26
Jun
04

Inverse Julia Set Fractal

Here is cubic Julia set fractal using the inverse Julia set technique. This technique is good for showing the edges of the set, but some regions are faint because they attract much slower

(* runtime: 4 seconds *)
InvertList[zlist_] := Flatten[Map[(# - zc)^(1/3){1, -(-1)^(1/3) , (-1)^(2/3)} &, zlist], 1];
zc = -0.5 - 0.05 I; zlist = {0}; Do[zlist = InvertList[zlist], {10}];
ListPlot[{Re[#], Im[#]} & /@ zlist, PlotStyle -> PointSize[0.005], AspectRatio -> 1, Axes -> None];

This can be improved using pruning. The following code was adapted from Mark McClure’s Mathematica code:
(* runtime: 4 seconds *)
InvertList[zlist_] := Flatten[Map[Floor[130(# - zc)^(1/3){1, -(-1)^(1/3), (-1)^(2/3)}]/130 &, zlist], 1];
zc = -0.5 - 0.05 I; zlist = Nest[InvertList, {0}, 8]; zlist2 = zlist;
While[zlist2 =!= {}, zlist2 = Complement[InvertList[zlist2],zlist]; zlist = Union[zlist2, zlist]];
ListPlot[{Re[#], Im[#]} & /@ zlist,PlotStyle -> PointSize[0.005], AspectRatio -> 1, Axes -> None];

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