I sloppily claimed that to define a polyhedron given a bunch of faces, it's necessary only to say which pairs of faces share an edge. This is not true! It's important to say *which* pairs of edges are identified, and also in which direction* each pair is identified. Thanks to Marc LeBrun for pointing out my error. (E.g., to identify the edges of a square by pretending to fold it along a diagonal gives a sphere instead of the usual torus you get by identifying opposite edges by a translation. And identifying each pair of opposite edges by a reversal results in the projective plane.) Puzzle: What surface do you get if you identify opposite edges of a square of side = 1 by a translation followed by sliding along each edge by an amount equal to 1/2 (mod 1)? In other words, if the square is [0,1]x[0,1] * identify (0,y) with (1, y + 1/2 (mod 1)) for each y in [0,1] and * identify (x,0) with (x + 1/2 (mod 1), 1) for each x in [0,1]. —Dan