Jared is right to point out that the board is not spherical, but it's not
a torus either. It would be a torus with a half-twist if a1->d1 were
joined to e8->h8 and e1->e4 to a8->d8, and it would be a sphere if a1->d1
were joined to h1->e1 and a8->d8 to h8->e8. But instead we have a1->d1
joined to e1->h1 and a8->d8 to e8->h8. This board is a compact nonorientable manifold with Euler characteristic 0, i.e. a Klein bottle.
Of course, chess on a Klein bottle has got to be at least as cool as chess
on a sphere, right?
Of course, chess on a Klein bottle has got to be at least as cool as chess on a sphere, right?