Patterns pop up all the time in nature, and they are typically very appealing to us. From appreciating the look of a flower to enjoying the tidiness of one’s room, order and structure are pleasing to the eye and soothing to the brain.
Numbers, on the other hand, may be slightly less appealing to some of us — or at least, they don’t seem to hold the same beauty as patterns in nature. But in fact, it is because of numbers that such patterns arise. These numbers follow their own pattern — mathematically, one would call this set of numbers a “sequence”— and there is one sequence, in particular, that shows up all over the world, across many fields and species. This is the famed Fibonacci sequence.
Most of you have likely heard about this sequence and may have even come across it in a course or two (for me, I remember a mathematics course sophomore year where we proved a special property of the sequence that I will describe a bit later). But I hope I can convince you of its ubiquity in nature and, as a result, its beauty.
Let’s begin by defining the sequence. The Fibonacci pattern is that each number in the sequence is the sum of the previous two numbers. This means we need to set the first two values of the sequence — these are typically chosen to be 0 and 1 — but after that, the sequence makes itself. Following 0, and 1, the next several values in the sequence are:
1, 2, 3, 5, 8, 13, 21, 34, 55, …
…and so on.
The first application of this sequence is not all that beautiful, but it is fairly practical, especially if you are traveling! If you pick a number in the sequence and jump to the next one, that is a good approximation for the conversion from miles to kilometers. Runners may recognize that 5 kilometers is about 3 miles, but this goes for all the higher numbers in the sequence too! If you don’t believe me, you can Google “convert miles to km” and plug in some other numbers.
This perhaps may be a little easier than multiplying miles by the conversion rate, which is about 1.609 km/mi. But why does this work? The answer is that the ratio between a number in the Fibonacci sequence and its next-lowest neighbor approaches “the golden ratio” as we continue writing terms in the sequence. The golden ratio is about 1.618, so we’re very close to approximating kilometers from miles this way!
Where else does the golden ratio show up? Well, pretty much everywhere. It is how many types of flora, including sunflowers, daisies, pinecones, and pineapples, to name a few, are patterned. We see this by observing the first several values of the Fibonacci sequence showing up in such patterns. An example is given below, here with a chamomile flower.
Fibonacci numbers also show up in fauna, specifically for bees. Honeybees generally follow the pattern that eggs laid by unmated females are male, while eggs fertilized by a male bee are female. With this information, one can trace the pedigree of male bees – they have 1 parent, 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. So, we again have a Fibonacci sequence!
One other place that not necessarily the Fibonacci sequence, but the golden ratio, shows up, is much more man-made: this final example is credit cards. These cards are designed so that the ratio of the longer side to the shorter side is the golden ratio. If credit cards have an aesthetically-pleasing size to you, this may be why!
The main point of going through all these examples is that numbers can show up in very
unexpected and disparate situations — but, nevertheless, they help us understand patterns in those situations, and furthermore, how such patterns can extend across several areas of nature. Perhaps the most miraculous part of this story is that nature seems to be able to understand math as much as we can, and can utilize math to make life more orderly, and, yes, more beautiful.
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