The Golden Section Hypothesis: A Critical Look

Alabama Hills with Storm, #5 © Mike Spinak


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There’s a prevalent belief about composition among photographers and other visual artists. In academic circles, the belief is called “the golden section hypothesis” or “the golden section theory”, though most of it’s adherents don’t know the name. Today, I’m going to thoroughly scrutinize the golden section hypothesis, and – as a nonbeliever – I’m going to make a case against it. This also touches upon the rule of thirds, which is often used as a simplified approximation of the golden section hypothesis, to make it quicker and easier.

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What is the golden section?

The golden section is an irrational mathematical constant, where a+b is to a as a is to b – which works out to a being approximately 1.618 times the value of b.

It’s perhaps more popularly known as the golden mean, or golden ratio. Indeed, it has many names, including the golden proportion, the golden number, the golden cut, Phi (not to be confused with Pi), the divine ratio, the divine proportion, the harmonic ratio, the mean of Phidias, the cosmic code, the fingerprint of God, and more. Also, “Fibonacci” is commonly used interchangeably with “golden”, “divine”, and the rest, because ratios of successive numbers in the Fibonacci sequence (wherein each subsequent number is the sum of the previous two) converge toward the golden ratio.

The golden mean’s derivative mathematical constructs also go by many names, especially combinations of the words above, such as the Fibonacci spiral, the Phi rectangle, and the harmonic proportion – plus some other unrelated names, such as the rule of sticks, and the eye of God.

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What is the golden section hypothesis? What does this arcane mathematical constant have to do with photography?

A lot of people consider the golden mean key to composing art. Many believe that pictures should, by default, incorporate some construct of the golden mean into their structures, except in special cases where there’s a compelling reason to break the “rule”.

“The golden section hypothesis” is the belief that a visual form is most aesthetically pleasing when possessing golden mean proportions of some type.

To read an example of someone propounding the golden section hypothesis, click here.

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How do photographers and other visual artists compose with the golden mean?

First, photographers (and other artists) derive a wide variety of mathematical constructs from the golden mean, for the sake of composition guidelines. They divide a line segment according to ~1.618 (the golden section). They make a rectangle where the long sides are ~1.618 times as long as the short sides (the golden rectangle). They make an isoceles triangle where the two long sides of the triangle are ~1.618 times the length of the short side (the golden triangle). They make a triangle where the longest side is ~1.618 times as long as the second longest side, which is ~1.618 times as long as the shortest side (Kepler triangle). They make a logarithmic spiral which gets wider by a factor of ~1.618 for every quarter turn of its rotation (the golden spiral). And so on, with numerous others, such as the golden rhombus, Bakker’s saddle, Saint Andrew’s cross, and Bouleau’s armature of the rectangle. Some artists also like more oblique and esoteric constructs, such as division of the visible light spectrum by the golden mean, or segments thereof.

Photographers, and other creators of visual content, use these constructs to incorporate golden mean relationships into their work. One of the two most common ways to do this is by positioning the primary element of the picture approximately where the tightly curled “end” of the golden spiral would fit into the frame, like this or this. Further, if the photographer can also arrange the picture so that some of the picture’s lines roughly follow the spiral’s lines, like this, then that’s sometimes considered even more aesthetically pleasing. The other most common way to incorporate the golden mean is to imagine a grid superimposed on the picture-to-be, based upon vertical and horizontal 1:1.618 divisions of the picture – and then place focal points approximately on the spots where the grid lines intersect, like this. (Again, it’s often considered an added bonus if the photographer can also arrange some of the picture’s lines to sort of follow the grid’s lines.) The various other constructs listed above are similarly used most often as guides for where to place subjects and/or lines in pictures. Occasionally, they’re used for choosing relative proportions – such as composing with the background building ~1.618 times as tall, in the picture, as the person in the foreground. Or, they’re used to choose color combinations for a picture’s palette. Or … There are many ways to work the golden mean into pictures; they’re all looked upon favorably by many adherents, as long as they have some evident approximation of a ratio of ~1:1.618.

Here are a few video examples of people applying the golden mean to composition:

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Is this belief widespread? Is composing this way popular?

Yes, very much so.

Applying the golden mean to composition has been adopted quite widely, and has seemingly become elevated to established orthodoxy. YouTube has a plenitude of such videos as the ones above, if you care to look. Similarly, a Google Search on the topic will bring up more than one and a half million listings. Most basic photography instruction books discuss composing with the golden mean. Adobe Lightroom – perhaps the most popular photo processing software on the market, currently – has several golden section overlays built into the program. (You can cycle through overlays for a golden ratio grid, a golden spiral, and a Saint Andrew’s cross, by pressing the letter “O”, while in crop mode.) There are websites dedicated to seeing your pictures superimposed with golden mean overlays, such as this. There are also software applications available for overlaying anything on your computer screen with a variety of golden mean derived visual constructs, such as this. And golden mean calipers, such as this, are available online from hundreds of different makers.

In fact, it’s not only popular for composing, it’s also used by many for analyzing compositions. Analysts deconstruct pictures by drawing various line and pattern overlays upon pictures, and addressing such issues as: Does the picture conform to some derivative of the golden mean? Which one? How? How closely?

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Why do people believe the number 1.6180398874989484820458… is the key to composing art? Why do people believe the aesthetics of beauty can generally be boiled down to that number?

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The reasons why people believe this notion vary, depending upon whom you ask. Here are some of the reasons I’ve been told, and some of the reasons I’ve read:

It’s everywhere in nature; it’s the most prevalent spiral in nature; it’s the most prevalent ratio in nature

The proportions of the human body are based on the golden ratio; people like golden rectangles because they’re the proportions of the human visual field; our genetic code is pre-programmed to recognize the golden mean as beautiful; it’s where the human perception of beauty comes from

It’s nature’s definition of beauty; it’s the way nature designed our concepts of beauty; the vast majority of beauty in the world is based around that proportion

The sacred geometry of the golden section was used by God to create the universe

Images composed this way will end up visually pleasing, due to the ratio of ~1:1.618; it’s the key to showing beauty

The golden ratio provides a sense of harmony and balance; it’s due to the golden ratio that some things are more aesthetically pleasing than others; the golden mean is the universal way of defining the perfect proportions of any object

Applying the golden mean to composition has been popular since antiquity; all the ancient peoples of the world used it; it’s been used for millennia, and remains popular, today

The golden section is the most aesthetically pleasing point at which to divide a line; the golden rectangle is the most aesthetically pleasing rectangle; the golden spiral is the most aesthetically pleasing spiral; it is generally those proportions which make an image beautiful

It’s based in mathematics; it’s just a scientific observation; an image with a golden ratio is more likely to be attractive to the eye; it just works

But these reasons just move the questions back a step, without getting to the heart of the matter: Where do they get these numerous extraordinary claims about the golden mean? Why do golden numberists (the term academics use for people who espouse fantastic notions about the golden mean) have these beliefs?

Again, the reasons why people believe these notions vary from person to person. Some people believe it simply because it’s what they were told. Some people believe these notions out of a need for order and understanding. Some people find an amazing story irresistible. It appears many people want to believe there are numbers with magic properties.

In the most sympathetic instances, it’s often because a case has been presented to them, showing a seemingly compelling body of evidence; and they’ve been persuaded. Mostly the same set of the examples – nautilus shells, DNA, hurricanes, galaxies, sunflowers, pine cones, the Parthenon, and so on – get used to dazzle people with the golden mean, every time the case is presented. To show you the case in favor of these remarkable claims about the golden mean as fairly and flatteringly as I can, I present to you a video which someone recently used to try to persuade a bloc of disbelievers, which included me.


