Polyhedron	Vertices	Edges	Faces
	Tetrahedron	4	6	4
	Cube	8	12	6
	Octahedron	6	12	8
	Icosahedron	12	30	20
	Dodecahedron	20	30	12

There are other ways to think about the polyhedra, though. A particularly useful one for our purposes has to do with the idea of valence. Let's define valence as follows:

So, for example, in a cube, all the vertices have valence = 3 (since three edges meet at each vertex) and all the faces have valence = 4 (since all the faces are squares, or are bounded by four vertices.) Using these definitions, the five regular polyhedra are:

	Polyhedron	r = Vertex Valence	n = Face Valence
	Tetrahedron	3	3
	Cube	3	4
	Octahedron	4	3
	Icosahedron	5	3
	Dodecahedron	3	5

The diagrams in the left column are called Schlegel diagrams. You'll notice that in each diagram, every vertex has the same number of edges coming out of it and every face has the same number of vertices that bound it.

Look at the Schlegel diagram for the cube again. If you count the number of vertices, you get eight, just like you'd expect. Similarly, if you count the number of edges, you get 12, again, as you'd expect. But if you count the number of faces, you get five. Where's the sixth face? It turns out that in a Schlegel diagram, you count what's not in the diagram as a face as well. That should make sense in a way. Isn't the rest of the plane on the outside of the Schlegel diagram also a region bounded by four edges?

Schlegel diagrams are going to be helpful when we make the jump to the fourth dimension. Looking at a Schlegel diagram of a three-dimensional solid is like looking at a two-dimensional shadow the solid casts on a plane.

If you'd like to make an octahedron, you'd place four equilateral triangles at each vertex, then fold them up. You know you can place four equilateral triangles with room left over to fold up as four 60 degree angles makes 240 degrees. There's still 120 degrees left over to allow for folding.

Placing five equilateral triangles at a vertex (which uses up 5 * 60 = 300 of the 360 degrees at the vertex) folds into an icosahedron. You can place six equilateral triangles at a vertex, but that uses up all 360 degrees, and there is no room left over to fold. Six equilateral triangles at each vertex creates an infinite planar grid of triangles. Clearly, you can't place seven equilateral triangles at a vertex.

You can put three squares at a vertex with room to fold. When you do, you get a cube. You can't put four at a vertex and still have room to fold. Four squares at each vertex creates an infinite planar grid of squares.

Each interior angle in a regular pentagon is 108 degrees. Placing three at a vertex uses up 324 degrees. There's still room left to fold, and when you do, you get a dodecahedron. Clearly, you can't put more than three at a vertex. You can put three regular hexagons at a vertex, but that takes up all 360 degrees. Again, you get an infinite hexagonal grid. Just as clearly, you can't fit 3 heptagons, or octagons, or any other regular n-gon at a vertex, as the interior angles are larger than 120 degrees.

The following table should make clear that this new way of thinking about polyhedra is just a restatement of the valence way of thinking about them. I've included both the folding instructions, vertex valence (r) and face valence (n) in the table:

If you'd like to get a view of the regular polyhedra in action, click to run a polyhedron applet I wrote which displays the five regular polyhedra. The applet allows you the following options:

Thinking about making regular polyhedra (three-dimensional shapes) by taking regular polygons (two-dimensional shapes) fitting them around a vertex, and folding them up in the third dimension is going to be helpful to us in considering how to construct four-dimensional objects.

In the next lesson, I'll introduce you to rings of polyhedra called kaleidocycles. You have to understand what kaleidocycles are like and how they behave in three dimensions before we can talk about what they're like in four.