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 yychuls@gmail.com. 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.

RESOURCES:

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/4325

Euclid, 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>.

http://www.phimatrix.com/

**GOLDEN RECTANGLE ANSWER: **

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

WOW! Thank you for this truly ‘exorcizing’ pile of information! Very, very worth the read. Especially for any student who’s ever fallen into the clutches of a Fibonacci-fanatic pseudo teacher.

Thank you again for your generosity

It’s a pleasure for me Shelia. I am very glad to know that you enjoyed it. 🙂

Hi Anthony,

I’m the developer of the PhiMatrix software you show above to illustrate your point that both poorly composed images and well composed images will align with the lines of golden ratio templates. Unfortunately, your use of the program was very misguided, which had a direct impact on the validity of your conclusions. Rather than use the basic golden ratios which are the core of the program, and the basis for many good designs by many of its users, you choose the most complicated templates that the program offers. These are the armatures, which are buried deep in the program’s menus and which each contain dozens of lines. With enough lines based on any ratio or methodology, any image, good or bad, will of course match some of them. That, however, is not the point of the program and not how good designers apply it. Beyond that, the complicated armatures you chose for your illustration were in many cases not even those based on the golden ratio, and were just included in the program at the request of some users because they had seen them in other art instruction resources.

There’s an unfortunate pattern that is typical among those who seek to debunk appearances and applications of the golden ratio. Some claim that the evidence doesn’t even exist, when the truth is that they haven’t researched thoroughly enough to find it. Others do a very cursory review of one particular aspect of the golden ratio and then come to inaccurate conclusions that don’t reflect a thorough knowledge of the topic. See some of my articles in the links below that challenge the validity of the statements made by those who you referenced in your article.

I’m not someone who thinks the golden ratio is everywhere as some magical, mystical universal constant that is the secret to everything. I see it as providing some very interesting insights into design in nature, and as a very useful tool in the design arts. Articles by skeptics that completely debunk it with misinformation are just as damaging as the ones by zealots that fanatically extol it with misinformation. It’s a very broad, diverse topic and few people fully grasp it. I’ve been running my sites on it at http://www.phimatrix.com and http://www.goldennumber.net for almost twenty years and have interacted with thousands of people on the topic yet I still learn new things all the time. I appreciate your interest and the overall quality of your writing abilities, but I’d encourage you to push beyond some of the sources you used who say it has no validity and value, and instead learn more from those who have a deeper understanding and appreciation of it.

Best regards,

Gary Meisner

https://www.goldennumber.net/golden-ratio-myth/

https://www.goldennumber.net/golden-ratio-misconceptions-by-george-markowsky-reviewed/

https://www.goldennumber.net/fast-company-design-john-brownlee-golden-ratio/

https://www.goldennumber.net/golden-ratio-design-beauty-face-evidence-facts/

https://medium.com/@quick_brown_fox/why-the-golden-ratio-matters-583f6737c10c

Greetings Gary,

It is nice to hear from you. Thank you for taking the time to share your thoughts on my analysis of the role of the extreme and mean ratio in the visual arts. I hope that you will not hesitate to let me know if there exists any mathematical, scientific, or historical inaccuracies in my writing. I do go to great lengths to verify as much as I can with numerous sources and I indeed seek out interdisciplinary reinforcement when possible. As I am sure that you are aware, my full list of resources is indeed available following the main essay.

I am also sorry to hear that you feel that our use of the program was misguided. Unfortunately, I do not remember seeing any notice in the program documentation that stated that only the armatures below a certain level of complexity should be taken seriously. If you like, I would be happy to re-conduct the experiment publicly and limit myself to armatures built on the aforementioned ratio. I can assure you that the team of international participants thoroughly enjoyed it the last time around.

Without putting forth the deluge of issues that many artists and mathematicians have with the “research” that you circulate, I could definitely begin to be swayed with some substantiated answers to some very basic questions. I would start here:

What do you believe is the primary function of the extreme and mean (golden) ratio within the specific context of pictorial composition (as opposed to “random” of course)? What is the metric by which success is evaluated for this function? Remember, the concepts of beauty, the idea of meaning, and the art experience itself are human assignments. As such, Your answer should dwell in that context.

What neural or psychological mechanism, or even aspect of evolutionary biology, would cause us to find preference with the particular ratio where spatial organization is concerned?

You see, there ARE some very interesting biological relationships for some numbers that still have little impact on such complex behaviors. For example, did you know that the specific neurons for orientation preference in our visual cortex self-organize into structures that resemble pinwheels? More remarkable is that across a broad range of species, the average density of this pinwheel is roughly equal to pi. Now while that all might sound pretty impressive, do you know what that has to do with our spatial preferences relative to a frame? Absolutely nothing.

I’ll look forward to reviewing any answers (with peer-reviewed substantiation of course) that you can muster. Thanks again!

PS-if you like I’d be happy to publicly share anything you put forward. 🙂

I cam back to these articles after a few months with the hope that your questions have been addressed by the developer above. Sadly I still see no new development, I would have been very interested to see how one would answer (with arguments) such questions.

I was hopeful that the developer would follow up as well but I didn’t expect much. The problem here is that current research in fields like empirical aesthetics, vision science, perceptual neuroscience, etc…is just not supportive of such claims (without significant misrepresentation and misunderstanding.) While it is indeed a fun soup of magical thinking, pareidolia, confabulation, and misrepresentation—it is sadly lacking anything resembling adequate substantiation.