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Why does it matter? Why bother refuting it?

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It matters to me mainly because it shifts the concept of composition to an entirely different paradigm, which I consider wrongheaded. When people adopt this wrongheaded paradigm, doing so often damages their abilities to create and/or understand significant compositions.

To my view, the golden section hypothesis (and its rough approximation, the rule of thirds) merely addresses the issue of spatial arrangements, while masquerading as help with composition, itself. It’s strictly addressing the placement of subjects, and addressing this from a strictly graphical perspective. “Composition” is (or, at least, at it’s best, can be) something much richer and much more fluid. Composition is creation of meaning through all kinds of thoughtful employment of all aspects of a picture in juxtaposition with each other.

The art of photography is not fundamentally about the pleasing spatial arrangement of objects within a two dimensional space; it’s about communication. “Composing” pictures by the golden mean switches the whole enterprise from creatively constructing and communicating emotionally and intellectually engaging meaning, to hollowly making pretty or striking designs through rote application of a formula. Likewise, analyzing pictures by the golden mean switches the enterprise from trying to understand what pictures express, to checking how they compare to the dictated formula.

Those who compose by the golden section hypothesis tend to reduce visual communication down to two options: 1) incorporate golden proportions the vast majority of the time to supposedly make pictures pleasing and beautiful; or 2) “break the rule” on rare occasions to supposedly create dissonance. In my experience with photographers who believe in the golden section hypothesis, very few are aware that visual communication can actually express specific ideas and experiences, in varied, rich, articulate ways – not just “pleasing” or “dissonant”. Similarly, I’ve seen numerous occasions where golden section hypothesis adherents completely missed the most readily apparent visual communication in pictures they were looking at, while fussing over whether the pictures incorporated  golden section proportions. And I’ve often seen them mark up copies of masterpieces with various lines and squiggles, believing that this equated to understanding.

Even those golden section hypothesis adherents who realize that visual communication can be rich and articulate, tend to fall into a trap of letting the “rule” impose upon them rigidity, followed by frigidity.

The golden section hypothesis – and its approximation, the rule of thirds – take the rich enterprise of creating art, and diminish it into mere visual design. They take the roles of the artist as creator of objects with meaning, interpreter of the world, and communicator, and diminish them into the role of technician of formulaic, mechanized constructs. They take the roles of the art observer as thinker and participant in the exploration of meaning, and diminish them into tester of pattern accuracy.

Adoption of the golden section hypothesis usually leads to the subsumption of holistic art by mere visual design. All in the name of ideas which aren’t even true (as I’ll show), and don’t even work for the limited prescriptive purpose of effectively making visual designs pleasing (except by random coincidence).

Unfortunately, in this environment, wherein the golden section hypothesis has been elevated to established orthodoxy, even those who see that it’s false can’t always dismiss it, without paying a price. Choosing to not employ the golden mean in one’s photos can sometimes have unfortunate consequences. I’ve encountered a photography instructor who states that he’ll grade people low for not obeying golden section composition rules in his class; a judge for an industry photo contest for professional photographers who opines that entries which violate golden section rules are “crap”; and a head of a photo studio who considers potential employees unqualified for professional photography if they don’t compose by the golden mean.


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Can’t people compose both pleasing golden section design, and construction and communication of meaning, at the same time?

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If I’m right about my contention that the golden section hypothesis is simply false (as I’ll show), then there’s no reason why people should. If simply false, then, at best, the golden section hypothesis is an irrelevance and a non-sequitur. People often say “It’s important to know the rules before you break the rules. Once you do, it’s just a guideline – meant to be broken, when there’s a compelling reason”, but if it’s false and doesn’t work, then it has no merit whatsoever, and is not even a sensible guideline – just an added encumbrance. Suggesting to a photographer to use the golden section as a guideline, when composing, is like suggesting to a physician to give their examinations and consultations in iambic tetrameter. It’s hampering, and for no good reason.

This non sequitur is problematic in its lack of context – lack of context of the actual scene around the subjects at the moment of the picture’s creation, lack of context of the photographer’s creative intent and/or the viewer’s interpretive intent, and lack of context of everything about the picture, itself. Suggesting where or how to place subjects in a frame, in absence of consideration of an individual photo’s overall construction of meaning, is like suggesting to put the denouemont on page 542 of a novel, without taking anything else about the book into account. It’s arbitrary and preposterous.

In practice, the two composition paradigms are likely to be in opposition to each other: a primary concern with design will usually stifle most traces of art; a primary concern with artistic expression will involve every aspect of your picture, and thereby override and subsume design for its own sake. “Designing” photographic compositions will usually lead to frigid photographs, or perhaps even sterile ones. Composing a good picture begins with something worthwhile to express, then communicates it effectively, by whatever (unique to each individual situation) means work. Subject placements derive from these foundations. Intent is the guide. One tries to make subject placement, line flow, lighting – every aspect of the picture – integrate with the picture’s content. Insisting a priori that elements in the picture have to be placed here, not there, can change the juxtapositional relationships of a picture’s elements in ways that change meaning and expression.


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Then why does it look like a lot of the pictures on this website seem to follow golden section composition or rule of thirds composition? Are you perhaps unconsciously following these rules, due to an innate understanding of what’s aesthetically pleasing, and just not realizing it?

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No, I’m not. It’s an easy mistake to make, though. Understanding the sources of this error will help you understand some of the main reasons why the golden section hypothesis and the rule of thirds are so believable, and so popular, despite being wrong.

I have literally thousands of pictures online, and many times this number in the cue, to be put online. If one were to look through my pictures with an eye toward finding cases in support of the claim that I follow the rule of thirds and/or the golden mean in my work, one could find hundreds of pictures seemingly in conformity – such as this one:

Rebel Tulip, Mount Vernon, Washington © Mike Spinak

Conversely, if one were to look through my pictures with an eye toward finding cases in support of the claim that I tend to center my pictures, one could again find hundreds of pictures seemingly in conformity – such as this one:

Passion Flower (Passiflora caerulia) © Mike Spinak

Take a look at this illustration I made:

The bullseyes represent rule of thirds intersection points; the hollow circle represents the center area. Different people might consider larger or smaller diameter circles to fall within their personal interpretations of the rule of thirds intersection points, or the center, but hopefully we can agree that I illustrate these accurately enough, for the sake of discussion.

Please note that there’s four times as much rule of thirds intersection point area within the frame as centered area within the frame. (The same would be true with golden section areas.)

Imagine that I attach an intervalometer to my camera, set to trigger a shot once every 30 seconds. and imagine that I mount my camera to a tripod on wheels, and take it for a long, aimless walk each day, towing it behind me – so that it takes just random snaps.

Strictly by random chance, the pictures I come back with will most likely have a ratio of about four “rule of thirds compositions” to every “centered composition”, simply because there’s four times as much rule of thirds area within a frame as there is centered area.

Thus, even if good compositions are equally likely to have subjects fall anywhere in the frame (as could accord with my comment that the golden section hypothesis and the rule of thirds have no merit, whatsoever), then we should still expect random chance to distribute four times as many good compositions along rule of thirds intersection points as the number of centered good compositions – strictly because there’s four times as much area within the frame.

If you add in the golden mean, too, then you’re adding yet an additional few times more area in the pictures (since the golden spiral can be flipped vertically and / or horizontally, and also flipped left or right). Taken together, there’s about six to ten times as much rule of thirds and golden mean area within the frame as centered area within the frame, depending upon your interpretation of the appropriate size diameters of each area.

