Fractals & Chaos Recap for 12/20

The continued fraction sequence for T = pi is [3;7,15,1,292,…]. The very first rational approximation we get for pi is 22/7, which has three digits of accuracy after only one iteration. This is a very good approximation of pi, which is why the 7 spiral arms we see in Complex Paint are so persistent. You’ll notice that the next term of the sequence, a_2 = 15, is followed immediately by a_3 = 1, the lowest value we could possibly add. This suggests that the second rational approximation, 333/106, is very bad. Even though it gives us 5 digits of accuracy, we don’t even see 106 of anything in Complex Paint. Instead, the third rational approximation, 355/113, produces seven digits of accuracy with a next term of 292, meaning we “lose” very little accuracy by stopping our continued fraction there. The pattern we get in complex paint is 113 spokes.

Finally, this also gives us a way of saying that PHI, the golden ratio, is the most irrational number. Its continued fraction sequence is [1;1,1,1,1,1,…], the worst possible sequence we could get. This is why in Complex Paint we never see spokes, and why we can see multiple spiral patterns within the same value of R. None of the rational approximations we create are “good” approximations.

This fact about Phi is also why the Golden Ratio comes up so much in nature: sprouting leaves or seeds in rotations  around a central stem by a quantity of the golden ratio will guarantee that your seeds or leaves don’t line up. All leaves get some sun exposure, and you’ve maximized the quantity of seeds. See this Math is Fun page for a great explanation of this phenomenon, or Vi Hart’s series of videos on the topic starting here.

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