Platonic solid

A Platonic solid is a convex regular polyhedron all the faces of which share the same regular polygon and having the same number of faces meeting at all its vertices. Compare with the Kepler solids, which are not convex, and the Archimedean and Johnson solids, which while made of regular polygons are not themselves regular.

There are five Platonic solids, all known to the ancient Greeks:

                                                            faces meeting 
 name              face polygon   faces   edges  vertices   at each vertex   symmetry group
 tetrahedron        triangle       4        6       4             3                Td
 cube (hexahedron)  square         6       12       8             3                Oh
 octahedron         triangle       8       12       6             4                Oh    
 dodecahedron       pentagon       12      30       20            3                Ih
 icosahedron        triangle       20      30       12            5                Ih

That there are only five such three-dimensional solids is easily demonstrated. To have vertices, there must be three of the faces meeting at a point, and the total of their angles must be less than 360 degrees; i.e the corners of the face must be less than 120 degrees: this rules out all the regular polygons except triangles, squares, and pentagons.

• triangular faces: each vertex of a triangle is 60 degrees, so a shape should be possible with 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
• square faces: each vertex of a square is 90 degrees, so there is only one arrangement possible with three faces at a vertex, the cube.
• pentagonal faces: each vertex is 108 degrees; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron, and that exhausts the list of regular 3-dimensional solids.

Note that if you connect the centers of the faces of a tetrahedron, you get another tetrahedron. If you connect the centers of the faces of an octahedron, you get a cube, and vice versa. If you connect the centers of the faces of a dodecahedron, you get an icosahedron, and vice versa. These pairs are said to be dual polyhedra.

Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, (based on ideas regarding the music of the spheres etc.) and identified the five platonic solids with the five planets - Mercury, Venus, Mars, Jupiter, Saturn and the five classical elements. (The Earth, moon and sun were not considered to be planets.)

Uses

The shapes are often used to make dice. 6-sided dice are very common, but the other numbers are commonly used in role-playing games.

The tetrahedron, cube, and octahedron, are found naturally in crystal structures.

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