So, even with random distribution, we should expect somewhere in the range of six to ten times the number of golden section or rule of thirds pictures as centered pictures. This is what people see, and misinterpret. If they see five or ten times as many pictures which accord with these rules as pictures which don’t, within someone’s body of work, then they tend to think they’re seeing patterns of following the rules, rather than realize when they’re seeing random distribution within a context of more available “rule following” area than “rule breaking” area. Likewise, if they see five or ten times as many good pictures in the world which “accord with the rules”, compared to the number of good pictures which “break the rules”, then they tend to think that following these purported rules is five or ten times as likely to lead to a good picture, instead of realizing that it’s just a matter of random distribution and square area covered.

The amount of golden section area and rule of thirds area within the frame becomes even much greater, yet, for those who also count anything that falls along the lines, in addition to the intersections and the end points of the spirals. By the standards of those who count all the possible lines for the golden section grid, plus the rule of thirds grid, plus golden spiral lines in any orientation, plus Bakker’s saddle lines in any orientation, plus golden triangle lines in any orientation, plus Saint Andrew’s cross lines, etc., it becomes almost unavoidable that the subject will be placed somewhere with golden mean significance, no matter where in the frame the subject is placed. (This is even more so when considering that many also consider loose approximations to be valid cases. For examples, see here and here.)

In fact, mathematics is virtually unlimited in its possible derivations from a mathematical constant. The section, grid lines, rectangle, triangle, spiral, rhombus, etc., in this article, derived from the golden ratio, are just a small sample of the literally infinite constructs which could be derived from it. Any competent mathematician could easily derive a golden mean construct to match absolutely any subject placement within a frame.

Furthermore, whether to see one kind of composition or another can often be a matter of interpretation. I could show you pictures where I could say the composition is centered on the face, while you could say that the eyes are approximately following rule of thirds or golden mean placement. Or, I could show you pictures where I’d say that the centrally placed mouth is the main element of the picture, while you might insist that the eyes are the main element of the picture, and then say that they’re placed approximately following the golden mean. For example:

Angry Elephant Seal Cow (Mirounga angustirostris) © Mike Spinak

There is no “real” correct answer, in these cases, only interpretation – and people who want to believe in the “rules” ideas and look for the patterns will always find them.

Golden section adherents will often state that those who flaunt that they’re breaking the rules are usually following the rules, without realizing it. They’ll add that we naturally compose this way, because it’s what beauty is – even when we can’t identify the hidden golden section variation within what we’re seeing. The reality is that they’re doing the things discussed above, adding up to retroactively applying the golden section to any works they like, in a way so broad that it becomes literally meaningless. At that point, what it really boils down to is: If they like a picture, they’ll come up with a way to claim it fits the golden section hypothesis. If they don’t, then they’ll often find a way to claim the opposite.

If you go selectively looking through all of my pictures, you can find plenty of them that will fit “The Rules” – whatever set of rules you happen to believe and search by. That’s just the way random distribution works, over a decent sized set of pictures. After all, I don’t go out of my way to avoid placing subjects where the golden mean or rule of thirds advise, any more than I go out of my way to place them where them where they advise.

I would never say that I don’t have plenty of pictures which can viewed as following the golden mean or the rule of thirds. Rather, I’d say that I compose in a way which is entirely orthogonal to these rules, and I’d also say that accordance (or absence of accordance) with these rules has no bearing upon the merit (or lack thereof) of my compositions or my pictures.

I would also never say that I’m breaking these rules, nor that I’m suggesting others should break the rules. To say that I am or to suggest you should would still be operating within the paradigm of these rules being real and having some relevance. They aren’t and don’t.

Let me clarify this point with an analogy:

During Ayatollah Khomeini’s time as Supreme Leader of Iran, he codified numerous rules and guidelines about how to urinate and defecate, and how to clean yourself, afterward. Among them:

• One must not face the direction of the sun nor moon while evacuating

• One should keep one’s head covered, and have the weight of one’s body carried on the left foot, while evacuating

Now, suppose for a moment that you went ahead and defecated into a toilet while facing the direction of the moon. Most likely, your choice of the direction to face while defecating was heavily influenced by the direction the toilet was mounted. Practical considerations of the circumstances of the scene actually mattered. Meanwhile, breaking Khomeini’s rules about how you empty your bowels probably never even factored into your consideration. If someone pressed you on the matter, perhaps you might say that those codes and guidelines were just some arbitrary rules that someone created, which actually had nothing intrinsic to do with sound defecatory protocol.

Likewise, with the purported rules of composition; they are just some arbitrary rules someone created, which really have nothing intrinsic to do with good compositional protocol. Ignoring someone else’s erroneous fixations as you go about your business is not “breaking rules”, by any grand sense of the word “rule”. Those rules have nothing to do with the way the world works. There’s no sound principle underlying those rules. They’re not explanatory, nor predictive. As governing laws (or even rules of thumb or loose guidelines) of how the world works, more specifically how photography and all visual arts work, those rules aren’t real – so you can’t really break them, since they don’t really exist.

Instead of concerning yourself with following or breaking those kinds of rules, I suggest a completely orthogonal way of creating pictures, prioritizing communication of meaning first and foremost, subsuming design into this process, and leaving the rote design concepts that are strictly for the sake of aesthetics out of the equation entirely. Sometimes the results (of composing to make pictures which speak) might accord with the “rules of composition”, and sometimes they might not. Let the chips fall where they may, without obsessing over subjugating inspiration to fictional rules and guidelines.

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Is the golden ratio everywhere in nature? Is it the most prevalent ratio? Is it the most prevalent spiral? What about the nautilus shells? Hurricanes? Galaxies? Whirlpools? Embryos? Our DNA?

No, there is no special prevalence of golden ratios and golden spirals in nature, as far as I can tell. How can anyone disagree with such a large and compelling body of persuasive evidence, such as the presentation in the video, above? Read on, and find out.

NAUTILUS SHELLS

Let’s start with the nautilus shells, because they’re the golden numberists’ flagship example. Nautilus shells are shown in both the first and last videos, above. They’re featured on the covers of books about the golden section, such as this, this, this, this, and this. They’re the golden numberists’ most favored example.

While golden numberists usually selectively focus exclusively on the shell of the chambered nautilus (Nautilus pompilius), there are many thousands of species of spiral shelled mollusks which they conveniently overlook  – such as this, this, and this.

But let’s get back to the chambered nautilus shell, itself – the prime example that the golden ratio has a special role in nature and aesthetics.

For the sake of reference, here and here are a couple pictures of chambered nautilus shells.

The spirals of chambered nautilus shells are not the golden spiral. To my eye, they’re not even that close. See for yourself. Here’s a comparison of a golden spiral (left) to an actual nautilus shell spiral (right). Here’s a golden spiral overlayed onto a picture of a longitudinal-section cut chambered nautilus shell. Here are more details about the specs of golden spirals and chambered nautilus shell spirals. In brief, measurements of nautilus shell spirals have found them to range from 1.24 to 1.43, with an average ratio of about 1.33 to 1. In the words of Clement Falbo, a mathematician who studied the spirals of chambered nautilus shells, “It seems highly unlikely that there exists any nautilus shell that is within 2 percent of the golden ratio, and even if one were to be found, I think it would be rare rather than typical”.

