Re: possibility of emulation

[Eric McDonald <mcdonald@phy.cmich.edu>]

Jim Kingdon wrote:
>>Xconq on a Klein Bottle map, anyone?
> 
> Oh, that's different from hex vs non-hex (and probably easier).

Yes, I know. That's why I mentioned it in a different paragraph. 
However, the best type of tiling might change with the topology that one 
is dealing with.

> 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).

Right. All that you need to do is set the width of the playing area 
larger than the circumference.

> 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).

Tori simply require two aligned folds of opposite edges. By aligned, I 
mean that each pair of corners that are to be joined share a common edge 
prior to being joined.

> But twist one of the joins as we did for the Moebius
> strip.  

An anti-aligned fold.

>Actually, it is probably easier to implement in xconq than it
> is to visualize.

I would agree.

> 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.

If the AI was properly done, it could have an advantage in the Klein 
Bottle case, just because the poor humans might get disoriented.

>>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).

I would consider it to be pretty much of a necessity in presenting the 
playing surface to the player.

Eric