In thinking about "continuous" chess, I started thinking about pawn lines as a continuous line across a board of width 8 units. I then started thinking about Fourier transforms of that line, and it seemed natural to paste the left chess board boundary to the right one (without a twist!) -- hence a cylindrical chess board. (It makes a lot less sense to consider a toroidal chess board for a number of reasons.) I think thought -- let's go back to the "regular"/discrete chess board & consider a cylindrical board with the left & right boundaries pasted together. What do the chess moves look like? Pawns on either the left or right boundary can attack the other boundary. Bishops' diagonal lines extend beyond the left & right boundaries & continue on the other side. Knights aren't trapped next to the left & right boundaries, but can jump across them. A rook can attack a piece from both sides, if it and the other piece are the only ones on its rank! A Queen can also attack from both sides on its rank, as well as follow diagonals like bishops. A King can't be pinned against one of the sides of the chess board, although it can still be pinned against the top or the bottom. Q: Is cylindrical chess still pretty even for white & black? I can imagine that the answer is still yes, because while all the pieces have a bit more freedom, so does the King, so it is harder to trap him. Perhaps there would be a larger number of draws?