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 [email protected].  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.

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.

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http://www.phimatrix.com/

GOLDEN RECTANGLE ANSWER: 

The rectangle containing the ratio closest to phi is on the bottom row, third from the right.