Q:

How can I demonstrate triangulations of surfaces with real hands-on objects?

For a math circle activity on Euler Characteristic, I'd like the students to be able to triangulate some surfaces and count the number of vertices, edges, and faces of the result.
There are a lot of ways to get a nice big physical sphere that the students can draw on... but what about other surfaces? Have any of you found a nice, preferably cheap way to do this?
To be clear, I am demanding that the end result be a physical object here, and am simply not interested in software that will do this for us. Thanks!

A:

I have used polydrons with 5th-grade students
through to college students:

 
 
 
 
 

A store in Massachusetts used to sell them,
but recently I've had to purchase
them from England.

A:

Sphere: No problem.
Torus: Swim Ring.
Double Torus: Figure Eight (rock climbing), but could be to small for your purposes.
$n$-Torus: Craft them yourself fronm Polystyrene and cover them with paper to draw upon.
Non-orientable, bounded surfaces: Hmm, not possible, if you really want to draw triangles upon them.

A:

Instead of finding a pre-built object and drawing triangles on it, we've settled on just building the objects up from scratch using Zometools, which is a more engaging plan anyway.
There is a small problem where when you build a torus out of these toys, it's hard to tell which "faces" are faces and which ones are not, so the solution is to stick your arm through the "hole" in the torus to make sure you don't count the faces that are not really there.

