Chess on a Klein bottle

Horizontal glide-reflection between ranks

The version of the Klein bottle which is closest to the standard chess board is obtained by connecting every square of the uppermost and lowermost rank to the square obtained by shifting horizontally with half of the board width. This corresponds to a glide-reflection below rank 1 and above rank 8 with a distance of 4 squares. In addition, the board has a vertical translation by 16 squares.

Pieces are set up as in standard chess.

Horizontal glide-reflection within ranks

Another Klein bottle is obtained by a glide-reflection within ranks 1 and 8 by 8 squares. This doubles the ranks 2 to 7 and gives a vertical translation by 14 squares.

Here the pieces of each player are flanked by a line of pawns. Note that even though there are 16 pawns of each color, they all stand in one line. Pawns always move away from their friendly pieces towards the opponent, where they are promoted.

Vertical glide-reflection

Finally, one can also use a vertical glide-reflection by 14 squares and a horizontal translation by 8 squares, which means gluing the opposing edge points of a Möbius strip, to obtain this variant.

Here the pieces of each player are flanked by a line of pawns on either side. Pawns always move away from their friendly pieces towards the opponent, where they are promoted.

Horizontal or vertical glide-reflection within ranks (with modified pawns)

For this 8 × 8 board, there is a horizontal glide-reflection in ranks 3 and 7.

Alternatively, one may have a glide-reflection in files b and f.

In both cases the starting position achieves that the only pieces under attack are the pawns, which are attacked by the queen and bishop. There is no castling, pawns step orthogonally in any direction or capture diagonally in any direction and are not promoted.

Diagonal glide-reflection within cells and corners (with modified pawns)

For the modified pawn version, one can also conceive a version with a diagonal glide-reflection by (4, -4) and a diagonal translation by (8, 8).

Here one uses a slightly different starting position compared to the previous version, which is chosen such that on this modified board only the pawns are attacked by queen and bishop. Again there is no castling, pawns step orthogonally in any direction or capture diagonally in any direction and are not promoted.