a superimposed series of “Golden Rectangles”. George Markowsky cautions readers against this rather nebulous (but common) mechanism of investigation. He writes, “In some cases, authors will draw golden rectangles that conveniently ignore parts of the object under consideration. In the absence of any clear criteria or standard methodology, it is not surprising that they are able to detect the golden ratio.”
…I will call such unsystematic searching for phi the Pyramidology Fallacy. Pyramidologists use such numerical juggling to justify all sorts of claims concerning the dimensions of the Great Pyramid.”
Brian Dunning of Skeptoid writes: “Perhaps the best-known pseudo-scientific claim about the golden ratio is that the Greek Parthenon, the famous columned temple atop the Acropolis in Athens, is designed around this ratio. Many are the amateurs who have superimposed golden rectangles all over images of the Parthenon, claiming to have found a match. But if you’ve ever studied such images, you’ve seen that it never quite fits, at least not any better than any other rectangle you might try. That’s because there’s no credible historical or documentary evidence that the Parthenon’s designers, who worked more than a century before Euclid was even born, ever used the golden ratio in any way, or even knew of its existence”.
Now, rather than spend any more time challenging GR “application” in architecture, sculpture, poetry and music (all of which hold claims of GR use in design)—let us examine some historical claims regarding visual artists “using” the Golden Ratio for pictorial composition. (I place parenthesis around the word “using” here (as well around related terms) as advocates seem to have great difficulty in stating what the Golden Ratio actually does.)
It is claimed by many that countless artists throughout history have attempted to “apply” the Golden Ratio into the design of their works and indeed, a few left evidence that they have. Salvador Dali, Paul Serusier, Juan Gris, Giro Severini, Le Corbusier, Jay Hambidge, Maxfield Parrish, George Bellows, Denman Ross, and Al Nestler are among the more well-known visual artists that have documented at one time or another that they have indeed “used” or experimented with the Golden Ratio. Of the group, Dali, Serusier, Gris, and Severini all “seem to have been experimenting with GR for its own sake rather than for some intrinsic aesthetic reason.” states mathematician Keith Devlin in his 2007 paper, “The Myth That Will Not Go Away.” Mathematical Association of America. Devlin goes on to state that “…the Cubists did organize an exhibition called “Section d’Or” in Paris in 1912, but the name was just that; none of the art shown involved the golden ratio.”
Many other artists like Da Vinci, Botticelli, Michelangelo, Rafael, and Seurat are also said to have employed the golden ratio in their work. While it is possible that these artists may have experimented with the GR, there is no credible evidence to support the claims that they, in fact, did. Much like with the Parthenon, geometric overlays showing arbitrary intersections (confirmed via pareidolia) are offered as “proof”. Again, as Markowski states “In the absence of any clear criteria or standard methodology it is not surprising that they are able to detect the golden ratio.”
To demonstrate the problematic (and pareidolic) nature of this strategy, I posted a call on social media for examples of terrible composition. I then proceeded to use a program called PhiMatrix, which is “design and analysis software for Windows and Mac, inspired by Phi, the Golden Ratio.”, to add the same overlays used to “verify” GR use in masterworks, to the submissions. What I found was that the armatures did in fact, intersect with the image content in the same way that they do with masterworks. So does this prove that these geometric armatures are in fact a means to bad composition? Using the rationale of many golden section hypothesis proponents—it would seem so.To further confirm historical “use”, German scientist Gustav Fechner (a scientist whose work we will explore in depth shortly) conducted a detailed analysis in 1878 of 10,558 images from 22 European art galleries. Unfortunately for golden ratio proponents, Fechner found that the typical ratio of painting height-to-widths clearly deviated from the “expected” golden ratio. (It should also be noted here that some of the most common standard sizes for artist canvases today do not adhere to the Golden Ratio. They are (in inches) 8×10″, 5×7″ 6×8″, 11×14″, 9×12″ and 12×16″ holding ratios 0.8, 0.714, 0.75, 0.786, 0.75 and 0.75 respectively.
So are there any viable reasons to believe that the Golden Ratio holds any true aesthetic quality that can be applied visual art? The answer to that question would again turn our attention to biology.