Is an average ratio of 1.33 to 1 close enough? The spiral of the chambered nautilus generally about triples in radius with each full turn, while a golden-ratio spiral grows by a factor of about 6.85 per full turn. By my standards, that’s nowhere close. In the words of John Sharp, another mathematician who has studied the spirals of chambered nautilus shells, [the nautilus spiral] “is a very long way from the one for the Golden Section rectangle. In fact, … it is clear that the Golden Section rectangular spiral and the Nautilus spiral simply do not match. There just aren’t enough turns with the Golden Section spiral.”

So, upon further examination of the usage of chambered nautilus shells as an example of golden spirals all around us in nature, we learn:

• The golden numberists are often extremely selective and biased in their perception, ignoring the thousands of clear non-instances of golden spirals in mollusk shells to tell us “They’re all around us in nature” when they think they’ve found ONE instance

• The golden numberists are slipshod in their work to demonstrate their case, not even bothering to check their crown jewel example

• The golden numberists perpetuate false information

• The golden numberists can’t tell the difference between a golden spiral and a distinctly different non-golden spiral

• By being unable to tell a golden spiral from a different spiral – mistaking one for another, latching onto an incidence of a non-golden spiral as a celebrated example of the beauty of golden spirals – golden numberists are demonstrating that their believed innate aesthetic preference for the golden spiral is false

• Since the golden numberists seemingly haven’t managed to find even one single legitimate case of a golden spiral among the innumerable species of spiral shelled mollusks, golden spirals are apparently not remarkably disproportionately more prevalent in nature than non-golden spirals

The golden spiral is a member of a larger class of spirals: logarithmic spirals. It is true that logarithmic spirals of all kinds are commonly found in nature. It has to do with constraints of geometry upon the way organisms can grow in size. In the case of spiral shelled mollusks, such as many cephalopods and gastropods, a logarithmic spiral is the natural outcome of a continuous linear pattern of growth that overlaps upon itself, for evolutionary advantages of efficiency and safety. The distribution of types of logarithmic spirals seems fairly even, throughout nature (as the various spiral shelled mollusks pictured, above, demonstrate). Golden spirals, along with all other kinds of logarithmic spirals, do sometimes occur – just not in outstanding prevalence.

Rather than the golden mean’s spiral, the average chambered nautilus shell spiral is actually a much closer match to the spiral for an equally interesting ratio, the silver mean. (I’ll see if I can get an illustration of the silver spiral online.)

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HURRICANES

The spiral formations of hurricanes fall across a broad range of rotational curves, from those more tightly twisted than a golden spiral to those more loosely twisted than a golden spiral. Among the random distribution, some of them occasionally approximate golden spirals, while most don’t. If you look carefully at this picture of Hurricane Katrina, you can see plainly that it is a more tightly twisted spiral than a golden spiral; whereas, if you look carefully at this picture of Hurricane Earl, you can see clearly that it is a more loosely twisted spiral than a golden spiral. Moreover, all hurricanes change shape over time, starting out as looser and more poorly formed tropical storms, then tightening as they strengthen, and then loosening and breaking up as they wane.

WHIRLPOOLS

Similarly for whirlpools: they cover a range of rotational curves, which happens to include golden spirals, among others. Golden spirals are not at all widespread among the distribution of whirlpool rotational curves. Some are more loosely twisted, while most whirlpools are more tightly twisted than a golden spiral. Don’t just take my word for it: next time you take a bath, next time you wash dishes in the sink, and next time you flush the toilet, you can see for yourself. Or, if you prefer, you can look at some of the numerous long exposure photos of foam and/or leaves swirling in whirlpools, online. Here’s an example of one that’s more loosely wound than a golden spiral, and here’s an example of one that’s more tightly wound than a golden spiral. Here, here, here, and here are more examples that are more tightly twisted than a golden spiral.

Golden spirals are not in the majority; they are a small minority of the spirals in hurricanes and whirlpools. I don’t see the fact that a small percentage of hurricanes and whirlpools briefly resemble the golden spiral for a moment of their existence as indicating anything meaningful about any underlying natural principle favoring golden spirals in gases or liquids.

GALAXIES

Like hurricanes and whirlpools, galaxies can be found in a wide range of spirals, with some small percent approximating the golden spiral, while most don’t. Our own home galaxy, The Milky Way, doesn’t. See for yourself. The Milky Way has a pitch angle of about 12 degrees. The golden spiral has a pitch angle of about 17 degrees. Spiral galaxies, in general, have pitch angles from 10 degrees to 40 degrees.

EMBRYOS

I tried looking for golden spirals in embryos. I just don’t see it. Look at the results for a Google search for embryo pictures, and see for yourself. I even tried a Google search for the terms “Golden Spiral Embryo”; and the best results still appear to me to be a big stretch. In my view, that’s an unconvincing example of embryos exemplifying golden spirals in nature.

DNA DOUBLE HELIX

I’m not sure why golden numberists refer to what they’re purporting here as “golden spirals” or “Fibonacci spirals”, when they’re ignoring the double helix’s spiral characteristics. They’re discussing the double helix only in profile, not from a front angle. To make sure this is clear, it’s like discussing the Milky Way’s spiral from this angle, rather than from this angle. In any case, the claim that there’s a golden ratio of the double helix’s width and cycle length appears to be untrue, as far as I can determine. The DNA double helix has a width at any given point of 1.7 nanometers; the double helix’s spiral has a cross section width of 2 nanometers, and the double helix’s cycle length is 3.4 nanometers. This gives a ratio 1:1.7. That’s not a golden mean. You might think 1:1.7 is close, but it’s really not that close. If the cycle length had been 3.3 nanometers, or 3.2 nanometers (which would produce ratios of 1:1.65, and 1:1.6, respectively), then one might have a case to say “close enough”. However, as it is, saying that the DNA double helix’s width to cycle length is a golden ratio is just wrong.

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By now, you might be noticing a pattern, that all of these golden spiral claims are erroneous. While golden numberists claim that the golden spiral is uniquely aesthetically pleasing and preternaturally prevalent, it appears that they have little idea what one actually looks like, and they also tend to mistake most logarithmic spirals they see for valid examples of a golden spiral – perhaps even anything that vaguely hints at being spiral-like. It’s inaccurate to call all logarithmic spirals of any proportion “golden spirals” or “Fibonacci spirals”.

I recently pointed out to another photographer that her photo of a nautilus shell, with the title “Infinite Fibonacci (the Golden Ratio) – Cross Section of Chambered Nautilus Shell” was not a golden spiral nor Fibonacci spiral. She replied by telling me that it was an approximation, and adding text to her picture calling it “nature’s most perfect approximation of a Fibonacci spiral”. With an average ratio of ~1.33 versus the ~1.618 of the golden ratio, and an average of about tripling in radius for each full turn versus the golden spiral nearly septupling in radius for each full turn (i.e., ~3.2x versus ~6.85x), it’s actually a very poor approximation. Golden numberists often try to evade refutation with vague “approximation” claims, but trying to get around clear mismatches between the spirals by claiming that they are “approximate” is less than forthright.

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Selective data mining and pseudoscience

Hypothetically, let’s suppose that the DNA double helix actually did have a golden ratio proportion of width to cycle length (even though it doesn’t). Or suppose the Milky Way was a golden spiral, or human embryos were. Would it be sensible to interpret that fact as a profound indicator of nature’s own definition of beauty, or a divinely mandated proportion, or some such? In my opinion: no. It’s simply the finding of an unsound technique that can be used to find any pattern or proportion you want from coincidence, within a virtually infinite set of arbitrarily chosen data points.