Interesting article. A few considerations come to mind:

1. So far this series of articles has been addressing how compositional systems relate to the results of modern scientific studies of perceptual preference and what we do with visual information once it reaches the visual cortex.

I find this focus on cortical processing as the foundation for a discussion of composition problematic. Our pre-cortical visual processing isn’t passive. It’s a multi-step biological refinement system for comparing visual relationships against a series of types of contrast thresholds and subsequently altering the signal that is passed along toward the VC. I usually think of the systems as looking for significant contrast. Any relationship above the contrast threshold is enhanced. Any relationship below the threshold is suppressed. This happens multiple times before reaching the VC where we then conciously start to use the information to make decisions and predictions.

Much of what we manipulate as artists are these pre-cortical contrast relationships. Many of our compositional heuristics have evolved as seat of the pants systems for addressing these modifications. To me, any discussion of composition should address these processes.

2. Homo sapiens sapiens is far from the first hominid species on the planet. Consequently archeologists have had to look for ways to distinguish archeological sites for the different hominids. The biggest distinguishing elements of our sites tend to relate to two products of associative reasoning: material engineering and art.

The biological placement of our vocal chords in our throats is a choking hazzard but it also allows us to use our teeth, tongues, lips and soft palate to make complex sounds that we use to share and spread ideas and observations. These ideas are often about relational similarity. This led to both the development of the arts and the use of different materials to make similar tools. We didn’t just use a single type of stone. We used different stones, bone, wood, etc. based on our observations of their material properties and adapted our choice of materials to what was locally available and when possible to a material that best suited the task. It also allows us to tailor our immediate environment to our biological needs. We can live underwater because we make our own bubble of air.

Composition of visual information is an expression and celebration of our associative reasoning. We are intentionally manipulating visual relationships into structured systems. The point isn’t that we have a biological preference for any given system. The point is that we have a preference for structural systems. All of this debate about whether any compositional system is biologically preferred misses the point that what we prefer is organized systems of organizing visual information. The heueristics work not because they create a particular system but because they create a consistent structural framework. Sonnets and haiku both work because they provide structure and I can play off of that structure.

Who gives a rip what system the people used to build the Parthenon or the Pyramid or the Mona Lisa. The point isn’t the particular system. The point is that they had a system and tried to use it consistently.

If composition is an expression of biological preference I would argue that it the preference for structural depictions of visual relationships. And Aesthetics is a preference for elegantly restricted structural systems.

3. Too often our modern explorations of geometry revert to algebraic/numeric expressions of geometric relationships. It’s where all of the messy irrational numbers come from. We’re so obsessed with measuring the length of the sides of a square that we lose track of the fundamental fact that a square is an expression of area. The definition of the square root of two is the length of the sides of a square whose area is the whole number two. And this whole number is only meaningful in relationship to other geometric areas that adopt the same base unit system. By themselves they are irrelevant.

The point isn’t the length of the sides, the point is the area. And why is the area inportant? It’s important because of how our biology compares the relative size of objects. What is the tripping point where our system says that a size difference between two objects is significant? Below this threshold our brains say, “Screw it. These things are functionally the same size and we actively suppress and ignore the minor variance. Above the threshold, our brains amplify the apparent size difference so that we say, “This shape is bigger than that one.”

Which brings me to phi and Phi. My guess is that this proportion relates to that biological threshold. It isn’t a question of aesthetics or compositional structure. It’s a description of the proportional relationship where we start to pay attention.

Thank you again for even more interesting feedback Abbott. I am elated to see that you are engaging with the material with some significant depth. Revisiting these ideas is very helpful for me and I appreciate the exercise. Let’s see what you have here:

1. There may indeed be an issue with looking neuroscience/neurophysiology (or “cortical processing” as you put it) for a better understanding of how we engage with images, but I am not convinced that there is a better place to look. There may be some good points found with arguments against exploring complexity with a reductionist approach, as though we may be losing some access to the “gestalt” of the experience, but at the risk of putting forward an argument from ignorance—I have yet to hear the argument that accomplishes anything close.

There may be a little weaving of ignoratio elenchi and non-sequitur impacting your critique though. For example, you put forward an initial issue that seems to address an assertion that I do not remember explicitly stating or even subtly implying. I don’t remember asserting anywhere in my writing that our pre-cortical visual processing is a passive process. In fact I would go so far as to state that in some contexts it can be viewed as an absolutely active transformation process from phototransduction onward.

(Note: I do find great interest in your statement “Much of what we manipulate as artists are these pre-cortical contrast relationships. Many of our compositional heuristics have evolved as seat of the pants systems for addressing these modifications. To me, any discussion of composition should address these processes.” I would be very curious to review a list of these processes so that I can better address what you mean here.)

Second, I don’t remember any of my arguments or points resting on the idea that Homo-sapiens were the first hominids on the planet. In fact, I would go so far as to state that we can learn much about what we define as the art experience by looking beyond things like Acheulean hand axes and examining the behaviors of Bengalese finches, Bower birds, and Japanese pufferfish.

With all of that said, I do appreciate the ways in which you attempt to define the parameters of composition and the impact of considerations of contrast, but I still don’t see how turning away from the neurophysiology/neuroscience to look elsewhere would provide a greater insight. I’m waiting to see what your alternatives are (and I assure you that I will consider them with an open mind.)

Lastly, what I found most interesting in your response was this:

“Which brings me to phi and Phi. My guess is that this proportion relates to that biological threshold. It isn’t a question of aesthetics or compositional structure. It’s a description of the proportional relationship where we start to pay attention.”

To which I would state— prove it. 😉