Why Does The Fibonacci Sequence Appear So Frequently In Nature?

There are few sequences of numbers as famous as the one named after the Italian mathematician Leonardo Fibonacci. And that’s for good reason: from a relatively simple recipe, this set of numbers seems to touch on just about every aspect of life – not just in math, but also in the natural world around us.

And that seems weird, right? Why should one particular sequence of numbers, governed by a regular binary operation, turn up throughout nature? 

The answer is smarter than you might think.

What is the Fibonacci sequence?

If the name “Fibonacci” doesn’t ring a bell for you, then just think back to the first “tricky” number sequence you ever saw in math class. It goes like this:

If you can’t quite see what the rule there is, it’s this: each new number is simply the sum of the two preceding it. That’s more or less how it was first discovered, too, by medieval Indian scholars trying to figure out the ideal rhythms for poetry.

In the West, however, it would take a few more centuries for the sequence to turn up – and when it did, it wasn’t as a result of simple addition. In fact, it had more to do with multiplication.

“The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances,” explains Dr Ron Knott, a math communicator and one-time lecturer in the Departments of Mathematics and Computing Science at the University of Surrey, UK.

“Suppose a newly-born pair of rabbits, one male, one female, are put in a field,” he recounts. “The puzzle that Fibonacci posed was… how many pairs will there be in one year?”

Now, there are a few assumptions you have to make for this to work, which is why explainers of the problem usually point out that it’s an “idealized” – that is, biologically unrealistic – scenario. Firstly, ignore the fact that rabbits can die – they don’t, for the purposes of this exercise. Then, we have to assume that rabbits are not just capable of having babies at one month old, but absolutely certain to have those babies. Oh, and forget everything you know about inbreeding.

Then, Knott explains, “at the end of the first month, they mate, but there is still one only one pair.”

“At the end of the second month the female produces a new pair, so now there are two pairs of rabbits in the field,” he continues. “At the end of the third month, the original female produces a second pair, making three pairs in all in the field.”

“At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making five pairs.”

It continues in this way until you reach the end of the twelfth month, at which point there will be 144 rabbits happily hopping about – or, rather, 72 happily hopping about, and 72 that are heavily pregnant and probably rather tired. And the sequence of monthly totals that got us there will have looked like this:

Look familiar?

A measure of irrationality

From the start, then, the Fibonacci sequence was intrinsically linked to the natural world. But it turns up in far more places than just rabbit populations: you can see the sequence in the number of petals on flowers and the bracts of pinecones; in the branches of trees and the swirls of cauliflower florets; from the smallest snail’s shell to the grandest of Grand Design Spiral galaxies.

The question is: why? Why should this particular sequence of numbers – not the simplest you could come up with, but not all that complicated either – be so important to the natural world? 

A big part of the answer is explained by an area of math known as Diophantine approximation. Put as simply as possible, it’s the study of how irrational numbers can be, and some of its conclusions may surprise you.

Consider, for example, the “most irrational number”. Chances are, if asked which number was more irrational than any other, you’d either think it was a trick, and the question was nonsense, or you’d go for something like pi – not only irrational, but transcendentally so, and the subject of seemingly endless interest from computer scientists and mathematicians.

But in fact, the most irrational number is something much more demure: it’s φ – pronounced “phi”, and written numerically like this:

Now, it’s fair to say that this number doesn’t immediately look all that unique or interesting – so what sets it apart as “most irrational”? Well, the answer comes down to how close we can get to it using rational approximations – which, for the record, is “not very close at all.”

As an explanation, let’s look at π for a bit. You may have been taught at some point that it’s roughly equal to 22/7, and that’s true: it’s what mathematicians call the second convergent of the number, and it’s only 0.04 percent higher than the true value of pi. The third convergent, 333/106, is less than 0.003 percent out, and the fourth, 355/113, is just 0.00008 percent higher than the true value of pi.

While no fraction of whole numbers could ever describe pi exactly, we can definitely see that some combinations can approximate it pretty darn closely. But the same is not true for phi – instead, no matter how far down the list of convergents you go, there will always be a limit on how close you can get to the true value of the number compared to the amount of work you put in.

But here’s where it gets interesting. The convergents of phi – often known as the “golden ratio” – well, let’s see if you recognize them:

The nature of mathematics

Now, you may understandably be thinking at this point that nature doesn’t know shit about advanced number theory, and all of that must surely be a coincidence. But we promise, it’s not: “[T]he Fibonacci-like patterns and ratios found in many biological organisms, including in plants, truly are related to the Fibonacci sequence,” confirmed astrophysicist and science communicator Ethan Siegel in an article earlier this year, “both in a mathematically rigorous fashion and also for an evolutionary reason that makes perfect sense.”

So, think about the leaves on a plant. The plant’s energy comes from the Sun, so its goal as it grows is to maximize its leaves’ exposure to the sunlight. The obvious way to do that is to make sure new leaves grow a little way round the stem from the previous one – but how far round should it go?

Well, let’s try some examples. Halfway round won’t do; by the time you grow a third leaf, it will be directly underneath the first one, and won’t be able to see the sun. The same is true for a third, or a quarter, or a fifth of the way around – in fact, any rational fraction of the way around will eventually mean one leaf is entirely in the shade of another.

The answer, therefore, must be to go for an irrational fraction of a revolution – and the best of all must be the most irrational fraction. As we’ve seen, the best way to reach that particular value – after all, it’s physically impossible to do so exactly – is to use the Fibonacci numbers.

“If you keep putting out leaves at that key angle […] relative to the prior leaf, you’ll wind up with your leaf patterns forming a Fibonacci spiral,” Siegel explained. “That same mathematical property, encoded into pineapples, pinecones, and more, explains why biological organisms often display numbers found in the Fibonacci sequence.”

So, the ubiquity of the Fibonacci numbers isn’t just coincidence – it’s the result of a perfectly evolved optimization algorithm in nature. 

There’s just one caveat: sometimes, it really is just coincidence.

“While there are many spiral shapes that occur from purely physical, non-biological processes in nature — from whirlpools and vortices that form in bodies of water to the aerial shapes of hurricane clouds and clear lanes — none of these spirals are Fibonacci-like when it comes to the actual mathematical details of their structures on a sustained basis,” Siegel pointed out. 

“You may be able to take a ‘snapshot’ where one or more of the features exhibits ratios that are consistent with the ratios found in the Fibonacci sequence for a particular moment, but those structures don’t endure and persist.”

“The Fibonacci-like patterns seen in [most] spiral galaxies are inventions of our eyes, rather than a physical truth of the Universe.”

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