Suppose I wanted to find some seemingly remarkable mathematical properties in DNA, which I’d purport indicate some deep and mysterious principle of the universe. I’d start by trying to get all the DNA measurements I possibly can – length of the double helix; width of the molecule; width of the spiral; distance between apexes; number of genes; cycle length; number of twists; total number of atoms; number of atoms per cycle length; number of protein molecules per cycle length; various kinds of ratios of Adenine, Cytosine, Guanine, and Thymine; molecular weight, and so on. With the DNA double helix (or the Parthenon, or the human body, or anything else), there is no limit to all the things to measure, except your imagination. Once I’ve compiled a nice big list of thousands of measurements, then I’d do an unsystematic search through all possible pairs of measurements, looking for something that fit with whatever I was looking for. I’d find it, easily, regardless whether I was looking for a golden ratio ratio, or a silver ratio, or a Pi ratio, or 1:2.5 ratio, or a 1:4 ratio. They’ll all be there, within the abundance of measurements from which to pick. Additionally, I could apply any kinds of calculations I want to any of my measurements, while calculating them by whatever factors suit my fancy. It would be a fairly simple (though, laborious) task to find “evidence” that our DNA prophesied all of the pivotal events of the 20th century, and just as simple to match our DNA up with Homer’s Odyssey.

This is the “magic” of pseudoscience.

There are lots of examples of this kind of methodology producing seemingly amazing results. Take, for example, the popular pastime for conspiracy theorists, to find parallels between the assassination of Lincoln and the assassination of Kennedy:

Lincoln was elected to Congress in 1846.
Kennedy was elected to Congress in 1946.

Lincoln was elected president in 1860.
Kennedy was elected president in 1960.

Lincoln was succeeded, after assassination, by vice-president Johnson.
Kennedy was succeeded, after assassination, by vice-president Johnson.

Andrew Johnson was born in 1808.
Lyndon Johnson was born in 1908.

Lincoln was sitting beside his wife when he was shot.
Kennedy was sitting beside his wife when he was shot.

Lincoln was shot on a Friday.
Kennedy was shot on a Friday.

Kennedy had a secretary named Lincoln.


Lincoln lost a child while President.
Kennedy lost a child while President.

Lincoln was shot in Ford Theater.
Kennedy was shot in a Ford made car.

Kennedy was shot in a Lincoln (Continental Limousine).

Lincoln was shot in a theater and his assassin ran to a warehouse.
JFK was shot from a warehouse and his assassin ran to a theater.

John Wilkes Booth was born in 1839†.
Lee Harvey Oswald was born in 1939.

And so on. You can find plenty more, if you want.

Does all this indicate something deeper and more mysterious about Lincoln’s and Kennedy’s assassinations? I don’t think so. Could it really be just coincidence? Definitely. These kind of connections are easy to find, everywhere, if you bother to sift through enough data.

I must point out that the last pair of coincidental facts is not accurate: John Wilkes Booth was actually born in 1838, not 1839. However, this error is a much smaller percent off than the chambered nautilus–golden spiral error. It’s much smaller percent error than the DNA double helix cycle–golden ratio error. In fact, it’s a much smaller percent error than when the videos above (and many golden numberists) use “1.6″ interchangeably with “1.618″.

It’s a lot easier to come up with these kinds of extraordinary cases, when we comfortably accept fudging the facts a bit.

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Has the golden section hypothesis been popular since antiquity? What about the Parthenon? And Leonardo da Vinci?

THE PARTHENON

It does not appear that the ancient Greeks purposely worked golden proportions into the Parthenon’s design.

Again, the notion that the Parthenon matches up with the golden ratio doesn’t hold up to examination. First, the Parthenon’s dimensions:

Eastern Width: 101 feet, 3.5 inches
Western Width: 101 feet, 3.9 inches
North Length: 228 feet, 0.8 inches
South Length: 228 feet, 0.7 inches
Height: 64 feet

So, The Parthenon’s length to width is a ratio of about 1:2.25. It’s height to length ratio is about 1:3.56. It’s height to width ratio is about 1:1.58 – kind of close, but well off by a few percent. The ancient Greeks would have built it much closer to the Golden Ratio, if they were trying to do that, just like the way they got the length on the North side of the Parthenon within about a tenth of an inch of the length on the South side.

Is a few percent close enough? Take a look at this diagram of the Parthenon with a golden rectangle over it, and see for yourself.

To my eye: No, not close enough. It arbitrarily cuts off the steps, on the sides, quite significantly. Or, you could raise the base of the golden rectangle above the pedestal steps, to the floor level, in which case the apex of the roof doesn’t even come close to touching the top of the golden rectangle. Or you could expand the size of the golden rectangle to include the edges of the steps, in which case all three corners of the roof don’t even come close to touching the top nor sides of the golden rectangle.

In short, it just doesn’t fit. You can see this on the last video, if you watch closely. Watch the video from about 3:05-3:12, and you’ll see that he excludes the steps when he places his “Fibonacci gauge” to show the width of the Parthenon, but then includes the steps when he places his “Fibonacci Gauge” to show you the height. That’s no way to achieve an accurate height-to-width ratio! Watch closely, and you’ll see that, every time he lays down the “Fibonacci Gauge” to demonstrate a golden ratio relationship, he’s being entirely arbitrary (in favor of the appearance of golden ratio proportions, of course) about what to include, and what to exclude. His entire demonstration is literally sleight of hand, and it’s false.

Besides diagrams, some of you might have seen photos of the Parthenon, with the golden rectangle superimposed upon it, like this. Allow me to point a few things out about this picture, which apply to many of these types of pictures.

• The picture is taken from below, not from head-on. Because of this, the top of the building is farther away than the bottom of the building, causing the top to appear smaller than it is, in relation to the bottom. You can see a more obvious example of this kind of perspective effect in this picture of the Empire State Building. If the Parthenon picture was taken from straight on, rather than looking up at it from below, then the left and right corners of the roof wouldn’t come close to fitting within the rectangle.

• The decisions of what to include within the rectangle and what to exclude from it are entirely arbitrary. The line comes down to somewhere in the middle of the pedestal steps, neither including the bottom steps of the Parthenon’s pedestal, nor excluding the top ones. The pedestal’s steps also overflow out of the sides of the golden rectangle, considerably. The left corner of the roof doesn’t quite fit in the rectangle, either. It’s not a close fit; it’s just made to look like one by arbitrary inclusion and exclusion.

• The red line for the golden rectangle is drawn to be unnecessarily thick. Golden numberists often use thick lines in their demonstrations because they cover up a wide margin of error. In this case, the lines are seven pixels thick, accounting for 14 pixels of the golden rectangle’s 420 pixel height, which allows the line thickness to cover ~3.33% margin of error.

The Parthenon does not seem to have golden mean proportions.

POPULARITY OF THE GOLDEN SECTION HYPOTHESIS IN ANTIQUITY

While the ancient Greeks prolifically wrote down records of their thoughts about aesthetics, and while they kept good records of many things, there’s nothing to corroborate that they believed the golden mean was especially aesthetically pleasing, nor that they worked it into their architecture. They were aware of the golden mean, and its derivative spirals, rectangles, sections, etc. However, they appear to have made no comments about any special aesthetic properties of the golden ratio. Euclid explains how to calculate it in his book Elements; it’s a step in the construction of the regular pentagon. The name he uses for (what we call) the golden mean is “Division in Extreme and Mean Ratio”. It’s just another one of the calculations he discusses in the book. That’s all it was to the ancient Greeks, generally, as far as is known. The notion that the ancient Greeks widely held the golden ratio in special aesthetic regard, and worked it into their architecture, is unsubstantiated myth, based upon errors.

