There have been several variants of chess designed, where the edges of the board are assumed to touch. One of the chesses, where the board actually forms a torus is this one. The following description of the game was send to me by one of the inventors, Richard Gadsden, who invented this game with a friend, Phil Roberts, as a variant of Cylindrical Chess.
The game was invented around 1990, and played by the inventors at school. Gadsden wrote about the game:
It's a bit like cylindrical chess, but even worse.
We played several games, and it's fun, but I've regularly put someone into check and not realised. This is a problem.
You need a normal (8x8) board, and an extra set of pawns on top of a normal full chess set.
You set up like this:
The front rank of pawns may only move forward one square as their first move, but the rear rank may open with a two or three square move.
Pieces may wrap around not only the sides of the board, but also off the front and back, so the four corners are all adjacent.
Pawns effectively treat the board as cylindrical since they can only move forwards.
The game is strange (to say the least) as pieces tend to flit about like mad. Moreover it's very cramped and I've played several games where an unspoken agreement was made to exchange pieces and pawns at every opportunity, just to open the game up a bit.