Regular Polyhedra

One short project I completed this month with my geometry students was a four-day look at regular (and irregular) polyhedra. First we built the five Platonic solids out of toothpicks and gumdrops, and learned their names: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Toothpick and Gumdrop Polyhedra

Then we counted the faces, edges, and vertices. This revealed the pattern of Euler’s Formula, that for all convex polyhedra, $V - E + F = 2.$ The next day we investigated nets that could be used to fold up into our five regular polyhedra, then used the nets, with scissors and tape, to create them. After a brief extension of Euler’s formula to scribbles in the plane (where a vertex is an endpoint or place where the scribble crosses itself, an edge is any line segment or curve connecting two vertices, and a face is an enclosed 2-dimensional section, or the un-enclosed remainder of the page), we worked together as a class to prove that only five regular polyhedra exist. This is really an astonishing fact: since infinitely many regular polygons exist in 2-D, why shouldn’t there be infinitely many 3-D regular polyhedra? The proof connected back to our knowledge of regular polygons and their angles, as well as our work with regular tilings of the plane in our tessellation project. For the last part of the project, I had students summarize our proof in their own words.

The proof was certainly the hardest part of the project. Some students zoned out as we discussed it and worked to construct our proof, although I tried to keep every student engaged and contributing. Do you think there’s a way where students could could come up with this proof on their own or in small groups, instead of in a whole-class setting? What would I need to do to guide them to that point?