In the first installment of this series, we defined aesthetic qualities as the characteristics of a stimulus that elicit adaptive responses that have evolved to reinforce or discourage specific behaviors. However, many resources will define visual aesthetics much more narrowly, as the psychological assignment of beauty to certain visual stimuli. While the latter is arguably problematic, either definition will suffice for our assessment of the GR in regards to demonstrable aesthetic quality.
With either definition in play, it will be necessary for us to be able to perceive the Golden Ratio visually. In fact, the term aesthetics is derived from the Greek word “aisthetikos” which means “pertaining to sense perception.”
Markowsky points out in his aforementioned essay that The New Columbia Encyclopedia describes a “Golden Rectangle” as a rectangle whose length and width are the segments of a line divided according to the Golden Section. They go on to state that the shape occupies an important position in painting, sculpture, and architecture because its proportions have long been considered the most attractive to the eye. So let’s see if you can pick out this “most attractive” rectangle (I’ll reveal the answer at the bottom of the paper):
In an effort to determine whether or not the Golden Ratio (or Golden Rectangle) indeed presented aesthetic qualities, experimental psychologist Gustav Fechner presented test subjects with a similar challenge in the 1860s. He placed 10 rectangles before each a subject and asked them to select the “most pleasing” rectangle. The rectangles varied in their height/length ratios from 1.00 (square) to .40. The Golden Rectangle had a ratio of .62. Fechner reported that 76% of all choices centered on three rectangles having the ratios of .57, .62, and .67 (with a peak at the .62 “Golden Rectangle”.)
While this data may seem initially compelling, many mathematicians, including Mario Livio, have refuted the results of the experiment. Fechner was unable to explain a psychological basis for the preference and a significant number of experiments failed to replicate his results. Of the experiment, Livio writes,”Fechner’s motivation for studying the subject was not without prejudice. He himself admitted that the inspiration for the research came to him when he “saw the vision of a unified world of thought, spirit and matter, linked together by the mystery of numbers.” While nobody accuses Fechner of altering the results, some speculate that he may have subconsciously produced circumstances that would favor his desired outcome. In fact, Fechner’s unpublished papers reveal that he conducted similar experiments with ellipses, and having failed to discover any preference for the Golden Ratio, he did not publish the results.”
The design of Fechner’s experiment has also been criticized. “Several authors criticized Fechner’s test arrangement because the composition of the presented rectangles could have advantaged the selection of the medial one, which was the ”golden” (”trend to the mean” – phenomenon). The other points of [critique] are that the subjects were not [randomly] selected and could have been influenced in their decisions by knowing Fechner’s hypothesis.” -Do People Prefer Irrational Ratios? A New Look at the Golden Section–University of Bamberg.
Some that have attempted 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.”
The only aspect of Golden Ratio preference experimentation seems to be the inconsistency of their results. Christopher Green’s All that glitters: a review of psychological research on the aesthetics of the golden section contains a wonderful glimpse into a history of the many experiments testing the aesthetic preferences for the Golden Ratio. Green writes, “What has been found? Apart from the Fechner and Witmer studies—the ones that are consistently put forward by advocates of the golden section—the early study by E Pierce (1894) revealed a popular preference for the golden section (despite Pierce’s own efforts to minimize the finding). Angier (1903) found a preference for line divisions near the golden section on average but did not find that it was preferred by individuals. Haines and Davies (1904) found no sizable effect, but Lalo (1908), replicating Fechner’s procedure, found an effect nearly as strong as had Fechner himself. Studies by Thorndike (1917) and Weber (1931) revealed general trends in favor of figures with proportions in the range of the golden section, but nothing specific. Farnsworth (1932), on the other hand, found fairly strong support for a preference for the golden rectangle. Davis (1933) found modal preferences at √3, √4, and √5, but not at phi. Importantly, however, he was the first to suggest that the proximity of phi to other ‘basic’ proportions, such as √2 and √3, might be masking an otherwise reliable effect. Interestingly, although this kind of ‘Pythagorean’ attitude has not been popular in mainstream psychology, it was the metaphysical backbone of the psychophysics developed by Weber and Fechner in the 19th century. Thompson (1946), Shipley et al (1947), and Nienstedt and Ross (1951) all showed trends in favor of golden rectangles, but their use of median rankings, rather than modes or raw frequencies, make their results suspect in the eyes of many critics.
