I can't explain non-Euclidean geometry using the inner product and coordinate transformations to my family.. to illustrate the non-Euclidean metric. So I decided to use hand held models such as a sphere, hyperbolic plane (pringle potato chip) or torus.

Google searches produced nothing. Does anyone know of a math store that sells such objects?

I admit things like these may be a rarity and only available as prizes for math students.

Preferably a sphere would have triangles drawn on it. Varying sizes, with areas calculated in a legend. These would show the difference from a plane tringle area. Similar for a hyperbolic plane. Perhaps one is constructed using crochet.

Or a model of a stereographic projection of a hemisphere.

Jig saw puzzles would also be useful. 3D models that one assembles. Curved wood pieces, for example.

For a torus there could be some illustration in the surface that indicates why it is flat on the top but curved on the sides. Some type of tangent vector or osculating circle that clearly has one principle curvature equal to zero.

Or a sphere with complete longitude and latitude lines, including polar regions. Ideally one could have, inscribed in the surface, an inner product of two vectors with the resulting output shown as a distance. I.e..: what one one would obtain from the inner product of the metric tensor g with the two vectors. this would be the curved analog of a right triangle and its hypotenuse. IN a legend box g could be shown in terms of the First Fundamental Form. The two vectors could have assigned values for components. so the calculation would be numeric as well as have a visual representation.

My final idea is a basic tiling such as the half disc model of the hyperbolic plane. I know this is 2D but it could be a good printing on something durable.

Gary Willick