I was trying to develop a basic path-tracing algorithm for one of my projects when I encountered the following problem.
Suppose I have a spherical light source at the origin. A uniform arrangement of light rays is required to path-trace light from the light source. Here, uniform attributes to the flux of the light rays. If you take a unit sphere around the light source, then any two surfaces of equal areas must have the same number of light rays passing through it (approximately, of course.)
How can we get this uniform arrangement of light rays?
I tried the following way. Take any three neighbouring light rays, and the angle subtended by one with any other must be the same. Attempting to solve for this arrangement led me to no solution. Essentially, it boils down to attempting to tesselate a sphere using triangles (or hexagons), and this has an impossible Euler characteristic for a convex polyhedron.
$$\chi = V-E+F = F-\frac{3F}2+F = 0.5F=2.\\F=4.$$
Essentially, a tetrahedron is the only possible polyhedron.
Reframing the previous question, how at all can we tesselate a sphere using just one type of "polygon"?
Appending to it, is there a way to increase the number of rays (vertices) while keeping the uniformity intact?
Having read about the Platonic Solids, I am a bit sceptical about this. However, I am curious about how the folks at Graphics and Path-Tracing solve this issue since I do not have a background in that field.