eli5: Why is the Fibonacci spiral found everywhere?


It’s found in commonly through nature and often in design (whether intended or by coincidence).
Was wondering if anyone knew exactly why it was so common especially in nature.

In: 30

It’s because the sequence is usually found in optimized structures and systems. Nature tends to reach it through trial and error, but as long as the more optimal arrangements keep reproducing or surviving more than the least optimal, they will gradually select systems that are closer and closer to one represented by the Fibonacci sequence. As to why the sequence is found in optimized systems, it’s because the sequence is an offset of the golden ratio, and that is an important relationship ratio in a lot of different natural systems.

It’s super common because it’s what happens when you have exponential growth with discrete things.

Exponential growth is when the amount you gain depends on the amount you have. Just like multiplying cells…the more cells that are reproducing, the faster you grow.

But you can’t have half a cell…the integer version of exponential growth, which is basically all cell division or any kind of replication of discrete units, is the Fibonacci sequence. If it shows up in a spiral structure, which is super common for anything that grows attached to its predecessor like shells or flowers, you get a Fibonacci spiral.

It doesn’t. Not really.

It does appear roughly to systems that tend to compound as they grow but even then it’s not exactly the golden ratio.

People tend to misrepresent any logarithmic spiral as the golden spiral.

You can find the number 1.6 everywhere If you look enough.

Tldr: the movie nymphonaniac made us think that any logarithmic spiral is connected to the golden ratio. It’s not. It’s a myth.

It kind of isn’t. There are lots of *logarithmic* spirals in nature, but not all logarithmic spirals are Fibonacci spirals.

EDIT: this isn’t to say that Fibonacci numbers never turn up in nature. They are quite common in botany, for the patterns of seed growth in sunflower heads and pine cones, but it turns out that this is because logarithmic spirals with a rate of turn close to the golden ratio pack very efficiently. Also, the relationship between the Fibonacci numbers and the Golden Ratio is slightly deceptive. While the ratio between consecutive terms of the Fibonacci sequence does tend towards phi, this is not a particular property of that exact sequence, but is also true of all recursive sequences of the form

a, b, (a+b), ((a+b)+b)…

Check out [this video](https://youtu.be/ahXIMUkSXX0) and the two after it for a really great explanation about Fibonacci and Fibonacci-like spirals in nature.

Vi Hart is a lovely nerd