This week, I tackle what the so-called golden ratio is and where you can find it, focusing on what can be proven using math or logic. Next week, I’m going to explore the myths and theories about it, and whether artists or designers should ever use this ratio in their work.
The golden ratio is notorious for connecting math with art and design. There are whole websites dedicated to unearthing its appearances in famous works of art. But it’s also notoriously controversial, and there are plenty of people who say the whole shebang is overblown and overrated.
To separate myth from fact, one first needs to figure out what, exactly, the golden ratio is. In short, to derive it, you need a rectangle, an equal sign, and the name of an ancient Greek artist.
The golden ratio, first and foremost, is a statement of proportion, written as “A is to B as A+B is to A.” Or as Euclid described it in the Elements, “as the whole line is to the greater segment, so is the greater to the lesser.” It looks like this:
Turn that proportion into an equation, mix in a little algebra, and you end up with the number Φ (Also written “Phi”), an irrational number about equal to 1.618. Multiply any number by Φ and the ratio between the original and resulting numbers is the golden ratio. And that’s a vaguely interesting proportion on its own. But, much like everyone’s favorite irrational number, Pi, the fascination with Phi stems from how often it appears in places you wouldn’t expect.
The sneaky thing about the golden ratio is that, other than a number and a line segment, it can come in many different forms and with many different names. The “golden ratio” refers to the relationship seen in the line segment above. Dividing a line or rectangle using this ratio is also called the “golden section” or “golden mean.” Similarly, Φ is also called the “golden number.” It’s all been relabeled many other times on top of that, with names like “golden proportion,” “golden cut,” “extreme and mean ratio,” “medial section,” “divine proportion,” and “divine section.” Basically, scholars throughout history have adored this ratio, but they kept rediscovering and renaming it, unaware of how many other people had also shared their fascination.
Even its symbolic notation, Φ, is a subject of confusion and overlap. The use of this Greek letter is originally credited to Mark Barr, a mathematician from the beginning of the 20th century. Today, though, we don’t know about it from Barr. This story was relayed to popular culture secondhand, by Martin Gardner, longtime “Mathematical Games” columnist for Scientific American. He wrote that Barr had chosen this particular letter to honor ancient Greek artist Phidias, who used the golden ratio extensively in his work. The twist? Barr didn’t actually agree with that, writing in his paper “Parameters of Beauty” that it was unlikely Phidias had used the golden ratio.
But wait, there’s more!
One of the most common formats of Phi is the “golden rectangle,” whose side lengths are related by the golden ratio. As seen below, because of the properties of the ratio, dividing the golden rectangle into a square and a rectangle produces a rectangle of the same “golden” proportions. Keep doing this ad infinitum, and the result is a particular logarithmic spiral often referred to as—you guessed it!—the “golden spiral.”
As if those aren’t enough different forms for the golden ratio, it’s worth noting that Φ is inextricably tied to the Fibonacci Series. The Fibonacci Series is a sequence of numbers where each number is the sum of the previous two. It reads like this: “0, 1, 1, 2, 3, 5, 8, 13, …” and on and on forever. It makes a nice, pretty curve, and the number sequence shows up in nature all the time.
What’s more, if you take the ratio of two adjacent numbers in the Fibonacci Series, you’ll get a number not so far away from Phi. In fact, the farther along the Fibonacci Series, the closer the ratio becomes to “golden.” So almost wherever you find the Fibonacci Series, you can find the golden ratio, and vice versa.
If you’re the kind of person who enjoys doing geometry for fun, you’ll soon learn that Phi tends to show up all the time where you’d least expect it. For example, want to inscribe a square in a semicircle? Just use a golden rectangle!
Trying some arcane magic and looking to draw a pentagram? You’ll need a “golden triangle” or two, where the ratio of the big side to the small one is Phi.
Is your pentagon lopsided and irregular? It’ll actually still contain the golden ratio.
What about the 3-4-5 triangle? (Making life easier for trigonometry haters everywhere, circa 300 BC). It, too, related to the golden ratio.
I could go on, but the point is, my cup floweth over with examples of the golden ratio in geometry. I was actually unable to unearth a good explanation for just why it appears in so many shapes. There is one subject, on the other hand, where Phi’s ubiquity is very explainable, and that is plant growth.
Plants, you see, having stems that are approximately circular, tend to grow their leaves, petals, seeds, etc. in spirals. When deciding what kind of spiral to use, these plants want nothing more than to maximize their use of space. In leaves, this means showing as much possible leaf to the sun, so that the plant gets the most possible nutrients and has the best chance of survival. In seeds, this means using space efficiently so that there are no pesky gaps that could have been devoted to producing potential offspring.
The absolute most efficient way to do this turns out to use the angle Φ. If any rational fraction were selected, like 1/3 or 1/5 or 1/20, the arrangement of leaves or petals would eventually complete a whole circle and overlap with the first growth, blocking sunlight or leaving wasteful gaps. Phi is the more efficient angle to use in growth because it will never produce this 100% overlap.
Phi is clearly a very important number in the natural world. It’s possible that this importance stems from its bizarre title: Φ is the “most irrational number.” Just how could one number possibly be more irrational than another? Well, essentially, irrational numbers can be approximated using rational number fractions. From there, the “error” of these approximations can be calculated. All other irrational numbers, like Pi or √(2), are statistically closer to rational.
It is at this point in the story of this ratio, which has fascinated so many for so long, that fact gives way to opinion and debate begins. Phi is a constant in many geometrical phenomena and also occurs in many plants and other natural forms, quite possibly because it’s the most irrational number. But does it also exist in ancient art and the human face? Do people really find it to be the most pleasing proportion? And more importantly, does this mean you, dear reader, should be thinking about the golden ratio if and when you draw, paint, or design something?
These are questions over which math and logic do not preside, and so they have been the subject of countless debates. In next week’s post, I’m going to examine the various claims made about the golden ratio, looking at the semantics and statistics of just how special this proportion really is.
I made a list, with summaries, of a whole bunch of science or psychology papers about the golden ratio. You can skim it here to see all the hard evidence—and don't worry, it's a short list.
Math youtuber Vi Hart offers a funny and eloquent explanation of the whole plant-Phi-Fibonacci phenomenon, and I’d highly recommend it. Make sure you get the whole story with parts one, two, and three.
Taste the golden ratio with homemade Fibonacci Lemonade according to this recipe.
For a huge list of examples of the golden ratio in geometry, with links to geometric proofs, try here.
And finally, for a glimpse into next week's content, you can see a website promoting the mystical properties of the golden ratio here, and a quick overview of some of the opposition here.
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