I strongly believe in the use of physical models, whenever possible, to introduce mathematical concepts. For example, when teaching non-Euclidean geometry to high school teachers, I like to have them create triangles on actual physical spheres, using rubber balls, push pins, and rubber bands to create geodesics. It is easy to “discover” that the sum of the angles in spherical triangles is greater than 180 degrees, and that the excess of a triangle (the sum of the angles minus 180 degrees) is additive and hence proportional to the area of the triangle. These demonstrations can be easily done for specialized triangles, so the student becomes familiar with the geometric fact before thinking about how it might be proven.
What I would like to do next is to show that the sum of the angles in a triangle on a surface of negative curvature is less than 180 degrees. This leaves me with two problems.
(1) How can I make (or obtain) a physical model of a simple saddle surface for experimentation by students. Ideally, models should be cheap enough so that I can supply each pair of students with a model to work with.
(2) How can students draw geodesics on such a surface? Rubber bands are not going to work here, because a band stretched between two points on the surface will not necessarily lie on the surface. This problem is sort of mathematical, because I think a good understanding of the nature of geodesics should lead to discovering a way of having students create them on a surface of negative curvature.
I am aware of some very good software that uses the Poincare disk model to do geometry on the hyperbolic plane, and I plan to use the software when I teach. But I want students to have real physical experience first.
Anyone have any ideas?