The Singularity you refer to is indeed a very interesting game, and perhaps the most elegant design of a board with an irregular topology. I would second its nomination for being featured, except that we cannot feature a variant for which we don't even have a rule-description page. But perhaps I should make one.
The board is very interesting. The way it is represented, with the circles inside the rectangular grid, is probably not the best way for getting intuitively clear moving. The problem is that the cell corners where a circle touches a line (so on the mid-line of the board) suffer a degenerate distortion, making two different directions coincide and the angle at the corner collapse to zero degree. There are other representations that do not suffer from this.
The singularity is created by taking a 12x6 rectangular board (of which two 2x2 corners are cut away) and fold it back on itself to connect the left half of the upper edge to the right half. If you do that with a sheet of paper, it warps it into a cone, where the singularity the top of the cone. Projecting the checkered pattern on a plane perpendicular to the cone axis (i.e. looking at it from a large distance above the top) would produce a pattern where all corners keep a finite angle.
The 12x6 rectangular board can even be mapped analytically (locally angle-preserving) way by consider it the upper part of the plane of complex numbers, (with the singularity at 0), and square those numbers to make the grid fill the whole plane. This increases the size of the squares as one moves away from the singularity, though.
It is even possible to completely 'rectify' the board, by duplicating it and join it to a rotated version of itself. So that a 12x12 'four-player' board is created, with 2x2 areas cut away at the corners. The extra condition is that the position should always be point symmetric around the singularity. (So that the same player controls armies that start opposit to each other, and in each turn makes the mirrored move in both armies.) This is similar to representing a cyclindrical 8x8 board by a 16x8 board, requiring the right half is a replica of the left half, to get better visualization of what goes on at what would otherwise have been the wrapping edge.
This shows that the rule for diagonal moves through the singularity is really an irregularity (allowing the Bishop to change its shade): one would have expected the diagonal to continue into the mirror quadrant, and the move thus to effectively bounce back from the singularity. But instead it bounces off from the singularity at a 90-degree angle, onto the other main diagonal of the 12x12 board.
The Singularity you refer to is indeed a very interesting game, and perhaps the most elegant design of a board with an irregular topology. I would second its nomination for being featured, except that we cannot feature a variant for which we don't even have a rule-description page. But perhaps I should make one.
The board is very interesting. The way it is represented, with the circles inside the rectangular grid, is probably not the best way for getting intuitively clear moving. The problem is that the cell corners where a circle touches a line (so on the mid-line of the board) suffer a degenerate distortion, making two different directions coincide and the angle at the corner collapse to zero degree. There are other representations that do not suffer from this.
The singularity is created by taking a 12x6 rectangular board (of which two 2x2 corners are cut away) and fold it back on itself to connect the left half of the upper edge to the right half. If you do that with a sheet of paper, it warps it into a cone, where the singularity the top of the cone. Projecting the checkered pattern on a plane perpendicular to the cone axis (i.e. looking at it from a large distance above the top) would produce a pattern where all corners keep a finite angle.
The 12x6 rectangular board can even be mapped analytically (locally angle-preserving) way by consider it the upper part of the plane of complex numbers, (with the singularity at 0), and square those numbers to make the grid fill the whole plane. This increases the size of the squares as one moves away from the singularity, though.
It is even possible to completely 'rectify' the board, by duplicating it and join it to a rotated version of itself. So that a 12x12 'four-player' board is created, with 2x2 areas cut away at the corners. The extra condition is that the position should always be point symmetric around the singularity. (So that the same player controls armies that start opposit to each other, and in each turn makes the mirrored move in both armies.) This is similar to representing a cyclindrical 8x8 board by a 16x8 board, requiring the right half is a replica of the left half, to get better visualization of what goes on at what would otherwise have been the wrapping edge.
This shows that the rule for diagonal moves through the singularity is really an irregularity (allowing the Bishop to change its shade): one would have expected the diagonal to continue into the mirror quadrant, and the move thus to effectively bounce back from the singularity. But instead it bounces off from the singularity at a 90-degree angle, onto the other main diagonal of the 12x12 board.