# Kleinian 3group Minkowski 4d Sphere Projection:4 cusps;

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 \ I have trouble sleeping on cold nights here, so I read math books to put me to sleep.Robert Gilmore's Lie Groups, Lie Algebras and Some of Their Applicationshas a physics projection of Pauli matrices to SL(2,C) used in a Nuclear magnetic resonance calculations.He leaves out the 4th dimension,so I thought of this Minkowski Determinant {1,1,1,-1} signature group as a 4d projection.I get a very neat 4 cusp fractal in 3 group by using a disk matrix{{1 + I, 1}, {1, 1 - I}}as the third Kleinian group matrix:s = {{1 + I, 1}, {1, 1 - I}}; s = {{0.5+ 0.5 I, 0.7071067811865475}, {-0.7071067811865475, 0.5- 0.5 I}}; s = {{0.5- 0.5 I, -0.7071067811865475}, {0.7071067811865475, 0.5+ 0.5 I}}; s = Inverse[s.s.s];(* Mathematica*) (Minkowski 4d 2x2 group)s = {{1, 0}, {0, 1}};s = {{I, 0}, {0, -I}};s = {{0, 1}, {-1, 0}};s = {{0, 1}, {1, 0}};In:= Clear[a, b, c, d]m = as + bs + cs + ds;In:= d = d /. Solve[Det[m] == 1, d][]Sqrt[-1 + a^2 + b^2 + c^2](Unit sphere projection)a = Sin[t]*Sin[p]; b = Cos[t]*Sin[p]; c = Cos[p];(* two matrices at{Pi/4,3Pi/4} 90 degress apart*)N[m /. t -> Pi/4 /. p -> Pi/4] {{0.5 + 0.5 I, 0.707107}, {-0.707107, 0.5 - 0.5 I}}N[m /. t -> 3Pi/4 /. p -> 3Pi/4] {{0.5 - 0.5 I, -0.707107}, {0.707107, 0.5 + 0.5 I}}(end)