Fractal Pi

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Offline FractalDave

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« on: September 13, 2018, 10:36:50 PM »
Fractal Pi



https://fractalforums.org/index.php?action=gallery;sa=view;id=1348

Just a bit of fun after iterating x+sin(x) to calculate Pi to test UFs arbitrary precision. this is part of the Mandy Set for z+sin(z)+pixel with z start as 3.1415926
Although not visible here the Set for this includes many Minibrots of the z^3+c form which I found quite interesting - why ^3 ?
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Offline claude

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« Reply #1 on: September 13, 2018, 10:48:14 PM »
why ^3 ?

the taylor series has 3 as the smallest non-linear power:
\[ \sin(z) = z - \frac{z^{\mathbf{3}}}{3!} + \frac{z^5}{5!} - \cdots \]
this means that in the renormalization/tuning near periodic points, the power 3 dominates (as the deltas are small, so higher powers get much smaller much more quickly, and constant and linear powers don't do anything interesting) and you get cubic minibrots

there is probably a more precise derivation of renormalization with equations proving this observation, but I'm too lazy to work it out

Offline FractalDave

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« Reply #2 on: September 14, 2018, 12:05:12 AM »
Thanks Claude - that makes perfect sense !!


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