Stereographic projection

This model is featured in figure 3.12 of Visualizing Mathematics with 3D Printing.

Video.
Stereographic projection is a map from the sphere to the plane: Imagine straight lines going from the north pole of the sphere down onto the plane. Each line hits the sphere in one point and the plane in one point. Stereographic projection maps that point on the sphere to the point on the plane.
If you put a light source at the north pole of this sculpture, the rays of light effectively do stereographic projection! The curves on the sphere cast shadows, mapping them to a straight line grid on the plane.
Stereographic projection is also available at Shapeways.

Update: added a version of the file with a wider base for easier printing.