# The Five Platonic Solids

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.
No other possibilities form a closed convex solid. For example, four squares or three hexagons at each corner would result in a flat surface, like floor tiles.

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.

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.

`                faces edges vertices`
`tetrahedron      ___   ___    ___`
`cube             ___   ___    ___`
`octahedron       ___   ___    ___`
`dodecahedron     ___   ___    ___`
`icosahedron      ___   ___    ___`