From: tuc@calico.cs.wisc.edu (Brian A. Cole)
Newsgroups: rec.games.abstract
Subject: Re: Chess on spherical boards.
Date: 12 Nov 92 20:45:00 GMT
Article-I.D.: cs.tuc.721601100
Organization: U of Wisconsin Madison - Computer Sciences

The chessboard we are talking about here is this one:
] Mark-Jason Dominus (mjd@saul.cis.upenn.edu) writes:
] ]     A good version of spherical chess would have all the squares be the
] ] same, each adjacent to exactly eight others, or so it seems to me.
] ] Perhaps you could accomplish it this way:
] ] 
] ] +--+--+--+--+--+--+--+--+
] ] |A8|B8|C8|D8|E8|F8|G8|H8|
] ] +--+--+--+--+--+--+--+--+
] ] |A7|B7|C7|D7|E7|F7|G7|H7|
] ] +--+--+--+--+--+--+--+--+
] ] |A6|B6|C6|D6|E6|F6|G6|H6|
] ] +--+--+--+--+--+--+--+--+
] ] |A5|B5|C5|D5|E5|F5|G5|H5|
] ] +--+--+--+--+--+--+--+--+
] ] |A4|B4|C4|D4|E4|F4|G4|H4|
] ] +--+--+--+--+--+--+--+--+
] ] |A3|B3|C3|D3|E3|F3|G3|H3|
] ] +--+--+--+--+--+--+--+--+
] ] |A2|B2|C2|D2|E2|F2|G2|H2|
] ] +--+--+--+--+--+--+--+--+
] ] |A1|B1|C1|D1|E1|F1|G1|H1|
] ] +--+--+--+--+--+--+--+--+
] ] 
] ]     First make file A adjacent to file H; square A,i (for some number i)
] ] is orthogonally adjacent to H,i and diagonally adjacent to H,(i+1) and
] ] H,(i-1) when these exist.  You now have a cylinder.
] ] 
] ]     Now pinch the ends together: x,g8 (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.

Ok, on with the post.

Mark-Jason Dominus (mjd@saul.cis.upenn.edu) writes:
] ... tuc@calico.cs.wisc.edu (Brian A. Cole) writes:
] >   I don't think this behaves the way you want it to.  Consider the bishop
] >   starting in F1 that wants to move one sqare "southeast". 
]
]     A1.  The bishop could also have moved to H2 or G3 instead.  Or,
] going `southwest', to C1, D2, or E3.  Note that the white-square bishop
] has just turned into a black-sqare bisop.  That's the breaks.

yes, but I was talking about how many places the bishop could go that
were only _one_ space away.  The problem is that in this board, the
obvious notion of a "diagonal" is ambiguous.  So you either have to
allow the bishop to move in either of the two "southwest diagonal"
directions, or you have to (arbitrarily) limit it to one of them, and
such a limit changes the topology of the board (see below).

] >   "squares" (actually closer to triangles) at the "pinched" ends meet three
] >   other spaces at edges and meet seven (!) others at its corners (five of
] >   these at the pole).
]
]     I must be missing something.  I think E1 is adjacent to D2, E2, F2;
] D1, F1; and H1, A1, B1, and no others, which makes eight total, as it
] should be.

Well, there's two seperate things going on here.  One is the topological
shape of the board, and the other is the game we will play on it.

First, consider the shape of your board.  It _is_ a sphere, but the
"pinching" of the end causes _all_ of the squares in the bottom row
to touch each other at a point.  B1 shares an edge with A1, B2, and
C1; and share a corner with A2 and C2 (in the usual way) and with
D1, E1, F1, G1, and H1 (because of the pinch).  Again, the board
_is_ a sphere.

OK, now consider the rules of the game.  The rules don't handle what
to do with a square that meets 5 (!) squares at one of its corners
instead of just one, so the rules need a patch.  Fine, you just simply
declare that
    A1 meets E1 and doesn't meet C1, D1, F1, or G1
    B1 meets F1 and doesn't meet D1, E1, G1, or H1
    etc.
This is a reasonable solution, and it makes the game quite playable
(except for the diagonal problem).  The problem is that THE BOARD IS
NO LONGER A SPHERE (in fact, it isn't even continous, which is why the
diagonals are strange).  Again, this doesn't mean that the game we are
playing fun or worthwhile, it just means we're not playing it on a sphere.

This does makes sense, right?  We can't take to regions of a shere that
share a corner, alter the sphere so that they don't share a corner any
more, and still expect to have a sphere.

] >   Again, I think the only way (up to isomorphism) to do it is this way:

]     Any time you try to bend a chessboard into a funny shape some
] properties are going to change, to talking about an `only way' without
] first discussing the properties you want to preserve is a little silly.

You must have missed the post to which I was referring with the
word "Again".  I'll include the relevant part of it here:

# I think that this spherical board (or ones isomorphic to it) is the
# only one in which these properties hold:
#   1. each square shares one edge each with four other squares.
#   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).
# 
#         a d c b b c d a
#         g . . . . . . j        Squares with the
#         f . . . . . . i        same letter touch.
#         e . . . . . . h
#         e . . . . . . h        Think of folding the
#         f . . . . . . i        corners up till they
#         g . . . . . . j        meet above the center.
#         a m l k k l m a

It happens to preserve other "nice" properties as well.  For instance,
diagonals are well defined, with one-color bishops.  Also, if you pick
any two adjacent squares for the white king and queen to start on, and
then put the other 30 pieces down in the correct squares (relative to
the king and queen), the game is mathematically the same no matter which
initial squares you picked.

] The board I've shown has some desirable properties (uniformity,
] diagonals are great circles) and some undesirable ones.  (Diagonals are
] each half black and half white.)  Yours has some good points and some
] weak points too. (Left `k' is adjacent to right `k' in two different
] directions, for example.)

"diagonals" on your board are ambiguous without further clarification, but
that's not the point.  The point is that it is not a sphere, but some other
strange (perhaps more interesting) shape.

brian
-- 
~~~~~~~~~~~~~~~~~~~~~~~~~~     _______    _____     ~~~~~~~~~~~~~~~~~~~~~~~~~~~
Brian A. Cole                 |_   _  |  |  ___|    Most people think disc golf
  tuc@cs.wisc.edu               | | | |__| |___     is a joke.    It is not.
  ...!uwvax!cs.wisc.edu!tuc     |_| |__________|    That gives us the element
                                                    of surprise.