From: tuc@calico.cs.wisc.edu (Brian A. Cole)
Newsgroups: rec.games.abstract
Subject: Re: Chess on spherical boards.
Date: 11 Nov 92 23:19:28 GMT
Organization: U of Wisconsin Madison - Computer Sciences

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.

everything is ok up to here.

]     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.

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".  Where does it
end up?  I see two possibilities:

1. bishop faces "south", takes a step, rotates 90 degrees inticlockwise,
   and takes another step.
2. bishop faces "east", takes a step, rotates 90 degrees clockwise, and
   takes another step.

The problem is that these are inconsistent.  The lead to different
destinations.  Consider:

1. Bishop is stading in F1, and is facing "south" (toward B1).  It takes
   a step, so in is now in B1 facing B2.  It turns 90 degrees anticlock-
   wise, so it is now still in B1, but now facing A1.  It takes another
   step, ending up in A1.
2. Bishop is stading in F1, and is facing "east" (toward G1).  It takes
   a step, so in is now in G1 facing H1.  It turns 90 degrees clockwise,
   so it is now still in G1, but now facing C1.  It takes another step,
   ending up in C1.

The problem is that going over the poles (created by pinching the ends
together), the notion of direction gets skewed.  The reason is that the
"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).


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

        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

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.