Re: possibility of emulation

[Jim Kingdon <kingdon at panix dot com>]

> Xconq on a Klein Bottle map, anyone?

Oh, that's different from hex vs non-hex (and probably easier).

We all know flat maps (edges on left, right, top and bottom) - the
xconq default.

xconq has the start of a cylindrical map (edges on top and bottom, but
if you go off the left edge you appear on the right with the same y
coordinate).

There are plenty of computer games with a doughnut-shaped map (like
cylindrical, but wrap around top/bottom as well). (Mathematicians call
this one a torus).

Now, let's try the Moebius strip.  Make it like the cylindrical map,
but when you go off the left edge you appear on the right with a y
coordinate of "size - origY".  So imagine following the top edge to
the right (east).  As you go off the right, you get to the bottom left
corner, and if you keep going east, you follow the bottom edge, and
then appear at the top left.  Note that you are following the edge,
and without moving to the other edge, there is only one edge.

Ready for the Klein bottle?  Join both edges (as we did with the
doughnut).  But twist one of the joins as we did for the Moebius
strip.  Actually, it is probably easier to implement in xconq than it
is to visualize.

There are some nice pictures at
http://www.jcu.edu/math/vignettes/Mobius.htm
Looking at the pictures of rectangles with arrows on them should show
how it would look on the xconq screen.

What all this does for gameplay, I don't know.  Might have a big
effect, given how easy it is to sneak past the AI at the edge of the
map.  But maybe the AI would just have another weak spot - I'm not
sure whether this weakness of the AI is a matter of geometry or just
of the AI not protecting its flanks/rear in general.

> local plane approximations of multi-dimensional saddle curves or
> spheroidal pieces. (There probably couldn't be any decent,
> comprehensible representation beyond a certain zoom factor, so there
> would be no world map.)

That would be interesting too (in mathematical terms, it is a question
of changing the geometry, not just the topology).