Humanity has always been in search of patterns. They make us feel comfortable. They give us meaning. Whether they be in the deepest, most conceptually difficult topics like string theory and quantum mechanics, or even in simple things like the behavior of the person we are talking to, we love to seek patterns, and do so sometimes against our better judgement. Nature and mathematics are no exceptions to that list when it comes to pattern-seeking. In fact, it is at the forefront of it. We have evolved to notice patterns and be alerted when something isn’t right.
One of these so-called patterns that has fascinated mathematicians and individuals alike for centuries is the golden ratio. Also known by the Greek letter Phi, it can be defined by taking a line and breaking it into two separate pieces. If the ratio between these two new portions is the same as the ratio between the original line and the now larger piece, the portions are said to satisfy the golden ratio.
The value that satisfies this equation is roughly 1.618… It’s an irrational number, meaning we don’t know how to represent it using the ratio of 2 whole numbers. In fact, we can’t even write the number fully, it’s unending. It’s effectively the same as pi in that regard, and so we instead use the dots to represent the non-terminating nature of this number.
But what makes this ratio so golden? Well, I should point out, the ‘golden’ part of the name is rather modern. The more original name that was given to this ratio was ‘the divine proportion.’ And right away, you can see that the term implies some sort of divinity - an extraordinary property that people must have noticed when they were dealing with this number. To further explain this fascination, we should begin by dividing the fascination into two parts - mathematics and aesthetics.
The golden ratio and the Fibonacci sequence appear in nature every day, and arguably the strongest evidence of “goldenness” in the golden ratio is in the floral arrangement of seeds.
Take this for example: if you were a sunflower, how much of a turn would you make before you make a new seed? If you don’t turn at all, well you just continue making a straight line of seeds, kinda boring. If you make half a turn, or a 180 degree turn each time, well now you have a line of seeds, but in opposite directions. 120 degrees gives you 3 lines, 144 degrees gives you 5 lines. There should exist some angle, some number of turns that, if properly executed, produces a pattern of seeds that is closely packed together with no gaps between them. Something like this, this seems more natural to nature, right? The number of “turns” needed to produce a spiral design like this is… well, the golden ratio. One seed placed every 1.618 turns, or every 137.5 degrees. This is known as the golden angle, and it is seen all throughout nature.
The idea is to arrange seeds in a way that can maximize the sunlight and rain that they receive, so that the genetic material can successfully be passed on to the next generation. If you don’t do this efficiently, evolution won’t be very kind to you. If you pack in the seeds too tightly, all the seeds won’t get the nutrients they need. If you pack them too sparsely, you’re just wasting space. This happens with not only sunflower seeds, but in plant leaves, tree branches, and more. In fact, it goes even deeper. If you were to count the number of spiral arms in both directions, left and right, you’ll find that they aren’t equal; however, they will both always be
Fibonacci numbers. The higher and higher the numbers go, the closer and closer the ratio between them approaches Phi.
The beautiful spirals that result are purely a creation of nature. These spirals are consistent across different flower types, and even the numbers of petals seem to be related to the golden ratio. The sunflower example is particularly interesting because it actually ties the aesthetic element of the golden ratio to the mathematics behind it; there is a reason why the golden ratio is used in flowers, and any other value doesn’t seem to work very well. But what does that really tell us about the underlying, and possibly grand design of the universe?
We are all somewhat familiar with the aesthetic elements of the divine proportion. The idea is that visually pleasing faces tend to have proportions that are somewhat close to the golden ratio. The closer you are to it, the more beautiful your face should be. This inclination of harmony extends to the rest of the body as well, with many claiming that Da Vinci’s ‘Vitruvian Man’ demonstrates that the human body owes its beauty to the golden ratio. The phenomenon also has its place in architecture, with claims thrown around that structures as old as the pyramids of Egypt are in some way shape or form inspired by the golden ratio... except for the fact that there’s hardly any evidence that the Egyptians even knew this ratio existed.
All of these beliefs contributed to the idea of the golden ratio and made it what it is today. The mathematical beauty is undeniable, it’s proven; however, it’s generally hard to verify any of these claims regarding it’s aesthetic purpose in our lives. For one, it is not sufficient for a structure to just have the golden ratio somewhere. It must be meaningfully incorporated and be the result of someone’s conscious or subconscious choice. What does meaningful incorporation entail? Why would you choose the ratio of, say, this pillar with that one, and not with another?
Then there are questions of measurement. Oftentimes, you see pictures thrown around the internet with a famous celebrity’s face with random lines drawn on them that claim to satisfy the divine proportion.
Long story short, they’re reaching pretty hard. Without proper definitions of the facial features they are referring to, like what a cheek bone is, where it starts and ends and so on, there is basically no scientifically verifiable structure to what they’re claiming. Even if you do find that a certain arrangement and placement of measurements do wind up representing the golden ratio very well, it’s never quite exactly that.
Phi, the golden ratio value, is irrational. It’s unending. You can come close, but you can never truly achieve it just by drawing some lines.
