Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new vertices onto the sphere.
Some sequences of tesselations converge to the sphere in the sense that the maximum distance between the polyhedra and the sphere converges to 0. Some sequences possibly converge more strongly if the density of vertices converges to the uniform distribution over the sphere (this is not the case with the sequence offered as an example above).
- Are there convergent sequences of tesselations where the number of distinct triangles involved in each tesselation is bounded for the whole sequence, and if so, what is the lowest bound?
Rephrasing: is there an integer $k$ such that $\forall \epsilon > 0$ there is a polyhedron with triangular faces which approximate a unit sphere within $\epsilon$ and has at most $k$ distinct faces.
- If so, do some of these sequences also let the density of vertices approach the uniform distribution?
Clarification: we are looking at the number of distinct triangles. For example, an isocahedron has exactly 1 distinct triangle.
Further clarification: the rephrasing above is not quite accurate. As commenters point out, a simple rasterization of the sphere yields arbitrarily good approximations with just one triangle.
We are looking for polyhedra where the vertices lie on the sphere itself or, at least, convex polyhedra.