LEONARDO DA VINCI

Leonardo Da Vinci is the top Renaissance painter that golden numberists, and especially adherents of the golden section hypothesis, like to point to, to show that Renaissance painters incorporated the golden mean into their beautiful paintings. He’s widely considered one of the best painters in history, as well as one of the smartest people in history. Some of his works are among the best known and most acclaimed paintings in the world. He was well versed in the golden section; indeed, he illustrated the book De divina proportione, for his friend Luca Pacioli.

Sounds good, so far – however, Luca Pacioli’s book did not make recommendations to incorporate golden mean proportions into works of art; and there’s no reason to believe Da Vinci believed in the golden section hypothesis, nor that he applied it to his works.

Golden numberists often purport that Da Vinci incorporated golden ratio proportions into Vitruvian Man. For example, one particular claim made about this picture, is that Vitruvian Man’s navel height / full height ratio is a golden proportion. It’s actually .604, not .618.  That’s kind of close, only off by a little more than two and a quarter percent. However, if Leonardo had intended to place the navel at the golden proportion, there’s no reason he wouldn’t have placed it spot on, slightly higher up.

They also purport that Da Vinci incorporated golden proportions into the Mona Lisa. You can find many illustrations showing how the Mona Lisa has golden mean proportions, such as this. What you can’t find, is sensible reasons behind the placement of the golden sections and golden rectangles in these illustrations. Look at that top line. It would’ve made just as much sense to place it at the top of her head, or in line with her eyes. Placing it where it is, in order claim it’s a golden section proportion, is completely arbitrary. By the methods used to place that line, I could make equally (il)legitimate claims of golden proportions with just about any painting or photo in the world. The golden rectangle’s placement arbitrarily excludes part of her face on her left side, in order to make her face fit. The size and positioning of the golden rectangles over her face are entirely arbitrary. With the flexibility to arbitrarily place and arbitrarily size the golden rectangles, like this, you could make them line up over literally almost any person’s face, and almost any animal’s, too, regardless whether live, photographed, painted, or sculpted.

Golden numberists also make these golden claims about Da Vinci’s unfinished painting of Saint Jerome. Putting aside that it’s not a perfect fit with Saint Jerome’s left elbow nor the top of his head, the golden rectangle placement completely ignores that he has a right arm sticking way out of the rectangle. (And, even if it was a perfect fit: does it really seem like that box around his body, that particular ratio of his sitting height and width in the picture, is the dominant feature of the composition, and is what makes it work?)

If you go through all of the claims about Leonardo Da Vinci incorporating the golden section into his work, they’re all arbitrary, tenuous, and unconvincing. Furthermore, Da Vinci is known to have drawn his pentagons from approximations, rather than calculating them with the golden section, which seems to go against the notion that he placed importance on the golden mean.

In sum, there’s no real reason to suspect that Leonardo Da Vinci incorporated the golden section into his work, and it’s not really visible in his body of work. The claims are nonsense.

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So, what are we left with, in regard to the claim that the golden section hypothesis has been popular for composing art through the millenia? It’s just not true. The claim that a visual form is most aesthetically pleasing when possessing golden mean proportions actually comes not from the ancients, but from Adolf Zeising, from his lengthily titled 1855 book, 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. Zeising’s book is the source of the body of mythology which has so taken golden section hypothesizers and golden numberists of today. All of these beliefs about its special aesthetic properties, its use in art and architecture since antiquity, its prevalence in the natural world, its incorporation into our body proportions and visual field shape and our DNA, its programming into human perception of beauty, etc., have sprung up since Zeising’s book.

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My own experience

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Photographing nature for a living, I can tell you that I run across a fair share of natural spirals. There has been no particular prevalence of Fibonacci spirals nor golden spirals among them. Here are a few examples of spirals I’ve photographed:

© Mike SpinakThis Passiflora tendril has two different kinds of spirals. One is kinda-sort-of-maybe close to a golden spiral, but has a tighter curve; the other one is nowhere close.

© Mike SpinakThis California slender salamander is curled into another spiral that’s tighter than a golden spiral.

© Mike SpinakThis spiral from dry ice boiling in water is yet another example tighter than a golden spiral.

And with all the spirals I’ve come across in nature, I haven’t come across a good match with a Fibonacci or golden spiral, yet.

I’ve also seen and photographed my share of spirals made by ancient people. None of them have been close to Fibonacci or golden spirals. For example, here’s a ~2,000 year old spiral petroglyph and pictograph:

© Mike SpinakThis is kind of close to an Archimedean Spiral, but nowhere close to a golden Spiral.

I haven’t seen nature show an outstanding prevalence of golden spirals, and I haven’t seen people historically show a predilection for golden spirals.


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Is the golden section hypothesis correct? Are visual forms most pleasing when possessing golden mean proportions? Are golden rectangles the most pleasing rectangles? Are golden spirals the most pleasing spirals? Are golden section subject placements the most pleasing for compositions?

The golden section hypothesis is a testable claim. It’s been tested. It’s been refuted. It’s false.

For examples:

• H.R. Schiffman, of Rutgers University, tested the golden section hypothesis in 1966, by asking subjects to “draw the most aesthetically pleasing rectangle” on a piece of paper. He found that their average length to width ratio for the drawn rectangles was 1:1.9, not 1:1.618.

• T. H. Haines and A.E. Davies also tested this by asking subjects to draw their preferred rectangles. They concluded that there was no clear preference in their results. (The psychology of aesthetic reaction to the rectangular, 1904)

• George Markowsky, a mathematics professor at the University of Maine, has also investigated whether the golden rectangle is the most aesthetically pleasing rectangle, in 1992, by presenting participants with a set of rectangles of different proportions (including a golden rectangle), and asking them to pick the most pleasing; and he found that the most commonly selected “most aesthetically pleasing” rectangle is one with a ratio of 1:1.83.

• Dr. Keith Devlin, mathematics professor at Stanford University, performed tests tests on several of his classes, presenting them with a page of rectangles of various aspect ratios and asking them to pick out the most pleasing; he found that few picked out the golden rectangle as most pleasing.

• There used to be a web poll to select the most pleasing rectangle, at this address. (Unfortunately, it’s now gone.) It, too, has shown the notion that the golden rectangle is the most pleasing is false. Among the 8 rectangle choices, only 12% of the 1501 participants picked the golden rectangle as the most pleasing – whereas 30% picked a 1:1.33 rectangle as the most pleasing, and 16% picked the 1:1.5 rectangle as the most pleasing.

• There’s another “Most Pleasing Rectangle Poll”, still available, at this address. The 1:1.46 rectangle is in the lead for Poll 1 (the vertically oriented version of the poll), with 24% of the 11,370 participants, beating the Golden Rectangle’s 18%. In Poll 2 (the horizontally oriented version of the poll), the golden rectangle is in the lead, chosen by 27% of the 7,809 participants.

• In 1969, J. M. Hintz and T. M. Nelson, of the University of Alberta, tested the hypothesis that the shape of the human binocular visual field determines an aesthetic preference for golden rectangles. They found that there was no correlation between an individual’s visual field ratio and preferred rectangle ratio, and concluded “The results indicate that the proportions of the binocular visual field should not be used to explain preference for rectangles having golden-section characteristics”.

• In 1974, Michael Godkewitsch, of the University of Toronto, investigated the influence of presentation upon choices of the most aesthetically pleasing rectangle (which included the golden rectangle). His results showed that the order in which rectangles are presented, the orientation in which rectangles are presented, and other artifacts of presentation, will affect which rectangles people pick as most pleasing.

