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Author Topic: Where There's Smoke?  (Read 399 times)
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Pauldelbrot
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Posts: 2592



pderbyshire2
« on: May 03, 2013, 10:20:35 AM »

Where There's Smoke?



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=14009

There are no iteration bands here ... only smoke, flames, and minibrots. Fractal detail fills nearly all of the image.

The mapping used to generate this image is

z_{n + 1} = frac{z_n + frac{1}{sqrt{3}}a}{b(z_n^3 - sqrt{3}az_n^2 + cz_n + frac{1}{sqrt{3}}ac)} + d

and the image is in the a-plane with upper left corner 0.26422942305 - 2.1472630887i, lower right corner 0.31023820036 - 2.1817696717i, z0 = a, b = 0.073049 + 0.039934i, c = -1.804256 + 2.869793i, and d = -0.119048 + 0.083333i.

For most of the a values in the image, except in the minibrots, the dynamics possesses no attractor and the Julia set is the whole Riemann sphere. Elliptic Harlequin coloring (UF users have access to this via, IIRC, akl-m-math.ucl) gives the tools needed to visualize the dynamics without any attractor to target. The result resembles smooth iteration coloring, without need for convergence to any attractor, and makes visible the dendrites that completely occupy the space around the minibrots. This location is in a seahorse valley, shallow and on the double-spiral side, giving the usual shapes for such an area, but fattened to fill space with infinite smoke-swirl detail as there is no fixed attracting basin to capture most of the points as there is in the normal Mandelbrot Set's seahorse valley.
« Last Edit: May 19, 2013, 06:10:06 AM by Pauldelbrot » Logged

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