Newsgroups: rec.games.abstract
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: 12 Nov 92 22:25:51 GMT
Subject: Re: Chess on spherical boards.

mjd@saul.cis.upenn.edu ("[*] The Power in the Streets") describes
turning the chessboard into a sphere by first making the A and H files
adjacent to form a cylinder, then

> Now pinch the ends together: x,8 (for some letter x) is
> orthogonally adjacent to (x+4 mod 8),8 and diagonally to (x+3 mod
> 8),8 and (x+5 mod 8),8.

and follows up:

> Of course this isn't a sphere either, because the circumference
> is 8 squares in the east-west direction and 16 squares in the
> north-south direction.

More to the point, this isn't even topologically a sphere. For
instance, it can be covered with disjoint north-south circumferences,
but a true sphere cannot be covered with disjoint circles (or
homeomorphs thereof).  I think topologists would call this a
sphere with two cross-caps.

The idea of making file x extend into file -x (mod 8) (that is, files
a, b, c, and d extend to files h, g, f, and e, respectively) would
make the board a sphere.  And I like the suggestion of
t...@calico.cs.wisc.edu (Brian A.  Cole) that we also make rank x
extend into rank 1-x (mod 8) (ranks 1, 2, 3, and 4 extend into ranks
8, 7, 6, and 5, respectively) because it emphasizes the symmetry
between ranks and files.  However, I think Brian's claim that

> 1. each square shares one edge each with four other squares.

is incorrect, because H4 shares two edges with H5; similarly the other
edge centers have only three neighbors.  And when he says

> 2. moving one square "left" then one square "down" takes you the
>    same place as moving one square "down" then one square "left"
>    (and so on for other paths-that-should-go-the-same-place).

it should probably be mentioned that whey you go through the "top" or
"bottom" the "right" and "left" directions are interchanged, and when
you go through a side the "up" and "down" directions are interchanged.
(Does this mean that when White pawn at b8 captures f8, it is now
moving toward rank 1?  How about g8xa8/Q?

This has the unusual feature that some bishop paths bounce back when
they get to an edge:

  ...G3,H4,H4,G3,...

while others continue on:

  ...F1,G2,H3,H5,G4,F3,E2,D1,F1,G2,H3,....

Also, a knight on G3 has seven neighbors: F1, H1, H7, H5 (two ways),
F5, E4, and E2, and a knight on H4 has seven neighbors: G2, H7, G6
(two ways), G4, F5, and F3.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil