1. The chess board is the real projective plane: http://en.wikipedia.org/wiki/Real_projective_plane . So if a bishop is on h1 and moves forward and to the right, it comes out on a7 and would proceed to b6. A rook that moves down from f1 comes out on c8 and proceeds to c7.

2. The board starts out empty. Players begin with each of the eight non-pawn pieces in their "inventory". A turn consists either of moving a piece or of placing a piece from your inventory. You must place your king on the first turn.

3. All pieces except for pawns follow the same movement rules as in regular chess. Pawns can move one space in any orthogonal direction or capture one space in any diagonal direction.

4. Whenever one of your pieces other than a pawn is captured, you get a pawn placed in your inventory.

5. If one of your pawns captures a piece, your pawn transforms into the piece it captured (but remains its own color).

6. The object of the game is to capture your opponent's king.