Pillow Chess

In cylindrical chess, one plays on a regular chessboard with the pieces in their standard positions, but imagines that the left and right edges of the board are identified. So, for example, when a rook travels horizontally off the right-hand edge, it reappears in the same row on the left-hand side of the board. It is easy to make a cylindrical chessboard out of paper, by taping together the leftand right-hand edges. However, to avoid the pieces falling off, the simplest way to play cylindrical chess is to use the regular (flat) chessboard and just remember the edge identification. Like 3-dimensional chess, cylindrical chess is well over 100 years old. According to Pritchard [59] it dates to the early eighteenth century. Byzantine or Round chess is played on a cylindrical 4 × 16 board, and is possibly a thousand years old [59, 77, 53]. In chess on a torus, one identifies the left and right edges and also the top and bottom edges, as in FIGURE 1a. These identifications really do produce the torus; the left and right edge identification gives a cylinder, and the top and bottom edge identification amounts to connecting up the ends of the cylinder (see Barr [3] or Stillwell [70] for drawings of this construction). Once again, for playing toroidal chess, it is to best to use the regular chessboard and just remember the edge identifications. The origins of toroidal chess are not clear, but it goes back at least as far as Pólya’s 1918 paper [57]. One can also play chess on the Klein bottle; the situation is similar to the torus, but the horizontal edge identifications involve a reflection (FIGURE 1b): as a rook travels up from h8, it reappears in a1. For games on the Klein bottle, there are even stronger reasons to use the regular chessboard; the Klein bottle can’t be constructed in 3-dimensional Euclidean space without self-intersections [3]. The projective plane is another surface that is commonly represented by a square with edge identifications, as in FIGURE 1c. The projective plane is the space of straight lines through the origin in 3-dimensional Euclidean space. It can be obtained from the sphere by identifying antipodal points. Like the Klein bottle, it is not orientable and can’t be realized in 3-dimensional Euclidean space without self-intersections. See Barr [3], Stillwell [70] or Prasolov [58] for more information about representing surfaces. An alternative projective chess is obtained from the traditional board by adding squares at infinity [56, Chapter 6.16]. This is also called projected chess [59]. The “fairy chess” games of FIGURE 1 are easy to understand; conceptually they are really only a small variation on traditional chess since after all, the usual board can be