Why is the Fibonacci sequence found everywhere?
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Baader-Meinhof phenomenon, aka frequency illusion doesn’t explain it, but maybe you’re just noticing it more (like gta rare cars suddenly everywhere when you hop in one)
They arent “everywhere” in the way that pi and e are everywhere. They arent that fundamental. But there are some examples in nature where they show up. To understand those, we need to understand their relation to the golden ratio.
The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, … Every number is the sum of the two that come before it. While the Fibonacci numbers themselves obviously blow up to infinity, the ratio between two consecutive Fibonacci numbers actually doesnt. The ratio between the second Fibonacci number and the first is 1. The ratio between the third and the second Fibonacci number is 2. The next ratios are 1.5, 1.67, 1.6, 1.625, 1.615. If you continue doing this, the values approach the number 1.61803… (it has infinitely many digits). This number is called the golden ratio and its irrational meaning that it cant be represented as the ratio of two whole numbers (the Fibonacci process gets closer to it, but there are no two Fibonacci numbers whose ratio gives you *all* the digits). It has a not so obvious mathematical property: it is the “most irrational” number. What I mean is that if you try to approximate an irrational number with the ratio of two whole numbers, continuously tweaking that approximation to account for more and more digits, the golden ratio will be the number where you will have to tweak the approximation the longest to get more digits.
If you were designing a plant, an important aspect would be the leaf placement. If you place leaves above each other, the lower leaf wont get any sunlight. So you decide to put each leaf at an angle to the last placed leaf. You pick 1/5 of a full circle as the angle. But to your shock, you notice that the 5th leaf is right above the first! Because after 5 leaves (each adding an angle the size of 1/5th of a full rotation), you have completed one full rotation and are back where you started with. Trying other ratios is no use, 1/8 will be a completed rotation after 8 leaves, and 2/8 will be 2 completed rotations after 8 leaves (and that lands you in the same spot as 1 completed rotation). So we need to pick a number thats not a ratio of two numbers. Any irrational number will do. However, it’s still possible that a multiple of that number makes something like a 3.0002 rotation. In other words, the leaves are still *almost* fully covering each other. To minimize that risk, we need a number thats as little a ratio as possible: the golden ratio! If we pick the angle between consecutive leaves to be 1.61803… of a full circle, we minimize the chance that multiples of that angle add up to a whole number, minimizing the risk of two leaves almost completely covering each other.
Other examples Im aware of usually work similarly. I remember that certain animals had hibernation cycles in accordance to the golden ratio so that their a
wake phase wouldnt eventually coincide with the predators wake phase.
So really, it’s the golden ratio thats the real star. But to construct the golden ratio, you typically make something in a ratio of Fibonacci numbers, so thats where they come into play.
I want to add that Im not a biologist so I cant really verify these examples. I found them on Wikipedia and I do think that the math and overall logic makes sense though.
The other answers are good, but I also want to link to [this series of vhart videos on the topic from a while back](https://youtu.be/ahXIMUkSXX0?si=uzWSqMrLPh02bBAX).
She does a really good job at breaking it down and showing WHY it shows up everywhere, at least in plants, in a very ELI5 way.
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