The Golden Ratio: Nature's Favorite Number
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 behaviour 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 they go...