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 ___ ___ ___