Despite these claims, however, one could still argue about the idea of facial beauty and the importance of the golden ratio in it. All that matters is our perception of something. If the ratio of some facial feature is close enough to the golden ratio, so much so that we’re unable to distinguish it from the actual ratio itself, and we find it pleasing, wouldn’t you say that the aesthetic element of the golden ratio is justified?
To investigate this point, experiments have been conducted relatively recently using a geometric object that has the golden ratio written all over it. It’s the golden rectangle. It’s any rectangle that has sides a, b that satisfies the relationship I just described at the start of the video, except this time it’s a shape and not just lines.
The experiments include showing participants many different rectangles of different proportions, with one or more of them being golden rectangles. They were then asked to choose which rectangle they felt represented the ratio best. If the hypothesis that things with the golden ratio embedded in them are more attractive to us is indeed true, you would expect to see a significant number of participants choosing the golden rectangles. But, turns out, the evidence is not very great on this one.
Instead of choosing the golden rectangle, participants seem to prefer rectangles around that number, like 1.5 or 1.75. It’s close, but it already somewhat disproves the idea of the golden ratio. If a rough value around the golden ratio is so good, then the golden ratio itself isn’t so golden after all. But even if, say, the preferences do seem to be around that golden ratio, there are theories as to why that might be. People who are more aware of the golden ratio might be predisposed to having a preference toward it because of the place it holds in history and in mathematics. Besides, if you spend a lot of time staring at a screen with aspect ratios relatively close to that of golden ratio, something like a 16:9 monitor, you will unintentionally find them more attractive.
But now it would be a fair question to ask, well what started this golden trend?
There does seem to be a sincere focus on the golden ratio in the modern design language, especially with UI/UX design. While no designer in their right mind would swear by one single rule, the golden ratio has seemingly become a mainstay in graphic design. For example, a lot of websites tend to have headings that are 1.6 times the font size of the general text. Website layouts also incorporate golden rectangles. Again, once something like the golden ratio has been generally accepted as something divine, it’s very hard to distinguish whether it’s desirability stems simply from its familiarity in our lives or from some deeper meaning.
But it just so turns out that there is some deeper meaning, deeper inside us. There’s a body of research that focuses instead on deeper biology and how the golden ratio plays a crucial role in it. The cardiovascular system is of keen interest in this regard. This is because it is one of those biological systems that has very small differences across ethnic groups. So, if the golden ratio were to be found in such a system, we could claim that it is indeed a more universal phenomenon, rather than existing simply due to some ethnic variation or bias.
The cardiovascular system is deep in us, it runs our lives. There is no element of fabrication down there. Evidence found here would much better support the idea that there is some natural significance to the golden ratio, and in fact, there is. Your heart has many moving parts that help keep your body functioning. One of them being the left ventricle, it’s what helps pump oxygen throughout your body. Now, the average ratio of left ventricle horizontal and vertical lengths of over 1,400 Swedish and Chinese subjects was found to be 1.618 – awfully close to the golden ratio. Patients with left ventricle ratios that were further away from this value were deemed to have poorer functioning hearts, and they were. And as we age, our hearts tend to lose some of their power. Over time, your blood begins to pump slower and slower throughout your body.
Even the age-related cardiovascular decline was related to the golden ratio, with an early diastolic velocity decline of 1.6 cm/s per decade.
On the other side of the spectrum, the early diastolic flow velocity of muscles in the heart of a growing fetus increases by 1.6 mm/s per week. It’s remarkable how these and other metrics of cardiovascular health hover close to this special value. But even in these examples, there’s still a bias in choice of units. For example, why is this particular metric in mm/s and not inches/s? Such choices could alter these quantities and take away the divinity in them.
But why? If there isn’t so much evidence behind the divinity of the golden ratio, why do people have this obsession over it? Well, it’s the same reason why we once believed the earth was the center of the universe. We want to believe these things. Of course, people have moved on from those particular beliefs to more modern and sophisticated ones, like how the earth is flat and all. But the core philosophy remains the same. We love patterns, and we love them even more if they make us seem special. Why did the golden ratio only have to do with our faces and not those of other animals? What about the body proportions of a snake? It’s only human to only care about ourselves.
This also highlights the age-old question about whether math is invented or discovered. You see, the phenomenon of the golden ratio existed long before we had any idea of it. The flowers arranged themselves without us having a cognition of numbers or mathematics. Even when we tried forcing our hand on certain elements of the golden ratio to suit our narratives of existence, the golden ratio itself did not change. If the golden ratio were from a different time and it were named differently, we would still be talking about the same thing, only with a different name. It was what it was, and it will always be, whether we have any cognition of it or not. There are things, properties of math and science out there that exist currently, right in front of us perhaps, and we have zero idea what they are or what they entail. We may define things along the way, just as we have given the ratio a name, but the underlying mathematical properties remain unchanged.
In truth, the divine proportion owes its divinity not to some all-pervading transcendental property, but rather to the idea of beauty and perfection that we’ve imposed on it. This beauty we seek is quite possibly not concentrated in one golden number, and even if it is, the golden ratio is probably not the one. In trying to find perfection, we could nonetheless keep defining Phi. But remember, it’s irrational. Phi does not end, and neither does our search for beauty in the universe we live in.
- MA, MM