As a non-mathematician, one thing that helped me understand this stuff -- or at least, helped it seem not totally arbitrary which tilings "exist" -- was this ta

As a non-mathematician, one thing that helped me understand this stuff -- or at least, helped it seem not totally arbitrary which tilings "exist" -- was this table: https://en.wikipedia.org/wiki/Template:Regular_hyperbolic_tiling_table https://en.wikipedia.org/wiki/Template:Regular_hyperbolic_ti... (You gotta click [show] for some reason) These are all the possible tilings, organized by the number of sides of each polygon (Y axis) and the number of polygons meeting at each point (X axis). The table shows that all of the infinitely-many tilings are possible, but most of them (infinitely many) only work on the hyperbolic plane (cells with blue backgrounds). The cells with red backgrounds are tilings that work on the sphere, like {5, 3} (three pentagons around each point). And the cells with green backgrounds, the rarest of all, are the tilings that work on that knife-edge between the sphere and the hyperbolic plane: the flat, Euclidean plane. The green cells are at {6,3} (hexagontal tiling), {4,4} (square tiling) and {3,6} (triangular tiling). Anyway, just wanted to share this table because it's quite hard to find on Wikipedia and presents the subject in a way that I found enlightening and satisfying.