Schiffman (1966, 1969) failed to find any effect for the golden rectangle, confirming the growing suspicion that golden-section research was a wild-goose chase. Eysenck and Tunstall (1968) found golden-rectangle effects, especially for introverts, but used the dubious tool of mean rankings. Berlyne (1970) found similar effects among Canadian subjects, using mean rankings as well, but showed, as had Angier in the early part of the century, that this does not accurately reflect individuals’ preferences. The research of Hintz and Nelson (1970, 1971), however, revealed modal preferences quite close to the golden rectangle. Significantly, modes are not subject to the criticisms that have historically been made of means in this area of research. Godkewitsch (1974) claimed to show that historically established preferences for the golden rectangle were nothing but artifacts of poorly conceived experimental procedures, but he did not treat those studies in which other procedures and methods of analysis had been used. Still, the replication of Godkewitsch’s finding by Piehl (1976) boded ill for golden-section research. Reversing this apparent fate, Benjafield (1976) showed that a more carefully conceived experiment would give rise to the traditional effect for the golden rectangle, even when Godkewitsch’s criticisms were taken into account. The results of Piehl (1978) supported this conclusion, and golden-section research was restored to the psychological agenda. An interestingly parallel case occurred in the case of research on divided lines. McManus’s (1980) result, too, lent some additional, though inconclusive, credence to at least group preferences for the golden rectangle. Schiffman and Bobko (1978) claimed to refute the positive findings of Svensson (1977), but again, an experiment carefully conceived and conducted by Benjafield etal (1980) restored an effect that had been lost in less-exacting work. Boselie (1984a, 1984b) has argued that apparent preferences for complex proportions, such as square roots and <^ are, in fact, the result of subordinate simple proportions, 966 C D Green such as equality. Boselie (1992) showed that the 1.5:1 rectangle may also be preferred to (|>. The failures of Nakajima and Ohta (1989) and of Davis and Jahnke (1991) to find positive results are both beset by methodological problems.
Even with the many “positive-result” experiments that Green lists, he too seems to realize that the GR hypothesis ultimately remains unsubstantiated–stating that “if” such preferences do exist, they are “fragile”. Green writes” I am led to the judgment that the traditional aesthetic effects of the golden section may well be real, but that if they are, they are fragile as well. Repeated efforts to show them to be illusory have, in many instances, been followed up by efforts that have restored them, even when taking the latest round of criticism into account. Whether the effects, if they are in fact real, are grounded in learned or innate structures is difficult to discern. As Berlyne has pointed out, few other cultures have made mention of the golden section but, equally, effects have been found among people who are not aware of the golden section. In the final analysis, it may simply be that the psychological instruments we are forced to use in studying the effects of the golden section are just too crude ever to satisfy the skeptic (or the advocate, for that matter) that there really is something there.”
Many contemporary mathematicians and researchers seem to be increasingly dismissive of the Golden Ratio hypothesis. British mathematician and author Keith Devlin states that “the idea that the golden ratio has any relationship to aesthetics at all comes primarily from two people, one of whom was misquoted [Pacioli], and the other of whom was just making shit up [Zeising].” Devlin, the executive director of Stanford’s Human Sciences and Technologies Advanced Research Institute, has been debunking Golden Ratio myths for years now. He states, “It’s like Creationism. You can believe it if you want, but there’s no evidence…If you believe it, you’re not being scientific.”
In a recent interview with writer John Brownlee Devlin states, “Let’s put it this way, if someone comes along tomorrow with a scientific explanation for why the Golden ratio would play a role in aesthetics and whatever else, then we’d all revise our opinion,…But on the science side, there’s no evidence.”
Some modern researchers attempted to uncover psychological or neurobiological underpinnings that might support the golden section hypothesis. One such attempt came American psychologists L.A. Stone and L.G. Collins. They put forward a hypothesis that suggested the shape of the binocular visual field may determine the supposed preference for rectangles possessing dimensions similar to those of the golden section. Stone and Collins tested this hypothesis by simply asking study participants to draw pleasing rectangles. The “average rectangle” contained a length-to-width ratio of about 1.5. As with Fechner’s original experiment, subsequent experiments failed to replicate even these results. In 1966, Rutgers University’s H. R. Schiffman performed a similar experiment that resulted in an average length-to-width ratio of 1.9.