• In 1946, G.G. Thompson tested the rectangle preferences of preschoolers, 3rd graders, 6th graders, and college students, by presenting them with 12 rectangles and asking them which they liked best. The preschoolers showed no particular preferences. The 3rd graders and 6th graders preferred wider rectangles than the golden ratio. Thc college students preferred the 1:1.55, 1:1.6, and 1:1.65 (those rectangles closest to the golden rectangle, in the experiment). This experiment may indicate that rectangle preferences, if they exist at all, seem to vary by age. (The effect of chronological age on aesthetic preferences for rectangles of different proportions)

• In 1951, C.W. Neinstadt and S. Ross tested the rectangle preferences for people aged 61-91, and for college age subjects, by presenting them with rectangles and asking them to pick the most aesthetically pleasing. The young adults showed a weak preference for the 1:1.65 rectangle. (The 1:1.6 rectangle would’ve been the closest to the golden mean in the experiment). The elderly subjects preferred wider rectangles than golden rectangles. This experiment, again, may indicate that, if consistently predictable rectangle preferences exist at all, they vary by age. (Preferences for rectangle proportions for college students and the aged)

• In 1938, R.S. Woodworth found differences between the rectangle preferences of American and European subjects, undermining the notion of a universal preference for golden rectangles. (Experimental Psychology)

• In 1970, D.E. Berlyne investigated the rectangle preferences of Canadian high school girls and Japanese high school girls. (The Golden Rectangle and Hedonic Judgments of Rectangles: A Cross-Cultural Study.) He found that both Canadian and Japanese girls preferred the square to the golden rectangle, with 27% of Canadian school girls and 20% of Japanese school girls picking the square as their first choice, while only 9% of Canadian school girls and 5% of Japanese school girls picked the golden rectangle as their first choice. He also found that Japanese students first choice percentages were fairly close for all nearly-square rectangles, but dropped off sharply for 1:1.5 rectangles, showing a distinct preference for squares; while the Canadian students first choice percentages dropped off greatly after the square, and then peaked again for the 1:1.5 rectangle, with 18% picking it as their first choice. Overall, the Canadians and the Japanese showed different patterns of preferences, and the Japanese school girls appeared to have a marked cultural preference for square rectangles.

• In 2006, Steven E. Palmer, Jonathan S. Gardner, and Thomas D. Wickens, of the University of California, Berkeley, tested people’s preferences for subject positions within the frame, though 2 methods: 1) adjustment of existing photographs, and 2) free choice in taking photographs. They found that, with both experimental methods, for front facing subjects, preference was greatest for subjects positioned at or near the center of the frame.

These are just a small sampling of the body of research which has produced results in contradiction with the golden section hypothesis, over the last century or so.

In summary, results of tests of the golden section hypothesis vary widely, and are also strongly affected by artifacts of presentation. Age and culture also appear to influence ratio preferences. A definitive preference for golden rectangles over rectangles of other proportions has not been established, despite repeated testing. Preference for golden ratio positioning has not been established, either. The purported visual field basis for a preference for golden mean rectangles has been disproved. Many aspects of the golden section hypothesis have been put to the test, and every one that has been has not withstood testing. It seems the golden section hypothesis can’t be rigorously demonstrated. In the words of C. Plug: “A review of studies of the golden section hypothesis indicates that the attempt to determine preferences for certain ratios by asking subjects to rank a fixed series of stimuli does not lead to meaningful results”.

As Michael Godkewitsch sums it up: “…Aesthetic theory has hardly any rationale left to regard the golden section as a decisive factor in formal visual beauty”. (The golden section: an artifact of stimulus range and measure of preference, 1974)

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Is there anything special about the golden section or the Fibonacci series?

Some pine cones, and the flowers and seed heads of some plants, do, indeed, show sequential numbers in the Fibonacci sequence, and (approximately) show golden ratios, in the number of clockwise spirals to the number of counterclockwise spirals. This is a long way from “everywhere in nature”, but they’re definitely there in pine cones, sunflowers, pineapples, and some others.

These plant structures don’t actually tend to have golden spirals, as is commonly purported, but they do have Fibonacci ratios in their structures, nonetheless. (The rotational curves of these spirals don’t usually match golden spirals. The particulars vary widely, depending upon the pine species and the sunflower species; they’re usually visibly looser spirals than golden spirals. For example, here’s a picture of a pine cone, overlayed with its clockwise spiral lines; and here’s a pine cone, overlayed with its counterclockwise spiral lines. And here’s a picture of a sunflower seed head, with the clockwise and counterclockwise spiral lines drawn onto it.) It appears that this is one of the ways evolution has converged upon a solution to the closest-packing problem (such as: how to pack seeds as densely as possible, in as little space as possible, on a seed head, for efficient use of the plant’s resources). This doesn’t make it the only mathematical pattern widespread throughout nature, and doesn’t mean it possesses mystical properties of applying to everything, or being “Nature’s definition of beauty”. It’s not even the only way nature solves closest packing – nature has also shown an at-least equal tendency for solving closest-packing with square patterns, hexagonal patterns, etc.

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Besides this, the golden mean and the Fibonacci sequence definitely do have unique, fascinating, remarkable mathematical properties. As does Pi. And the square root of two. And 1,089. And the prime sequence. And the Platonic solids. And Pythagorean triples. And enough other mathematical gems to fill a very large library.

Yes, there are some Fibonacci spirals in the natural world. Some golden ratios in nature, too. This doesn’t make the golden mean extraordinarily prevalent, doesn’t make it particularly more common than the many other mathematical patterns throughout nature, doesn’t mean the ratio can be validly applied to everything, and doesn’t make it “Nature’s definition of beauty”. There are plenty of non-Fibonacci spirals, too, such as the cochleas in our ears, and the curls in my hair. And plenty of instances of many other ratios, too, such as the human finger-to-toe ratio of 1:1.

Math games, and the appearance of remarkable, governing principles of the universe

If someone wants to find or create the appearance of the remarkable by gathering, correlating, and touting unrelated data, it’s not hard to play this game. Suppose, for example, I wanted to make the same kinds of remarkable claims about the ratio 1:1 – about its mysterious prevalence in nature, about its special aesthetic appeal, about its divine origin and mandate, and so on. Saying things which are all true, I could start by pointing out that 1:1 spirals are in our DNA double helix (i.e., the double helix’s spirals generally don’t get wider, nor narrower, as they progress). Then I could mention how much this ratio is found in the proportions of the human body – such as: our maximum arm span is generally quite close to the same as our height. I could mention various approximately 1:1 spherical objects in the human body, such as our eyeballs. I could mention it’s use in beautiful classical architecture: For example, the Pantheon is built with 1:1 ratios. (Note, the Pantheon is not the Parthenon.) I could mention how much 1:1 ratios, both in circles and squares, are used in product design, and assert that we surely are naturally gravitating toward this specially pleasing ratio. The letter keys and number keys on my keyboard are 1:1 square; my hard disk is circular, in a square case; my cell phone screen is square; my water bottle has a circular cross section; my DVDs are circular; my doorknobs are circular; my mugs and glasses are all circular cylinders; and so on. I could note how this ratio is all around us in nature, and point out some bubbles in water, or snowflakes, or some persuasively spectacular looking cubic crystals, such as this, this, this, and this.

And so on. In a matter of minutes, I’d have as nicely developed a hoax as the remarkable claims about the golden ratio. I could come up with as many examples as you want, in as diverse of areas as you want. It’s very easy to find cases of patterns or ratios, of a wide variety of types. It doesn’t automatically mean that it’s some special governing principle of the universe. I could just about as easily do this with a 1:2 ratio, a or 1:? ratio, or with hexagons, or triangles, or whatever. It means nothing.

