Sometimes I’m excited by musical ideas. I play around until I find a chord progression or a bass line that stimulates me, and a week later I’ve expanded it into a new piece.
Other times my musical ideas bore me. Nothing has any verve. When I hit one of these dead zones, my mind starts drifting off in the direction of chess variants. Don’t ask me why that happens — I don’t know.
Thousands of chess variants have been proposed. (You can find a rich repository of them online.) Many employ exotic pieces. A smaller number change the topology of the board. You can play chess on a board of hexagons, or in a three-dimensional matrix, if you make suitable adjustments to how the pieces move.
Instead of (or in addition to) those changes, you can make an invisible connection between the left side of a conventional 8×8 chess board and the right side. A rook on b3 can travel through a3 and continue to h3 and g3, or vice-versa. This makes the board, topologically speaking, a cylinder. This well-known and quite playable variant is called Cylindrical Chess.
If you also connect the top and bottom edges of the board in the same way, you’ve constructed a three-dimensional surface called a torus. (That’s what mathematicians call a doughnut.) Playing chess on a torus would require some serious readjustments in the initial layout of the pieces, and probably other rule changes as well. If you don’t change anything else, in the initial position the two kings would be on adjacent squares, and both would be checkmated. With a slightly larger board and two rows of pawns surrounding each row of pieces, toroidal chess becomes at least conceivable.
Rather than connect the top of the board to the bottom and the left side to the right, you could warp the board so that the left side connects to the top and the right side to the bottom. Topologically, this would still be a flat, two-dimensional board, not a torus. With a standard opening layout of pieces, one of white’s rook pawns could capture a black bishop, and the black and white rooks would be adjacent to one another. Probably not a feasible opening layout, then.
Alternatively, you could connect the left side to the right with a twist. The rook on b3 would then move sideways through a3 and emerge not on h3 but on h6. (Again, the white and black rooks would be adjacent to one another.) This board would be a Mobius strip. Leaving that connection in place, you could also do the same thing with the top and bottom edges, twisting the connection so that the a file connects to the h file and so on. At that point, a simple two-dimensional 8×8 chess board has become a four-dimensional object — a Mobius strip whose edge (yes, a Mobius strip has only one edge) is warped into the fourth dimension so as to fold over and connect with itself. A Klein bottle, unless I miss my guess.
But why limit ourselves to a flat, two-dimensional chess board? What if we start with a three-dimensional playing area, perhaps 6x6x6 cells? (We wouldn’t want to make it 8x8x8. That would be too many cells.) If we simply connect the left and right faces, we have another torus. This time, though, the pieces are moving around inside the torus rather than on its surface. But we could also, at the same time, connect the front and back faces and the top and bottom faces. What we have now is a hyper-torus, a four-dimensional object.
Nobody is ever going to try playing chess on a hyper-torus, so I don’t have to worry about how to set up the pieces. It would be messy. I just think the board itself is interesting.