Inspired by this idea of finding a rationale for golden section preference in the mechanisms of biological vision, I returned to the eye-tracking work of Russian psychologist Alfred Yarbus (introduced in the second installment of this series) to see if there was any detectable saccade pattern that would seem related to the structure of an armature based on the golden ratio.
In the above examples from Yarbus’ experiments, I’ve applied the classic golden section armature in the manner that seemed to ‘fit” best (a “proof” in and of itself for many that these images were indeed “designed” with the golden ratio.) With the armature visibly in place, we can easily see that the recorded eye patterns do not seem to correlate in any way.
In a 2007 study titled “The golden beauty: brain response to classical and renaissance sculptures.” Di Dio, Cinzia, Emiliano Macaluso, and Giacomo Rizzolatti used an fMRI to explore the possibility of an objective, biological basis for the experience of beauty in art. The wrote in the 2007 study, ““The main question we addressed in the present study was whether there is an objective beauty, i.e., if objective parameters intrinsic to works of art are able to elicit a specific neural pattern underlying the sense of beauty in the observer. Our results gave a positive answer to this question. The presence of a specific parameter (the golden ratio) in the stimuli we presented determined brain activations different to those where this parameter was violated. The spark that changed the perception of a sculpture from “ugly” to beautiful appears to be the joint activation of specific populations of cortical neurons responding to the physical properties of the stimuli and of neurons located in the anterior insula.”
While this may once again seem like a reason to consider the possibility that the GR preference may be supported by evidence, a review of the experiment materials quickly reveals a rather comical methodology. Here is one of the images that the participants were presented with in order to assess “beauty” (incorporating the golden ratio):
As you can quickly see from this example of the experiment stimuli, the stimulus that carries the “golden ratio” does so with an arbitrary division at the navel and is compared against distorted versions of the same stimuli that “violate the ratio”. Using this strategy, you can conclude that any ratio is pleasing. Just pick one.
Another strategy for promoting the golden ratio hypothesis is to focus on the prevalence of phi in nature. And while the GR and Fibonacci sequence both crop up with surprising regularity in many living systems, some of the “occurrences” cited are indeed misrepresentations. Pinecones, pineapples, the pattern of sunflower seeds, the arrangement of leaves on a stem of plants (phyllotaxis), nautilus shells, hurricanes, whirlpools, the double-helix of DNA, and spiral galaxies are just a few of the natural phenomena that are associated with phi.Probably the most commonly used representation of phi in nature is the nautilus shell. While the shell is indeed a logarithmic spiral, (also known as an equiangular spiral or growth spiral) it is NOT a “golden spiral” (a golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch about 17.03239 degrees). It can be approximated by a “Fibonacci spiral”, made of a sequence of quarter circles with radii proportional to Fibonacci numbers.) Similar errors are made in regards to spiral galaxies and hurricanes.
One of the most fascinating TRUE occurrences of phi in nature is phyllotaxis. In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon “leaf” and táxis “arrangement”). Phyllotactic spirals form a distinctive class of patterns in nature. Although the first person to discover the relationship between phyllotaxis and the Fibonacci sequence was astronomer Johannes Kepler (1571-1639), and the name coined by coined in 1754 by Swiss naturalist Charles Bonnet, it would be nineteenth century German botanists Karl Friedrich Schimper(1830), Alexander Braun (1835), French physician/scientist Louis François Bravais and his brother, crystallographer Auguste Bravais (1837) that would “discover the general rule that phyllotactic ratios could be expressed by ratios of terms of the Fibonacci series and also noted the appearance of consecutive Fibonacci numbers in the parastichies of pinecones and pineapples”. -Mario Livio
So is the fact that phi can be found in sunflowers, pine cones, and pineapples some type of validation for the aesthetic claims surrounding the golden ratio? We might answer that by looking to how often these natural manifestations of phi appear in artworks. We have long since celebrated those things we have deemed “beautiful” on paper and canvas. Therefore, if phi holds significant aesthetic quality, and the aforementioned objects are natural manifestations of phi, and natural occurrences of phi are often put forward as “evidence” of likely biological influence, then it would seem reasonable that many celebrated artworks would feature pinecones, pineapples, and sunflower florets.
