## Some hyperbolic, Euclidean, and spherical worlds. October 9, 1998

These universes are populated with fake disk galaxies.
It'd be easy to make them, say, fade out a bit toward their edges,
to look more galaxy-like and less cartoon-like, though they'd still
be very unconvincing.

The hyperbolic world
looks densely populated, but in fact it's the least dense of the three.
Since apparent size shrinks **rapidly** with distance in
hyperbolic space, the moderately-distant "galaxies" appear very small,
and the most distant ones are not even visible.

In a Euclidean universe, apparent size shrinks linearly with distance.
The most distant, reddest galaxies (at distance about 4) are easily
big enough to see.

This spherical universe, though apparently sparsely populated, has the
highest density of galaxies of the three -- more than twice that of the
hyperbolic universe. Objects look smallest when at a distance of
1/4 of the sphere's circumference; more distant ones appear larger!
In this as in the other universes, redness increases with distance,
and in fact the color coding is the same in all three spaces.

## Well-ordered hyperbolic spaces

The view of the hyperbolic world above was made from a regular tessellation
of hyperbolic space, with each galaxy randomly perturbed from the center
of each tile. Here's what we see without the perturbation. The tiling
was chosen so that there are many infinite fault planes (some of the tile
edges have 90-degree dihedral angles). Viewed in hyperbolic space, a
plane looks like a finite disk, so there are lots of rings in these pictures.

In this version, I show the tile edges as thin blue lines.
Probably these "crystalline" artifacts would make all three such images
unsuitable for illustrations to your articles --
people might think the pattern was the characteristic feature of the space --
but I thought they looked nice.

## First tries at a hyperbolic world, 21 Sep 1998

Built by taking the tiling of the hyperbolic plane with 7 equilateral
triangles meeting at a vertex, then subdividing each triangle into
four smaller triangles at edge midpoints. A post stands at the center
of each of the resulting subtriangles. We view the whole thing from
a point in H3 slightly off the plane.

If you have geomview,
see the corresponding GCL file.

Another presentation of the same:

Corresponding GCL file

Stuart Levy, slevy@ncsa.uiuc.edu.