Why are the Fibonacci sequence and golden ratio so common throughout nature?

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Does it have to do with something that involves cell/DNA formation, or other topics in biology/chemistry? Or is this based on coincidence?

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The fibonacci sequence is a very logical way for a process to proceed. It makes sense that things may only be produced from things you already have, right? The fibonacci sequence is essentially exponential growth with inherent delay.

Let’s talk pairs of immortal bunnies. They are pregnant for 1 year and take 1 year to reach adulthood, and will always birth a male and female bunny as adults. (We shall ignore the practical implications of a bunny society like this for now)

So if we start in year 1 with 1 pair of baby bunnies.
Year 2 they’re grown up, we still have 1 pair of bunnies.
Year 3 the old bunnies had a pair of bunnies, so we now have 2 pairs.
Year 4 the first pair had babies, and the second pair is now grown up, so:
Year 5 2 pairs of new babies and 1 new adult.

You start to see the pattern here, every year we get as many new pairs of bunnies as we had in total the year before.
Regular exponential growth would give you just as many as you have times a particular number, but due to the delay incurred by the bunnies growing up, the effective growth factor decreases somewhat.

From this point on I’m not entirely certain anymore, so please correct me if I’m wrong and I’ll edit the comment accordingly: the golden ratio is this effective growth factor. The process described above is of course not how bunnies reproduce. You know how they are. Instead of having X kids every year in a very orderly fashion, it will be more of a continuous process, with one pair having kids now, another in a month, etc.
The golden ratio – as is my understanding – should be the growth factor of such a more realistic process.

The Fibonacci Numbers are a sequence to explain a growth over generations of species.
This Formula was developed by the Italian mathematician Fibonacci when observing the growth of rabbits over the number of generations. Think of it as the number of rabbit pairs in the
3rd generation is the number of rabbits pairs in the 1st plus 2nd generation (1+2=3), in the 4th generation it would be second plus third(=5), in the fifth 3rd plus and 4th(=8) and so on.
So nothing of cell, chemistry or DNA just a biological observation. Because it is of growing populations you find that sequence everywhere where things grow.

There is a theorem that Fibunacci numbers can be calculated by golden ratio.(Please don’t ask me to explain, I only happen to know). But that would also explain, why the golden ratio is so common.

Golden ratio is half math and half myth.

We look at nautilus shells and see Golden ratios, but we don’t look at trees, not find it, and disprove it.

We look at things like the pantheon and see elements that fit. But they are approximations of fit, and many other aspects of it do not fit at all.

It is confirmation bias, seeking to find patterns and only talking about the wins. Now designers insert those ratios into their designs, which in turns feeds our perception of balance.

There are geometric reasons why the Fibonacci sequence turns up in places like pineapples, pine-cones, and sunflowers – it turns out that efficient packing of similar sized features (spiky bits on a pineapple, or seeds in a sunflower) ends up displaying a variety of Fibonacci numbers simply because other numbers wouldn’t fit. It’s a bit like how honeycomb is hexagonal – other shapes wouldn’t fit.

The golden ratio is a geometric consequence of Fibonacci numbers, so it can come out of certain cases where Fibonacci numbers are involved.

To combine the answers of /u/WarlandWriter and /u/Ares1935, the nautilus shell is grown outward at a certain rate in a spiral, but the size of each successive section is larger as the creature inside grows. If the rate of growth outward and the rate of growth around match about right, then you end up with the same “exponential plus delay” type growth rate (i.e. an analogue version of Fibonacci numbers leading to a golden spiral) One important feature of the golden spiral is self-similarity, meaning that if you take the first chunk of the growth and just make it bigger, you get exactly the same shape as the older shell (i.e. a 2 year old nautilus shell is the same shape as a 3 year old nautilus shell, just smaller) This means that the *shape* of the wee critter is the same and they don’t have to change shape as they grow. Anything that meant that the growth rates became mismatched would be rather inconvenient and the resulting organism would likely lose the Darwinian game an not survive to make more non-golden-spiral nautilus shells.

Mathologer has [a good video](https://www.youtube.com/watch?v=_GkxCIW46to) on the subject of Fibonacci numbers / golden ratio / spirals in nature. Numberphile also has some good Fibonacci / golden ratio / golden spiral videos you might want to watch, too 🙂

Golden ratio is (√5+1)/2.

This number is special because it’s the most irrational number that exists.

A rational number could be expressed as p/q an irrational number could also be expressed as p/q to some degree of accuracy (take π and 22/7). Where p and q are relatively small.

However (√5+1)/2 cannot be easily expressed as small p/q.

I cannot easily type this but (√5+1)/2 is expressed at a continued fraction of 1+1/(1+1/(1+1/(1+1/…….)))). That corresponds to the smallest real number where the variability in the next term is extremely high.

TL;DR Watch this brilliant [Numberphile video](https://youtu.be/sj8Sg8qnjOg)