Zometools

The theme of PCMI 2014 was the mathematics of materials. Normally, it's a more "mathy" topic, like algebraic geometry, or Lie groups. But this year there were a lot of physicists, chemists, and engineers in attendance, which allowed me to understand more than I would have!

One of the subjects was the geometric structure of solids and crystals, and quasicrystals. People were studying how different shapes fit together (in both 2D and 3D). There was a lot of talk about the 5 Platonic solids, the cube, tetrahedron, octahedron, dodecahedron, and icosahedron. To help the students (and me!) visualize these structures, we had Zometools on hand, which are basically very fancy tinker-toys. I was able to bring some home, and here are a dodecahedron and icosahedron that I built using Zometools.

Notice that the dodecahedron has 12 (regular pentagon) faces, 30 edges, and 20 vertices, and 3 edges meet at each vertex. And the icosahedron has 20 (regular triangle) faces, 30 edges, and 12 vertices, and 5 edges meet at each vertex. These are called "dual" polyhedra. This is because if you take one of them and place a vertex at the center of each face and then connect those vertices with edges, you will obtain the other polyhedron!

This can be demonstrated mathematically by looking at something called Euler's formula (or the Euler characteristic):

F + V - E = 2

where F is the number of faces, V is the number of vertices, and E is the number of edges. This equality holds true for all polyhedra, not just regular polyhedra. (Well, there is one important restriction: the polyhedron must be "homeomorphic to the sphere," which basically means that it is concave, and has no parts sticking out.) For our two polyhedra above, they each have the same number of edges E, but switch F and V. This is what "dual" means, switching the roles of the faces and the vertices.

This simple formula is easy to prove. It is so easy to prove that there are (at least) 20 different ways to prove it! A very simple proof is on the Wikipedia page "Euler characteristic".

What about those Zometools? They consist of struts and nodes, and they are all shape-identified. The nodes are "universal" in that they have many different holes in which to place the strut ends, but they are all of different shapes (triangles, rectangles, pentagons) so that only one type of strut will fit in any given hole. This allows you (or forces you) to be able to make certain shapes quite easily. The blue struts have rectangular ends, and the red struts have pentagonal ends. I don't have any of the struts with triangular ends. There are also struts of different lengths.

Zometools are extremely fun to play with, and can be very addicting! And they are useful for researchers. In fact, they were used to develop some of the theory behind quasicrystals, for which Dan Shechtman won a Nobel Prize in Chemistry in 2011.