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 (for example, my cellphone light), the rays of light effectively do stereographic projection, and the curves on the sphere cast shadows, mapping them to a straight line grid on the plane. Shapeways shop. Video.
Wow ! Very clever indeed. Nice to see how well it works ! I think there are many other possibilites, maybe with mirrors..
Further adventures in stereographic projection! This time using a Mini Maglite LED torch/flashlight for the light source. Shapeways shop section. Video.