Here is the same fractal again using the inverse Julia set technique. The 4th dimension has been color-coded. This method is much faster, but some regions are faint because they attract much slower. See also my POV-Ray code and rotatable 3D version.
original version: Mathematica 4.2, 7/18/04; animated version: POV-Ray 3.1, 7/4/06
(* runtime: 5 seconds *)
Sqrt2[q_] := Module[{r = Sqrt[Plus @@ Map[#^2 &, q]], a, b}, a = Sqrt[(q[[1]] + r)/2]; b = (r - q[[1]]) a/(q[[2]]^2 + q[[3]]^2 + q[[4]]^2); {a, b q[[2]], b q[[3]], b q[[4]]}];
QInverse[qlist_] := Flatten[Map[Module[{q = Sqrt2[# - qc]}, {q, -q}] &, qlist], 1];
qc = {-0.2, 0.8, 0, 0}; qlist = {{1.0, 1.0, 1.0, 1.0}};
Do[qlist = QInverse[qlist], {12}];
Show[Graphics3D[{PointSize[0.005], {Hue[#[[4]]], Point[Delete[#, 4]]} & /@ qlist}, Boxed -> False, Background -> RGBColor[0, 0, 0]]];
Link: QJulia – a nice inverse quaternion julia program by Chris Laurel
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that is extremely pretty.
they need to make these kinds to where they wont crap up the computer everytime you want to make them an active desktop or screen saver.
at least, they do that to mine because it doesnt have a number cruncher.