Why is the Fibonacci sequence found everywhere?
In: 2220
The Fibbonachi sequence approximates and tends towards the Golden Ratio.
The question then is why the Golden Ratio shows up so often. It doesn’t show up *everywhere* but it does show up with unusual frequency compared with literally any other ratio of exponential growth. Why ~1.61? Why not 2.7 or 5.5 or 1.09?
If something occurs repeatedly when it would otherwise be arbitrary or random that generally means some process must be driving a bunch of different things towards the same target because it is optimal. So what makes the Golden Ratio Unique, and when is that property going to be beneficial?
The Golden Ratio is the **most* irrational of irrational numbers. It is the irrational number least well approximated by any integer ratio. If you use it to drive or sample a periodic process you will get the most aperiodic results.
So it makes sense that the golden ratio will show up in the real world in cases where periodic, repeated structures or events are detrimental to some process.
The classic example is growing branches for leaves on a plant as the stem grows upward. If the biological component that sets a new branch to sprout from the trunk spins around as it grows upwards, how often should it deploy a new branch?
Once per revolution would cause all the branches to be on one side of the tree, on top of eachother and blocking the light to those below. That’s bad. Maybe twice per revolution? Now you just have two stacks of branches on opposite sides of the tree – not much better. Maybe 3x? 5x? Any integer rate per revolution will just give you spokes like a bicycle wheel. What about a fraction like 3/5ths? You’ll still get spokes equal to the denominator. Any rational fraction is periodic and overall not optimal.
So try an irrational ratio! Every pi rotations, place a branch! Now no branch will ever sit directly above another branch. That’s an improvement, but pi is really well approximated by 22/7, so it’ll just look like 22/7 with a very slight fanning out of each spoke.
It will hardly be any better than 22/7. The Golden Ratio is the Least Well Approximated irrational number, so it will be the least periodic and get you as far from this ‘spokes’ problem as you can get.
So it’s not unsurprising that many plants deploy branches and leaves and petals and seeds at rates close to this ratio. This isn’t universal by any means but it’s an optimal arrangment when things benefit from not being periodic and thus many things evolve towards the same ideal.
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