I recently ran across, via this post on Division by Zero, a way to make hyperbolic space from paper, a project that I couldn't resist. In fact, it claims to be a hyperbolic soccer ball. Digging through Reader to find the link again, I found this other recent post about shapes relating to soccer balls, so thought I'd share it as well.
Anyway, the idea is that to typically make a soccer ball, you place a ring of hexagons around a pentagon, and iterate. The pentagons introduce some positive curvature in the process (hexagons alone have 0 curvature - they tile the plane), and you end up with something fairly spherical. If you place hexagons around a central heptagon though, you get negative curvature.
The directions to actually make one yourself are at The Institute for Figuring, and are available here (pdf). Following the basic instructions, I ended up with the following:
Which my cats only took fleeting interest in:
The directions suggested that you could continue adding more rings of hexagons, with heptagons in appropriate locations, and extend the model. I was curious to see what would happen when I did, so I printed out six more of the base sheets (3 of which gave the starting model, above). It's a bit of a monster: