You might have heard of Fibonacci’s sequence in passing or have seen references to the golden ratio in media, and if you haven’t, it’s actually quite an interesting concept to think on. In essence, the Fibonacci sequence is a series of numbers such that each number is the sum of the previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
This series of numbers dates back as early as 200 BC, presenting itself in Indian mathematics, but are named after Fibonacci, an Italian mathematician who brought the sequence into the western world in his book, Liber Abaci, where he used the series to describe the growth of a rabbit population — starting from one pair then mating them and the subsequent pairs which eventually increases to eight pairs and then upwards, with the number of pairs in each generation increasing according to the sequence.
Other than this funny rabbit situation, what’s interesting about this sequence is how often we see it manifesting in nature. If you take a look at a flower like a lily or a daisy and you decide to count the number of petals on it, chances are you’ll end up with a Fibonacci number—you might even end up noticing that a lot of flowers have five petals (AKA a Fibonacci number). Branching off of that, if we take a look at plants with leaf spirals, and should we measure the angle between one leaf and another either immediately above or below it, we may end up with an angle that we also find as a ratio of a Fibonacci number and the second number that follows after it in the sequence—for example, on oak we have the ratio ⅖ and for sunflowers we have ⅜.
Another common example I’ve seen is if we count the spirals on a pinecone or the seeds in the center of a sunflower, we’ll end up with a Fibonacci number, and even more curiously, should we count the number of spirals going the other way, we’ll end up with our two counts of spirals as two successive Fibonacci numbers.
If that isn’t enough to pique your interest—let’s take a look at the golden ratio. The golden ratio is an irrational number 1.6180339887…; that is, a really long number. Now looking back at our fun Fibonacci numbers, you might find it interesting to note that if we choose a Fibonacci number and divide it by its smaller preceding number, we’ll find a number resembling the golden ratio. As you continue down the sequence getting to larger numbers, you’ll find that the result gets closer and closer to the golden ratio. Furthermore, say we construct boxes with widths equal to the numbers of the Fibonacci sequence, and we connect them as follows: {SEE IMAGE AT THE TOP OF THE SECTION}
We see that by drawing arcs in each square, we find something called a Fibonacci spiral, which winds up being an approximation of the golden spiral, a spiral which widens by a factor of the golden ratio. You’ve probably seen this spiral before in an art class, as people have often attributed the aesthetics of an art piece to the use of the golden ratio; or maybe you’ve seen it in an image online of the spiral of a nautilus shell. Or, perhaps, you’ve seen a picture of a spiral galaxy with arms tracing out the golden spiral. People have even gone as far as to discuss how the golden ratio comes into play in the proportions of our bodies; that by comparing the lengths of our facial features or our hands and arms, we can get a number close to the golden ratio.
So, you might ask yourself, what kind of undefined mystical force is this? How is this series of numbers so important that we see clues of it in our world?
Well, a bunch of other people have had this question too, and better yet, people have looked deeper into some of these occurrences. And to answer that question—it actually isn’t some undefined mystical force, but rather just one of the many examples of how math presents itself in the natural world.
Let’s take a look at the spiral growth pattern for leaves that we talked about earlier; people have looked into this and have explained that these leaves grow in this pattern not because they adhere to this mathematical rule (surprise!) but rather because growing in this spiral pattern allows for the leaves to maximize the amount of sunlight each of them receive. As for the spiral patterns present in the seeds of a sunflower—this pattern simply allows for the maximum amount of seeds to be packed into it. And if we’re looking at things like the shape of a nautilus shell or the arms of a spiral galaxy, while sure these are spirals, not every single galaxy or every single nautilus shell can be exactly matched to the golden spiral.
So…maybe it’s not as omnipresent as it’s sometimes made out to be, but does that ruin the magic? I’d say — no! It doesn’t! There’s reason to why plants grow the way that they do, why art composed in a specific way is so compelling, and why —and not all of it can just be attributed to a single series of numbers; there’s nuance in our understanding of why things are the way that they are and it’s beautiful to see how math plays a part in that understanding. Everything, from the great vastness of our universe to the arms of a snowflake, can be described, in some way, using math.
It’s still interesting to observe how often you can find Fibonacci’s sequence occurring in nature — even if it actually isn’t as commonly present as it is made out to be. There’s beauty in finding where it exists, and as you continue to explore how math presents itself in the natural world, you might find other patterns coming up frequently and then, you get to question why that is. What’s wonderful about math is that it allows us to find logic in the things we observe in the physical world—giving building blocks on how to model things we otherwise don’t know how to explain. So, allow yourself to look for and appreciate the beauty of where you can find Fibonacci’s sequence in nature, whether it’s by the number of petals on a flower or by the spirals on a Nautilus shell, and you might find yourself delving deeper into how wonderfully poetic and natural math can be.
Sources
https://www.mathsisfun.com/numbers/fibonacci-sequence.
html https://math.temple.edu/~reich/Fib/fibo.html
https://en.wikipedia.org/wiki/Fibonacci_sequence
https://www.mathnasium.com/blog/golden-ratio-in-nature
https://science.howstuffworks.com/math-concepts/fibonacci-nature.htm
https://www.ijpam.eu/contents/2012-78-3/6/6.pdf
https://www.goldennumber.net/nautilus-spiral-golden-ratio/
https://www.jstor.org/stable/2686193?origin=JSTOR-pdf
https://stemettes.org/zine/articles/fibonacci-in-nature/
https://www.imaginationstationtoledo.org/about/blog/the-fibonacci-sequence#:~:text=Fibonacci%20numbers%20do%20appear%20in,when%20counted%20express%20Fibonacci%20numbers.