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Conclusion

Had you been persuaded to believe the golden section hypothesis? If so, were these the kinds of “evidence” which persuaded you? Nautilus shells? The double helix? Galaxies? Whirlpools? Hurricanes? The Parthenon? And so on? I suggest anyone who has been persuaded by such “evidence” look again, more critically, and then reconsider.

I contend that the golden section hypothesis, and the claims and beliefs of the golden numberists, are unsupported, and, in most particulars, refuted.

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These beliefs are false, without substance. They have no basis in reality. They are nothing more than superstition and hoax. They are not scientific observation based on evidence; they are mystical beliefs in numerology.

The real world is amazing, as is. Look through my gallery, at the top of this page, to see some of nature’s splendor. There’s no need for numerological embellishments to spice it up.

Many times, people feel a need for order, even if it demands belief beyond reason. People also have a natural tendency to recognize objects and patterns which aren’t really there, filling in the blanks, and stretching ambiguities and flawed data to fit. (Here’s a fun example: see Vladimir Lenin in a shower curtain stain.) And people have a tendency to ascribe meaning or significance to things which really mean nothing.

Combine these tendencies with (A) an appealing myth; (B) the flawed force-fit methodology of starting with a conclusion and then seeking data to support it; (C) selective perception; and (D) a high tolerance for large margins of error in the data – and anything can seem possible.

It doesn’t make it true.

Please spend a few minutes taking a really good look at this, this, this, this, this, this, this, this, this, and this. Do you think that the significance of those pictures is contained in those superimposed lines, and that you can understand the pictures through those lines? Is this the approach you want to use to conceive your creations? Structure, rather than substance? Is this how you want to view other people’s creations? Trying to find lines that fit, rather than interpreting to understand? Is this how you want others to view your creations? Marking them up in a game of find-the-pattern, rather than experiencing the meaning and emotion? Is this how you want to define beauty? The detectable presence of a derivative of ~1.618?

I hope not. Frankly, it saddens me.

Ironically, the only thing the golden section hypothesis accomplishes is transmuting art into lead.

Please don’t fall for it. Try to rescue your friends from it, too.

Thanks for reading.


Alabama Hills with Storm, #5, Near Lone Pine, Eastern Sierras, California

Rebel Tulip, Mount Vernon, Washington

Passion Flower (Passiflora caerulea)

Angry Elephant Seal Cow (Mirounga angustirostris), Piedras Blancas, San Simeon, California

Passiflora Tendril and Leaf Tip, Santa Cruz, California

California Slender Salamander

Dry Ice Boiling in Water, Exploratorium, San Francisco, California

Pictograph, Utah


All pictures and text are © Mike Spinak, unless otherwise noted. All pictures shown are available for purchase as fine art prints, and are available for licensed stock use. Telephone: (831) 325-6917.

  • Mark - Quite an extensive article Mike and an excellent resource. I am not sure I am convinced either way. I certainly see many golden mean compositions that I think lose effectiveness if composed in a different way. And certainly there are many cases that go the other way as you have pointed out. But when you think of all of the possible visual combinations in nature, it is quite easy to just pick examples that support either case. Until you have a survey of 7 billion people and a similar representation of the visual elements in nature, I think it will remain a bit mysterious.

    I tend to look upon the golden mean simply as guidance if you should find yourself lost in where to start if you know no other way. No doubt many photographers have experienced this, and especially those just starting out. If it then “clicks” for you when applying this principle, great. If not, well then keep on exploring. No one should be criticized for just trying to follow their mind’s eye.ReplyCancel

  • Tweets that mention The Golden Section Hypothesis: A Critical Look » Naturography – Nature Photography By Mike Spinak -- Topsy.com - [...] This post was mentioned on Twitter by Jim Gilbert, Mike Spinak. Mike Spinak said: The Golden Section Hypothesis: A Critical Look http://bit.ly/gPdkwJ If you like it, please share the link. Thanks. [...]ReplyCancel

  • Bill Henderson - I like the notion of great images and all the rest. One person’s great image may be another persons trash. Music had rules then along came rap, and guess what the rules gave way to street story telling in poetic rhyme which many “trashed”. But now it’s a billion dollar business and has changed music rules worldwide. Artistic approval is a moving scale depending on how it is used. For fashion, the goal is to get you to stare at the print ad. Some images are intended to create emotion, yet others tell a story. Putting rules in place will not hamper the gifted and may help explain the past to the rest of us.ReplyCancel

  • G Dan Mitchell - Excellent (and impressively argued and researched) article/post, Mike. A readers digest version might point out that:

    1. We enjoy detecting unifying/organizing patterns in things and in finding correlations. While this is interesting and useful (especially to photographers!) it says little or nothing about the value of the individual observations or correlations relative to others we observe or believe we observe.

    2. The cases for the golden section depend on an astounding amount of inductive logic, yet fail quickly when even a bit of deductive logic is applied.

    3. Many find art and the ways of artists to be mysterious, including artists themselves. In an attempt to make sense of why some art works – after all, we are reasoning creatures – many try to reduce it to principles in the domains they trust more, such as mathematics. Art often helps this along by (at least loosely) using materials in many cases that do incorporate mathematical relationships – such as harmonic and other relationships inherent in music, my primary field. Unfortunately, some who see these things fail or are unable to come to terms that art does not result from the application of these “rules” – it is much more complex and much messier.

    4. While we cannot “determine which rules” make an example of artistic expression effective, we most certainly can make a number of observations about “what is going on” in the work. I’m completely skeptical of “rules of composition” or other attempts to apply “rules” to creativity at these levels. (Which is not to say at all that I think art is better when it is naive!) On the other hand, I’m all for investigating and considering something like “observations about composition” and trying to see and understand how an image works on many levels, including the levels of design and composition.

    Take care,

    DanReplyCancel

  • Alan Majchrowicz - I think I agree with you even though I couldn’t get past the first paragraph. Way too much analyzing on a concept that should just come instinctively.ReplyCancel

  • Michael McKee - The golden mean has something very much in common with human beings, it’s often misunderstood. I share your objection for how the golden mean has been used in art but still think it is a beautiful concept that can save our stagnant and dying art world, a balance of unity and diversity. Few artists want every place in their image to look like every other place (unless they’re extreme minimalist), that’s too much unity. Diversity is needed to illustrate most ideas. Ideally a balance of unity and diversity of both the pattern and the illustration will work together to enhance each other in any work of art. That is the ideal expression of the golden mean and I believe all artist strive for it weather they admit it or not. I use the formula of the golden mean to create all of my art but it is not visible because it is the structure of my images not just the outlines. http://www.moydanart.comReplyCancel

  • Oleg Bari - Great articles and amazing photographs !!!ReplyCancel

  • The Golden Section and The Rule of Thirds « Ian Bonthrone – BA Photography Blog - [...] Spinak in his Blog “The Golden Section Hypothesis: A Critical Look” (http://naturography.com/the-golden-section-hypothesis-a-critical-look/) not only argues that the Hypothesis behind the Golden Section is flawed and proves that the [...]ReplyCancel

  • Fibonacci Spiral in a Nautilus Shell - [...] / golden spiralNautilus shell – one of the sacred geometry patterns in our livesNautilus ShellThe Golden Section Hypothesis: A Critical LookWhy do patterns like the Fibonacci spiral exist in natureNature by Numbers – Fibonacci [...]ReplyCancel