As someone who has spent a life in mathematics, I see a lot of attempts to ascribe mathematical concepts to real-world ideas in an overly simplistic way. The media misinterpreting a single medical study and reporting that a glass of red wine is equivalent to an hour at the gym does not mean you should forget the treadmill and buy more Malbec. Weathermen in Kansas do not expect the flapping of butterfly wings to cause tornadoes. But in photography, there's one incessantly perpetuated myth that drives me crazy.
In math, we give names to the really special numbers: π, e, etc. There's another number that has its own name: . Known as the "golden ratio," it's the number obtained when the ratio of two numbers is the same as the ratio of their sum to the larger of the two numbers. Check out the diagram from Wikipedia below:
In all my time in mathematics, I've encountered π and e constantly; they're deeply woven into the fabric of the entire subject and show up in places one would never expect them to constantly. On the other hand, I can't remember the last time came up. It's a neat number; it has some cool properties and interesting facets, but it's not the ubiquitous titan that the aforementioned numbers are.
And so, it surprises me that it has gained such a foothold in aesthetics. We hear about it constantly in photography; it's even included as a composition option when using the crop tool in Lightroom. But why? What's so special about it? Are we as humans really naturally drawn to this irrational number (meaning it has no representation as a fraction)? Or is something else at play? I propose it's the latter.
To me, the golden ratio in photography is an example of confirmation bias and the desire to ascribe something mystical. Confirmation bias is the tendency to accept information that supports a belief and reject or diminish the importance of information that undermines it. Horoscopes, luck, and politics all contain great examples of this. We hear it's good luck to perform some ritual, perhaps knocking on wood, and we remember that one time that something bad happened when we didn't, forgetting the hundreds of other times when life went on rather innocuously.
Take a look at the image below:
I'm pretty proud of my composition here. For reference, I've placed red dots at golden spiral points, green dots at rule of thirds points, and blue dots at golden ratio points. Can any one of the twelve points singularly explain why this composition works? Let's take a look. Beginning in the bottom right quadrant, I don't think anyone will argue that these three dots hold much significance, unless you're really into breakwalls.
Moving up to the upper right quadrant, things get a bit more interesting. At the golden ratio point, there's the corner of a fairly tall building, but it pales in comparison to the immediately adjacent Key Tower and Terminal Tower. The rule of thirds point is on a midtone section of the clouds and isn't especially interesting. The golden spiral point is relatively near the leading edge of the brightest portion of the image, but still an appreciable distance from it. Moving to the upper left quadrant, the analysis is about the same. The golden ratio point sits on an unspectacular building, while the rule of thirds point and golden spiral point sit on the trailing edge of the storm.
In the lower left quadrant, things get a bit more interesting. Both the golden ratio and rule of thirds points are over the open water, but the golden spiral point does hit the lighthouse, which is obviously the main and only foreground element. Is it in the most interesting spot, though? I would argue the most beautiful spot of the lighthouse is where the two buildings meet or the peak, not the rather plain side of one of them. And even so, does that lighthouse make the image? No. It certainly contributes, but the composition here is built upon the inclusion of foreground, middle ground, and background elements, coupled with a centered composition of the main subject, the skyline. The lighthouse serves to offset the bright sky on the opposite diagonal, and while interesting, it's not the main subject. To attribute it all to the golden spiral point being sort of near the lighthouse is a vast oversimplification, and it highlights another point: finding the golden ratio is often an endeavor in approximation, which undermines its validity. is a specific number; when we say something is an example of the golden ratio, it should be that number, namely 1.618033... A great example of this is the Parthenon, some of whose proportions are almost in a golden ratio, but not quite — enough off to not just be variations in construction. In math, is one number and one number only.
Here's another example:
In this case, it's almost there. But again, it's not quite: it's just off the center of her nose, and not quite vertical with her eyes. Was I thinking of the golden spiral when I cropped this image? Nope. Was I roughly thinking of the rule of thirds? A bit. My thinking was more along the lines of accentuating a tall, slender figure by placing her slightly above the upper third to increase the effect of her height. Was I magically drawn to some mythic number? Absolutely not.
Here's the issue: finding the golden ratio is often a huge example of confirmation bias. One of its main proponents, Adolf Zeising, thought that measuring the distance of a person's navel to their toes and dividing by their height was an example of its use in the human body, because that ratio is about 1.6. But why the navel? There's nothing special about that. In fact, give me any ratio and something as complex as the human body, and I promise you I'll find something close to that ratio in there. The truth is that we want something special, something "golden" to mean more, so we overemphasize the (statistically expected) times when it appears and ignore the vast majority of times it doesn't.
It's an attempt to find meaning between math and the arts, because wouldn't that be beautiful? But one must be careful, because math can be manipulated too. Did you know there's a nearly perfect correlation (R=.993) between divorce rates in Maine and per capita consumption of margarine in the U.S.? Is it meaningful? Of course not. There indeed are special relationships between math and the arts, but the golden ratio is not some transcendent thing, nor is it one of the most special relationships (not even close, I would argue). I get it, though: it's enticing. It's a spiral. It's a seeming yellow brick road to photographic brilliance. But it isn't a transcendent thing in mathematics, and its statistical consistency in art is dubious at best. Rather, I think it's a good representation of a simple fact: off-center composition is frequently pleasing. It gives a good approximation of a balance between being interesting and extreme. Studies that have tried to show as being the best approximation of that in the mind's eye have failed to conclusively agree with one another on that hypothesis.
So, what am I saying? Can you use it for a guide? Sure. Should you use it as the absolute rule at the exclusion of your own aesthetic preferences, compositional instincts, and basal feelings? Absolutely not, just like the rule of thirds (which, incidentally, likely came about as an approximation of the golden ratio). There is not some mythic power contained in , nor in any other ratio you care to toss out there. And worse, when you adhere to those while turning a blind eye to your artistic impulse, you censor your unique expression. Use them as a guide, nothing more.