Why is the Fibonacci sequence found everywhere?
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Actual mathematician here: the answers so far are all wrong, and so is your presumption. Fibonacci numbers are rare and there is no inherent mechanism in the universe for them.
Multiple answers have claimed it is all over nature, but examples of them appearing at all outside human works are pretty rare. They somewhat happen in sunflowers sometimes, if all the randomness of growth does not almost certainly screw it up. But that’s about it. Beyond those rare few examples, it is usually esoteric and/or made up.
All the spiral patterns are not Fibonacci based, either. They are simply what we call _logarithmic spirals_, which have effectively nothing to do with the numbers. They are based on exponential growth, but that could be the powers of 2, 3, the golden ratio (~ Fibonacci numbers), 5, pi, and most importantly and most common, e.
When you encounter individual Fibonacci numbers, it is random chance. Especially with small numbers like 1,2 and 3, they just as well could be a million other sequences. And it it looks like it might be the golden ratio, it almost always might just as well be 1.5, 1.6, square root of 2 or 3, or a lot of other options. The uncertainty is usually very high, and often we even know that it is definitely _not_ that one number.
Some posts even compared the golden ratio and the Fibonacci numbers to pi. But unlike the former, pi has a lot of reasons to be everywhere in a physical reality. For example, the laws of nature do not change when you rotate things, hence a lot of optimal arrangements are ones that don’t change with rotation as well: circles and spheres. Thus pi.
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