Rich asks: << Dan asks ...
Let T denote the torus obtained by identifying each pair of opposite edges of a regular hexagon.
How does this work? I can't get the edges to match up.
The regular hexagon's 3 pairs of opposite edge are each identified to become 1 edge. The 6 edges you began with end up as 3 edges. The 6 vertices you began with end up as 2 vertices. The identified space is locally Euclidean, and has Euler char = V-E+F = 2-3+1 = 0. Since this surface is orientable, it must be a torus. (Equivalently, tile the plane with regular hexagons so that the origin is the center of some tile. Denote the group of translations that preserve the tiling as G. Then the quotient R^2/G is a metric space that is exactly the same as the result of identifying the hexagon as above: a certain flat torus.) --Dan