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Author Topic: Fibonacci and the Mandelbrot set  (Read 14029 times)
Description: This is a Fibonacci-related area of the Mandelbrot set
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claude
Fractal Bachius
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Posts: 563



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« Reply #15 on: November 19, 2016, 10:26:36 PM »

the size-ratio of bulbs that are fibonacci neighbours do converge to phi^2 the higher the fibonacci period value gets?

Yep!  Great work with your conjecture, it's a pretty cool result smiley

(Note that I ran the code for 24 bulbs, but later bulbs start to deviate and go totally wrong around 32 when the bulb size is smaller than the resolution of double precision - so it'd be worth checking with higher precision at some point...)
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lkmitch
Fractal Lover
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« Reply #16 on: November 21, 2016, 05:08:33 PM »

Mandelbrot himself came up with a model for the size of the disks.  It's something like:

d = sin(m/n * pi)/n^2,

where m/n is the fraction associated with the disk (1/2 for the largest one, 1/3 for the big one at the top, 2/3 for the big one at the bottom, etc.).  For the disks in the series 2/3, 3/5, 5/8, 8/13, etc., the fraction converges to 1/Phi ~ 0.6180339, so there's probably some way to get the size convergence from Mandelbrot's model.

Note, the model is only approximate; the deviations from the model seem the be fractal!
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