So how many pinecones and pineapples have populated your favorite drawings and paintings? Now I would concede that sunflowers have long been in many celebrated paintings over the years–but take a moment to consider those representations. How many actually feature an accurate representation of the spiral floret patterns that exhibit phi? Or, have the artists opted to abstract the Fibonacci-laden central region to instead focus on the surround of colorful petals. You would think that if natural manifestations of a mathematical formula for aesthetic preference actually existed, these manifestations would find more “canvas-time” than they have.
So at this point I am hopeful that the nature of the Golden Ratio is a bit more clear for you. If not, you may want to consider the “Divine Circle Matrix”:
The Divine Circle Matrix is the oldest secret in the visual arts. For many years, creatives have measured aspects of nature in the hopes of uncovering the secrets of beauty. However, for centuries it has been known to a select few that the secret is found not in the external environment–but hidden within in the eye itself. Anatomically, the size of the human retina from ora to ora is 32mm (Van Buren, 1963.), while the fovea, which is the only part of the human eye that permits 100% visual acuity, measures 1.5mm (Polyak, 1941). These measurements reveal a ratio of 1.5:32, (or 3:64). If we look to the Bible we can find an astounding correlation: Lamentations 3:64: You will recompense them, O LORD, According to the work of their hands. Incredible!
So is there something special, or possibly divine, inherent to this root 1.5mm measurement? You bet there is! You can take any masterwork, apply a circle with the diameter of 3 (inches, centimeters, or any applicable unit) in the center (1.5 for each eye), and you will find that every circle tiled outward will capture an important aspect of the image. The more circles that capture important events in the image—the more beautiful the artwork.
Does that sound unbelievable? It should as it is complete nonsense. I manufactured this concept in about 10 minutes. To be clear–The math is correct, the biology is correct, and the quote from the bible is correct. So with a few bits of accurate math, several references to biology, a far-reaching coincidental connection to the bible, and a hint at a centuries-old standing secret and instantly we have a new divine formula for beauty. With a bit of collective effort, I am sure it would not be too long before this idea would “snowball”–soon being taught in a design class near you.
So if the Golden Ratio remains unsubstantiated after so many years of study—why does it persist? Keith Devlin answers this question best in his interview with writer John Brownlee:
“Devlin says it’s simple. “We’re creatures who are genetically programmed to see patterns and to seek meaning,” he says. It’s not in our DNA to be comfortable with arbitrary things like aesthetics, so we try to back them up with our often limited grasp of math. But most people don’t really understand math, or how even a simple formula like the golden ratio applies to complex system, so we can’t error-check ourselves. “People think they see the golden ratio around them, in the natural world and the objects they love, but they can’t actually substantiate it,” Devlin tells me. “They are victims to their natural desire to find meaning in the pattern of the universe, without the math skills to tell them that the patterns they think they see are illusory.” If you see the golden ratio in your favorite designs, you’re probably seeing things.“-Brownlee, John. Co.Design: The Golden Ratio: Design’s Biggest Myth w/ Keith Devlin.
Now if you can still stick with me, we have one more topic to cover in this installment before we close, Jay Hambidge’s Dynamic Symmetry. This still-popular work contains many of the same fundamental problems inherent to the Golden Section hypothesis so an examination of his efforts should not take long.
As I mentioned earlier in this paper, The Elements of Dynamic Symmetry was written by a Canadian-born American artist named Jay Hambidge in 1920. Dynamic symmetry is a proportioning system and natural design methodology that uses dynamic rectangles, including root rectangles based on ratios such as √2, √3, √5, the golden ratio (φ = 1.618…), its square root (√φ = 1.272…), and its square (φ2 = 2.618….), and the silver ratio (2.4142135623….).What you can probably glean from this brief description is that the ideas presented are just as nebulous and problematic as the golden section hypothesis explored above. Hambidge just goes a bit further to include additional related ratios. Therefore, when one’s pareidolia fails to produce the (hopefully) now familiar 1.618…, we can expand the search to include this ratio, or that one, because they all carry an aesthetic advantage. Again, like our “holy” ten, the math is correct—but there is absolutely no substantiating evidence that any of the ratios hold an aesthetic advantage.
