(7,3,2) triangle tiling (small)

A [larger version](http://www.thingiverse.com/thing:1608550) of this model is featured in figures 4.13 and 4.14 of [Visualizing Mathematics with 3D Printing](http://3dprintmath.com).

This 3D printed illustration of the hyperbolic plane is joint work with Saul Schleimer.
The hyperbolic plane has constant negative curvature - it is shaped like a saddle everywhere. The hyperbolic plane cannot be "seen" directly. This is similar to the problem faced by terrestrial mapmakers: any flat map of the earth necessarily distorts either size or distance.
This 3D print is a model, called the hemisphere model, of the hyperbolic plane. Lighting the piece, in three different ways, casts shadows of three further models: the Poincaré disk model (first image), the Klein model (second image) and the upper half space (third and fourth images) model.
Each triangle in the (7,3,2) tiling has angles pi/2, pi/3 and pi/7. This is very unlike triangles in Euclidean geometry, where the angles must add