On Fri, 22 May 1998, Gary V. Foss wrote:

> Ryan B. Caveney wrote:
> > On Thu, 21 May 1998, Gary V. Foss wrote:
> > >
> > > I like flat worlds, personally. They're easier to map.
> >
> > Oh, indeed! But all you really need is zero Gaussian curvature:
> > cylindrical worlds (or certain toroids) also have no distance distortion,
> > and have this nice property of having fewer edges. This is the approach
> > taken by many computer games.
> You lost me right after the word "zero" in the above message. I'm afraid I
> know more about water ballet than I do about cartography. More's the pity.

Ooops. Slight attack of jargonitis there, sorry.

What I mean is, any surface that you can make by rolling up a piece of
paper, but not stretching it, which in normal space is a generalized
cylinder (e.g., folding it into a U would also be a "cylinder", in the
differential geometry sense I'm referring to).

To do this with a torus (doughnut, inner tube), you need to fold it in
four spatial dimensions; and as long as we're talking "in fantasy you can
do anything", why not? I seem to remember an article in Dragon from about
1986 (unfortunately, I don't have a copy) in which it discussed a dungeon
with a room that was a 4D hypercube (also well-described in the Heinlein
short story "And He Built a Crooked House").

By "zero distance distortion" I mean that when you do a projection onto a
flat map of a spherical world, you necessarily have the problem that some
areas are represented very much out-of-scale with the others; for example,
on a Mercator projection (simplest kind), Greenland looks as big or bigger
than South America, whereas in reality it is much smaller. Thus the
notation 1"=300 mi is not helpful for such a map unless you specify the
position (latitude is enough, in this example) at which it holds, and how
it changes between positions. Find a National Geographic world map some
time: somewhere out in the middle of an ocean will be exactly the sort of
variable distance scale marker I mean, which will have multiple parallel
lines of scale disance representing a fixed actual distance, one for each
10 degrees of latitude, connected by solid curving lines for
interpolation. It works the other way, too: given a list of distances
between points, it is possible to say what the curvature of the underlying
surface is.

- --Ryan
(who really loves differential geometry;
cartography is just a sideline ;)