Essentially, Dynamic Symmetry is a larger net for catching fairies.
In his celebrated work on Phi, Mario Livio dismissed Hambidge efforts quite succinctly. “Another art theorist who had great interest in the Golden Ratio at the beginning of the twentieth century was the American Jay Hambidge (1867-1924). In 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. Few today take his ideas seriously.”
Let’s take a look at some of the claims in Elements of Dynamic Symmetry and see if they seem familiar :
DS: “The basic principles underlying the greatest art so far produced in the world may be found in the proportions of the human figure and in the growing plant.”
This claim should sound familiar. It is a simple rehash of the nonsense spewed by Adolf Zeising in the 1850s. Remember that Zeising felt that the Golden Ratio was the basis for ““the partition of the human body, the structure of many animals which are characterized by well-developed building, the fundamental types of many forms of plants,…the harmonics of the most satisfying musical accords, and the proportionality of the most beautiful works in architecture and sculpture…” Once again, while there are many proportions to be found in nature–the is no substantial evidence–no biological reason or rationale—to support that any one would be any more aesthetically pleasing than any other independent of context.
DS: “The principles of design to be found in the architecture of man and of plants have been given the name “Dynamic Symmetry.” This symmetry is identical with that used by Greek masters in almost all the art produced during the great classical period.”
“The analysis of the plan of a large building, such for example as the Parthenon, often is not so difficult as the recovery of the plans of many minor design forms.”
Within a few sentences of Hamidge’s introduction, we have jumped from evidence-via-natural-prevalence to evidence-via-assertions-of-historical-use. Anyhow, we can assume here that Hambidge is referring to the Classical period of 4th and 5th century BCE. So did the artists of this period use Dynamic Symmetry in “almost all art produced” during this period?
Just as I demonstrated with the Parthenon (created 438 BCE) above, the answer is “hardly”. The dimensions of the base of the Parthenon are 69.5 by 30.9 meters (228 by 101 ft). The height pf the Parthenon is 45 feet, 1 inch while the width is 101 feet 3.75 inches. The length is 228 feet and ⅛ inches. This gives us a width to height and length to width ratio of 2.25. Stuart Rossiter, renowned philatelist and scholar, gives the height of the apex above the stylobate as 59 feet.in his book Greece. This height produces a ratio of 1.71—closer but still outside of the range of the golden ratio.
Now we can go through every piece of Greek art from the classical period and measure EVERYTHING (like Fechner with his survey of over 10,000 works of art) but as Fischler stated in his paper, without supporting documentary evidence, we cannot definitively confirm that any design scheme was or was not used. There is no documentation that any greek artist from the classical period (let alone ALL) consciously employed the Golden Ratio in their work. Keep in mind that the extreme and mean ratio would not be formally defined for at least another hundred years with Euclid’s Elements.
After an introduction riddled with falsehoods and misrepresentations, Hambidge goes on to present the Fibonacci sequence, phyllotaxis, logarithmic spirals, root rectangles, whirling squares, diagonals, reciprocals, compliments, the gnomon, ratios and a host of other concepts that honestly do seem fairly interesting. The problem is that while the concepts are indeed accurate, the application of the concepts, and their functionality for pictorial composition, is never adequately addressed. The book’s only connection to pictorial composition seems to be via the misstatements and misrepresentations made explicit and implicit by the author.
I would argue though that you CAN use some of the geometries contained within Dynamic Symmetry to partition pictorial space—possibly to introduce some previously unconsidered locations for subject placement—however, the same could be said for tossing a handful of pebbles onto a flat surface to determine spatial placement.
In any case, I have seen both Hambidge’s work and the golden ratio hypothesis staunchly championed by a handful of contemporary colleagues, and unfortunately, some contemporary educators. None of them have been able to adequately answer (with substantiation) the questions “What does the Golden Ratio or Dynamic Symmetry actually do?” or, “How do these devices function?”
