A Matchmaker Julia set with an attracting 11-cycle and an attracting fixed point; the latter is close to becoming a Siegel disk. The image was rendered using adaptive BSM: each pixel is divided into a grid of subpixels that are calculated using a form of solid-guessing, recording only which of the two basins the subpixel landed in. Then subpixels that are not in the same basin as the subpixel below, to the right, or below and to the right are counted as boundary hits and the rest as misses. The pixel as a whole is colored based on boundary hits.
In this case, the subdivision depth was 4, meaning there were 17 subpixels across and down (so 256 possible values for the number of hits in a pixel, with a checkerboard pattern of basin assignment giving the maximum and all subpixels landing in the same attracting basin giving the minimum). This generates a greyscale image with potentially the full gamut of grey shades, but it was further subjected to standard 3x3 nonadaptive antialiasing (whose subpixels were the full pixels of the BSM algorithm).
The efficiency is less than perfect, with subpixels shared between adjacent pixels being calculated twice (and the corner ones four times), using the current software platform. The results, though, seem to be worth it, and without the hassle of implementing analytic DE for Matchmaker and contending with its tendency towards iffy performance with non-quadratic Julia sets. It won't work well near parabolic basins, but
c'est la vie.
Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.
Detailed statistics:
Name: Brambletwist
Date: January 2, 2010
Fractal: Matchmaker Julia set
Location:
a = 0.53256 + 0.31163
i;
b = 0.16737003 - 0.18180253
iDepth: Very Shallow
Min Iterations: 244
Max Iterations: 4635
Layers: 1
Anti-aliasing: 3x3, threshold 0, depth 1
Preparation time: 5 minutes
Calculation time: 14 hours (2GHz dual-core Athlon XP)
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