Thurston’s paper Shapes of polyhedra and triangulations of the sphere is really remarkable. I’m writing about it in my next column for the Notices of the American Mathematical Society. Here’s a draft ...
Besides Thurston’s paper, I recommend these others: P. Engel and P. Smillie, The number of convex tilings of the sphere by triangles, squares, or hexagons. R. E. Schwartz, Notes on shapes of polyhedra ...