- Known
to the ancient Greeks, there are only five
solids which can be constructed by choosing a regular convex
polygon and having the same number of them meet at each corner:
- The cube has three squares at each corner;
- the tetrahedron has three equilateral triangles at each corner;
- the dodecahedron has three regular pentagons at each corner.
- With four equilateral triangles, you get the octahedron, and
- with five equilateral triangles, the icosahedron.

It is convenient to identify the platonic solids with the notation {*p,
q*} where *p* is the number of sides in each face and *q*
is the number faces that meet at each vertex. Thus, the cube is {4, 3}
because it consists of squares meeting three to a vertex.

**Exercise: ***Give the* {*p, q*} *notation for all five
Platonic solids.*

**Answer: ***Always do these exercises yourself before looking
at my answer.*

Observe that if {p, q} is a possible solid, then so is {q, p}.

In nature, the cube, tetrahedron, and octahedron appear in crystals.
The dodecahedron and icosahedron appear in certain viruses and radiolaria.
Note that names such as *dodecahedron* are ambiguous; sometimes the
*regular* dodecahedron is meant and sometimes the word refers to any
of the many polyhedra with twelve sides.

**Exercise:** *Get to know these polyhedra and the relationships
between them by counting the number of faces, edges, and vertices found
in each of these five models. Make a table with the fifteen answers and
notice that only six different numbers appear in the fifteen slots.*

**Answer:** *Fill in this table before looking at my answer:*

faces edges vertices

tetrahedron ___ ___ ___

cube ___ ___ ___

octahedron ___ ___ ___

dodecahedron ___ ___ ___

icosahedron ___ ___ ___

P.S. (Platonic System) If you are using Netscape's Live3D VRML viewer, try this out.