Further Reading for Teachers

Regular Polyhedra

A regular polyhedron is a three-dimensional object built by putting together several faces according to these rules:

• All faces must be regular polygons of the same shape and size.

  Here are pictures of polygons you could use:

  Triangle	Square	Pentagon	Hexagon

• The same number of faces must meet at each vertex of your solid.

  For example, there are three triangles that meet at this vertex:

  

How many regular polyhedra can you build this way?

Only 5! Look at the construction of one vertex. Imagine having in your hand a regular polyhedron made out of cardboard faces taped together. Identify all the faces that meet at a given vertex. Untape and discard all the other faces. Remove the tape from one of the edges and flatten the whole thing out, as shown below:

Now, the idea is to figure out all the possibilities for "flattened-out" vertices. At least three faces are needed to build a 3-dimensional vertex.

Let's start with triangles. Here are all the possibilities:

They are the flattened-out vertices of the tetrahedron, the octahedron, and the icosahedron, respectively.

A similar analysis with squares yields the following possibilities:

Finally, pentagons may be used as faces to form vertices only in this way:

This is a flattened-out vertex of the dodecahedron.

Check that no other polygon will work.