## Even more hyperbolic, Euclidean, and spherical worlds. November 16, 1998

The hyperbolic and Euclidean images are the same as yesterday's,
so they're just here for reference. The spherical-space image has
more-distinct colors, and includes a nearly-symmetrical chain of spheres
extending off into the distance, per Martin Bucher's suggestion.
I've made this image with a rather wide field of view; it might make sense to
crop it a good deal smaller than the image given here.

Please note: it's not hard to render these at higher resolution if desired;
just let me know.

## Spherical space

Here's a spherical space tiled by regular dodecahedra,
three of which fit around each edge, with small spheres marking their centers.
Apparent size in spherical space shrinks more and more slowly with distance,
reaching a minimum at the yellow-white features lying 1/4 of the universe's
circumference away. Between 1/4 and 1/2 the universe's circumference,
apparent size *increases*, as shown by the chain of four spheres
extending from center (nearest, gray) to center-right (farthest, brown).
The deep reddish-brown dodecahedra appear largest in this picture,
but they're also the most distant.

(click for larger 5.9MB 1850x1850 .tiff image)

## Hyperbolic space

Here's a network of cube-like objects tiling hyperbolic space.
Like ordinary Euclidean cubes, they have six identical faces meeting
at 8 corners, with each face a regular 4-sided polygon. But their angles
are more acute than Euclidean cubes could ever have: 5 of these
shapes fit around each edge.

Notice how quickly the "pipes" at cube edges shink with increasing distance --
the most distant, deepest red features, at distance about 5 units, are almost
invisibly thin.

(click for larger 4.7MB 1950x1635 .tiff image)

## Euclidean space

In contrast, here's a tiling of Euclidean space by cubes.
They redden with distance in the same way as above,
and the farthest of them are also around 5 units distant. But in Euclidean
space, apparent size shrinks linearly with distance.

(click for larger 6.9MB 1950x1635 .tiff image)

See also
Tamara Munzner's test images, which are very simple and clear.

Previous attempts live here (Nov 15, pipes with marker spheres)
and here (Sep/Oct, galaxy confetti and hyperbolic telephone poles).

Stuart Levy, slevy@ncsa.uiuc.edu.