While the responses to this question may vary, most are little more than an exercise in deflection. Many put forward seemingly misunderstood concepts involving biological perception, cognitive psychology, and mathematics. This recent response to an inquiry into evidence for Dynamic Symmetry truly captures the attitude from those that will dogmatically believe in the fruits of the devices mentioned in this paper regardless of the scientific evidence to the contrary.
“…when grandmaster artists and designers like Maxfield Parrish and George Bellows extol the virtues of dynamic Symmetry in composition, you’ll have to forgive me for siding with them over an astrophysicist or anyone else attempting to diminish dynamic symmetry as powerful tool. Astrophysicists and researchers are not designers, side with them if you like, I won’t.”
Fortunately, in my experience, this attitude is not the prevailing one. As educational resources improve and the dissemination of information continues to grow, it would seem that the days of infusing science and mathematics with mysticism are coming to an end.
Bergamini, David. “the Editors of Life.” Mathematics. New York: Time Incorporated (1963).
Boselie F, 1992. “The golden section has no special aesthetic attractivity!” Empirical Studies of the Arts 10 1-18.
Boselie F, 1997 “The golden section and the shape of objects” Empirical Studies of the Arts 15 131-141.
Brownlee, John. Co.Design: The Golden Ratio: Design’s Biggest Myth w/ Keith Devlin. 04.13.15 http://www.fastcodesign.com/3044877/the-golden-ratio-designs-biggest-myth
Burton, David M. “The History of Mathematics, Wm. C.” (2000).
Calderone, Julia. The one formula that’s supposed to ‘prove beauty’ is fundamentally wrong. Tech Insider, Oct. 5, 2015 http://www.techinsider.io/the-golden-ratio-fibonacci-numbers-mathematics-zeising-beauty-2015-9
Corbusier, Le. The Modulor and Modulor 2. Vol. 1. Springer Science & Business Media, 2004.
Davis S T, Jahnke J C, 1991 “Unity and the golden section: Rules for aesthetic choice?” American Journal of Psychology 104 257 ^ 277
Devlin, K. “The Myth that Will Not Go Away.” Devlin’s Angle. The Mathematical Association of America, 1 May 2007. Web. 22 Aug. 2012. http://www.maa.org/devlin/devlin_05_07.html
Di Dio, Cinzia, Emiliano Macaluso, and Giacomo Rizzolatti. “The golden beauty: brain response to classical and renaissance sculptures.” PloS one 2.11 (2007): e1201.
Dudley, Underwood. “The Golden Section by Hans Walser Reviewed by Underwood Dudley.” Read This! The MAA Online book review column. Mathematical Association of America, 15 December 2001. Web. 17 April 2010.
Dunning, B. “The Golden Ratio.” Skeptoid Podcast. Skeptoid Media, 28 Aug 2012. Web. 9 Jul 2016. http://skeptoid.com/episodes/4325Euclid, Heath, T. L., & Densmore, D. (2002). Euclid’s Elements: All thirteen books complete in one volume : the Thomas L. Heath translation. Santa Fe, N.M: Green Lion Press. http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
Eydt, Rebecca A. “How Good is Gold? Recognition of The Golden Rectangle.” (2013).
Falbo, Clement. “The golden ratio—a contrary viewpoint.” The College Mathematics Journal 36.2 (2005): 123-134.
Fischler, Roger. “How to find the golden number without really trying.”Fibonacci quarterly 19 (1981): 406-410. http://www.herz-fischler.ca/ARTICLES/how_to_find.pdf
Fischler, Roger. “An investigation of claims concerning Seurat and the golden number.” Gazette des Beaux Arts (1980).
Fischler, R. “An examination of claims concerning The Golden Number. Seurat, Gris, Mondrian, Ozenfant, Jeanneret.” Gazette des Beaux-Arts.
Gardner, M., The cult of the golden ratio, Skeptical Inquirer 18 (1994) 243-247. http://www.cdlmadrid.org/cdl/htdocs/universidaddeotono/unioto/matematicas/goldenratio.pdf
Gillings, Richard J. Mathematics in the Time of the Pharaohs. Courier Corporation, 1972.
