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A 33 Metadisk group:
Group:
mu = -I/Sqrt[2];
a = N[{{1, 0}, {mu*N[Exp[I*2*Pi/3]], 1}}]
A = N[{{1, 0}, {mu*N[Exp[-I*2*Pi/3]], 1}}]
b = N[{{Sqrt[2] - I*Sqrt[2], Sqrt[3]}, {Sqrt[3], Sqrt[2] + I*Sqrt[2]}}]
B = Inverse[b];
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11/13/18
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Four views : one strange looking surface....
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A strange formation wrt a new hybrid version of an older fractal of mine.

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#Fractal #Art #Math #Space #Iteration
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Kleinian group hyperbolic 3 manifold projection onto 3d parametric coordinates ->33 metadisk:
(* Mathematica*)
Clear[x, y, z, t, p, a, b, A, B, f1a, f2a, f3a, f4a, w, z1]
(* hyperbolic cylinder parametric coordinates*)
x = Cos[t]Cosh[p];
y = Sin[t]*Cosh[p];
z = Sinh[p];
g1 = ParametricPlot3D[{x, y, z}, {t, -Pi, Pi}, {p, -Pi, Pi},
ColorFunction -> "CMYKColors", Mesh -> False, Axes -> False,
ImageSize -> 1000, Boxed -> False, PlotPoints -> 60,
PlotStyle -> Opacity[0.5], ViewPoint -> {5, 5, 5}]
mu = -I/Sqrt[2];
a = N[{{1, 0}, {mu*N[Exp[I*2*Pi/3]], 1}}]
A = N[{{1, 0}, {mu*N[Exp[-I*2*Pi/3]], 1}}]
b = N[{{Sqrt[2] - I*Sqrt[2], Sqrt[3]}, {Sqrt[3], Sqrt[2] + I*Sqrt[2]}}]
B = Inverse[b];
f1a[z1_] = (a[[1, 1]]*z1 + a[[1, 2]])/(a[[2, 1]]*z1 + a[[2, 2]]);
f2a[z1_] = (b[[1, 1]]*z1 + b[[1, 2]])/(b[[2, 1]]*z1 + b[[2, 2]]);
f3a[z1_] = (A[[1, 1]]*z1 + A[[1, 2]])/(A[[2, 1]]*z1 + A[[2, 2]]);
f4a[z1_] = (B[[1, 1]]*z1 + B[[1, 2]])/(B[[2, 1]]*z1 + B[[2, 2]]);
ComplexExpand[{f1a[x], f2a[y], f3a[z]}]
( cyclic Möbius projection of hyperbolic cyclinder by the Weeks hyperbolic 3 \
manifold*)
(Cyclic projection of complex variables: Im[x]→Re[y], \
Im[y]→Re[z],Im[z]→Re[x])

w = {0.` + (
1.` Cos[t] Cosh[
p])/((0.` + 0.35355339059327356` Cos[t] Cosh[p])^2 + (1.` +
0.6123724356957945` Cos[t] Cosh[p])^2) + (
0.6123724356957945` Cos[t]^2 Cosh[
p]^2)/((0.` + 0.35355339059327356` Cos[t] Cosh[p])^2 + (1.` +
0.6123724356957945` Cos[t] Cosh[p])^2) + (0.` - (
0.35355339059327356` Sinh[
p]^2)/((1.` - 0.6123724356957945` Sinh[p])^2 + (0.` +
0.35355339059327356` Sinh[p])^2)),
0.` + 2.4494897427831783`/(
2.0000000000000004` + (1.4142135623730951` +
1.7320508075688772` Cosh[p] Sin[t])^2) + (
2.9999999999999996` Cosh[p] Sin[t])/(
2.0000000000000004` + (1.4142135623730951` +
1.7320508075688772` Cosh[p] Sin[t])^2) + (
2.4494897427831783` Cosh[p]^2 Sin[t]^2)/(
2.0000000000000004` + (1.4142135623730951` +
1.7320508075688772` Cosh[p] Sin[t])^2) + (0.` - (
0.35355339059327356` Cos[t]^2 Cosh[
p]^2)/((0.` + 0.35355339059327356` Cos[t] Cosh[p])^2 + (1.` +
0.6123724356957945` Cos[t] Cosh[p])^2)),
0.` + (1.` Sinh[
p])/((1.` - 0.6123724356957945` Sinh[p])^2 + (0.` +
0.35355339059327356` Sinh[p])^2) - (
0.6123724356957945` Sinh[
p]^2)/((1.` - 0.6123724356957945` Sinh[p])^2 + (0.` +
0.35355339059327356` Sinh[p])^2) + (0.` - 2.4494897427831783`/(
2.0000000000000004` + (1.4142135623730951` +
1.7320508075688772` Cosh[p] Sin[t])^2) - (
4.000000000000001` Cosh[p] Sin[t])/(
2.0000000000000004` + (1.4142135623730951` +
1.7320508075688772` Cosh[p] Sin[t])^2) - (
2.4494897427831783` Cosh[p]^2 Sin[t]^2)/(
2.0000000000000004` + (1.4142135623730951` +
1.7320508075688772` Cosh[p] Sin[t])^2))}



g2 = ParametricPlot3D[w, {t, -Pi, Pi}, {p, -Pi^2, Pi^2},
ColorFunction -> "CMYKColors", PlotStyle -> Opacity[0.5], Mesh -> False,
Axes -> False, ImageSize -> 1000, Boxed -> False, PlotPoints -> 200,
PlotRange -> All]
Show[{g1, g2}, ViewPoint -> Top]
(end)
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