I have 2 mobius transformations that generate the tetrahedral symmetries (all 12 of them). This corresponds to the A4 subgroup of S4, I believe.
Expressed as matrices, they are:
U = [.5(1-i), .5(1-i); .5(-1-i), .5(1+i)]
V = [.5(1-i), .5(-1-i); .5(1-i), .5(1+i)]
U^3 = V^3 = (UV)^2
So I would like to tile my (Riemann) sphere with triangles. The angles of the triangle representing the fundamental region are PI/3, PI/3, PI/2 of course. Note that this region has an area equal to 1/24th the area of a sphere. Since I only have 12 symmetries, I expect to have 12 triangles with 12 empty triangles between them. Each non-empty triangle should touch another non-empty triangle at each vertex.
The problem I'm having is that I can't find a suitable fundamental region. I tried placing the triangle so that one leg is on the equator and another leg points to the north pole, but that doesn't result in a nice tessellation of the sphere with triangles. Instead the triangles overlap, or don't touch at the vertices, depending on placement.
So - how to calculate a fundamental region for the above generators for a (2,3,3) triangle group?
Finding an appropriate fundamental domain was not easy, but a triangle with vertices at these corners works: [0, -x + xi, x+xi] where x= -0.366025393.
In spherical coordinates (phi, theta) these corners correspond to:
[(0,0), (-PI/4,len), (+PI/4,len)] where len = acos((cos(PI/3.)+cos(PI/3.)*cos(PI/2.))/(sin(PI/3.)*sin(PI/2.))) = 0.955316618 rad.