22
Jun
05

Poincaré Hyperbolic Tiling

Poincaré Hyperbolic Tiling : Click here to download some POV-Ray code for this image.

Here is some Mathematica code:
(* runtime: 0.02 second *)
R = Sqrt[3]; Tiles = {Map[1.0 I{{#, R^2 - # Conjugate[#]}, {1, -Conjugate[#]}}/R &, {R + I, -R + I, -2I}]};
Tiles = Append[Tiles, Flatten[Table[Map[Tiles[[1, i]].Conjugate[Tiles[[1, #]]].Tiles[[1, i]] &, DeleteCases[{1, 2, 3}, i]], {i, 1, 3}], 1]];
Do[Tiles = Append[Tiles, Flatten[Table[Map[Tiles[[g, i]].Conjugate[#].Tiles[[g, i]] &, {Tiles[[g - 1, Ceiling[i/2]]], Tiles[[g, 2Ceiling[i/2] - Mod[i + 1,2]]]}], {i, 1, 3×2^(g - 1)}], 1]], {g, 2, 5}];
Tiles = Flatten[Tiles, 1]; n = Length[Tiles];
ToDisk[{{a_, b_}, {c_, d_}}] := Disk[{Re[a/c], Im[a/c]}, Abs[I/c]];
Show[Graphics[{Hue[0], Disk[{0, 0}, 1], Table[{Hue[Sqrt[i/n]], ToDisk[Tiles[[i]]]}, {i,1, n}]}, AspectRatio -> 1, PlotRange -> {{-1, 1}, {-1, 1}}]]

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