Godkewitsch, Michael. “The’golden section’: an artifact of stimulus range and measure of preference.” The American journal of psychology (1974): 269-277.
Green, C. “All That Glitters: A Review of Psychological Research on the Aesthetics of the Golden Section.” Perception. 1 Aug. 1995, Volume 24, Number 8: 937-968.
Ghyka, Matila Costiescu. The geometry of art and life. Courier Corporation, 1946.
Ghyka, Matila C. A practical handbook of geometrical composition and design. No. 2. Alec Tiranti, 1952.
Hambidge, Jay. The elements of dynamic symmetry. Courier Corporation, 2012.
Höge Holger, 1995 “Fechner’s experimental aesthetics and the golden section hypothesis today” Empirical Studies of the Arts 13 131-148.
Höge Holger, 1997 “Why a special issue on the golden section hypothesis? An introduction” Empirical Studies of the Arts 15 111-114
Höge, Holger. “The golden section hypothesis—its last funeral.” Empirical studies of the arts 15.2 (1997): 233-255.
Kepler, Johannes. The six-cornered snowflake. Paul Dry Books, 2010.
Livio, M. The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number. New York: Broadway Books, 2003.
Lucht, M. “Fee, Phi, Flim, Flam.” The Skeptic. 1 Jul. 2007, Volume 27, Number 2: 27-31.
Tung, K. Topics in Mathematical Modeling. Princeton: Princeton University Press, 2007.
Markowsky, George. “Misconceptions about the Golden Ratio.” The College Mathematics Journal 23.1(1992): 2-19. Web. 17 April 2010.
McManus, IC (1980). The aesthetics of simple figures. British Journal of Psychology, 71, 505-524.
McManus, I. C., Richard Cook, and Amy Hunt. “Beyond the golden section and normative aesthetics: why do individuals differ so much in their aesthetic preferences for rectangles?.” Psychology of Aesthetics, Creativity, and the Arts 4.2 (2010): 113.
McManus, Ian Christopher. “The aesthetics of simple figures.” British Journal of Psychology 71.4 (1980): 505-524.
Montucla, Jean-Étienne, and La Lande. Histoire des mathématiques: dans laquelle on rend compte de leurs progrès depuis leur origine jusqu’à nos jours,… chez Henri Agasse, 1802
Myres, John Linton. Herodotus, father of history. Clarendon Press, 1953.
Ohm, Martin. “Die reine Elementar-Mathematik.” Berlin: Jonas Veilags-Buchhandlung (1835).
Pisano, Leonardo, and L. E. Sigler. “Fibonacci’s Liber abaci.” A translation into modern English of Leonardo Pisano’s Book of calculation. Translated by LE Sigler, Springer, New York (2002).
Pacioli, Luca, and Venice De Divina Proportione. “1509.” Divina proportione (1969). https://archive.org/details/divinaproportion00paci)
Russell, P.A. (2000). Testing the aesthetic significance of the golden-section rectangle. PERCEPTION-LONDON-, 29, 1413-1422.
Schiffman, H. R. “Golden section: Preferred figural orientation.” Perception & Psychophysics 1.3 (1966): 193-194.
Seewald, Nick., http://goldenratiomyth.weebly.com/index.html
Socrates, G. Taseos. “Back in Time 3104 BC to the Great Pyramid.” (1990).
Sonnenburg, C., et al., Do People Prefer Irrational Ratios? A New Look at the Golden Section, University of Bamberg.
Stone, LeRoy A., and L. Glenn Collins. “The golden section revisited: A perimetric explanation.” The American journal of psychology 78.3 (1965): 503-506.
Yarbus, Alfred L. “Eye movement and vision, trans. B. Haigh.” ed: Plenum Press, New York (1967).’
Zeising, Adolf. “Neue Lehre von den Proportionen des menschlichen Körpers…” (1854).
Zeising, Adolf . Der goldene Schnitt. Engelmann in Komm., 1884.
The Cult of the Golden Ratio.” Laputan Logic. 19 January 2007. Web. 20 April 2010. <http://www.laputanlogic.com/articles/2005/04/14-1647-4601.html>.
GOLDEN RECTANGLE ANSWER:
The rectangle containing the ratio closest to phi is on the bottom row, third from the right.