Two chess variants, played on the chess board.
I can’t lay claim to the first, but I’m pretty sure that the second is my own creation – I certainly came up with it myself, but I don’t know if anyone else has thought of it in this form before. There is mention of something similarly-inspired on the Chess Variants Pages, but the game is very different to mine, and you need special equipment!
These variants both use the same board and pieces as regular chess. The movement and capture rules are the same as chess. The difference lies in the geometry of the board, where we make edge identifications to alter the space the game is played on. The first variant is called Cylindrical Chess, the second is Möbius Chess.