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 Applications
has 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:
s1 = {{1 + I, 1}, {1, 1 - I}};
s[2] = {{0.5 + 0.5 I, 0.7071067811865475}, {-0.7071067811865475,
0.5 - 0.5 I}};
s[3] = {{0.5 - 0.5 I, -0.7071067811865475}, {0.7071067811865475,
0.5 + 0.5 I}};
s[4] = Inverse[s1.s[2].s[3]];
(* Mathematica*)
(Minkowski 4d 2x2 group)
s1 = {{1, 0}, {0, 1}};
s[2] = {{I, 0}, {0, -I}};
s[3] = {{0, 1}, {-1, 0}};
s[4] = {{0, 1}, {1, 0}};
In[24]:= Clear[a, b, c, d]
m = as1 + bs[2] + cs[3] + ds[4];
In[26]:= d = d /. Solve[Det[m] == 1, d][[2]]
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)