This is my attempt to create the Kleinian Double Cusp Group on page 269 of Indra’s Pearls. Thanks to Dr. William Goldman for helping me get started with this. Here is some POV-Ray code for this fractal. See also my Double Spiral. You can also see this code on Roger Bagula’s web site.

(* runtime: 0.06 second *)
ta = 1.958591030 - 0.011278560I; tb = 2; tab = (ta tb + Sqrt[ta^2tb^2 - 4(ta^2 + tb^2)])/2; z0 = (tab - 2)tb/(tb tab - 2ta + 2I tab);
b = {{tb - 2I, tb}, {tb, tb + 2I}}/2; B = Inverse[b]; a = {{tab, (tab - 2)/z0}, {(tab + 2)z0, tab}}.B; A = Inverse[a];
Fix[{{a_, b_}, {c_, d_}}] := (a - d - Sqrt[4 b c + (a - d)^2])/(2 c); ToMatrix[{z_, r_}] := (I/r){{z, r^2 - z Conjugate[z]}, {1, -Conjugate[z]}};
MotherCircle[M1_, M2_, M3_] := ToMatrix[{x0 +I y0, r}] /. Solve[Map[(Re[#] - x0)^2 + (Im[#] - y0)^2 == r^2 &, Fix /@ {M1, M2, M3}], {x0, y0, r}][[2]];
C1 = MotherCircle[b, a.b.A, a.b.A.B]; C2 = MotherCircle[b.a.a.a.a.a.a.a.a.a.a.a.a.a.a.a, a.b.a.a.a.a.a.a.a.a.a.a.a.a.a.a, a.b.A.B];
Reflect[C_, M_] := M.C.Inverse[Conjugate[M]];
orbits = Join[Reverse[NestList[Reflect[#, a] &, C1, 63]], Drop[NestList[Reflect[#, A] &, C1, 63], 1], Reverse[NestList[Reflect[#, a] &, C2, 71]], Drop[NestList[Reflect[#, A] &, C2, 56], 1]];
Show[Graphics[MapIndexed[({{a, b}, {c, d}} = #1; {Hue[#2[[1]]/15], Disk[{Re[a/c], Im[a/c]}, Re[I/c]]}) &, orbits]], PlotRange -> 35{{-1, 1}, {-1, 1}}, AspectRatio -> Automatic];

This animation shows the set morphing into a single cusp group.

Links

• Explanation – by David Wright, coauthor of Indra’s Pearls
• Kleinian Gallery – beautiful Kleinian groups by Jos Leys
• Limit Sets – interesting animations by Jeffrey Brock, see his 3D bending
• Indra’s Pearls course – Kleinian groups with David Wright
• Swirlique – program for drawing Kleinian limit sets
• Gallery – includes some nice Kleinian groups, by Curtis McMullen
• Double Spiral – POV-Ray rendering by Jeffrey Pettyjohn