Platonic Solid

The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids (Steinhaus 1999, pp. 252-256): the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid in the last proposition of the Elements. The Platonic solids are sometimes also called "cosmic figures" (Cromwell 1997), although this term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot polyhedra (Coxeter 1973).

The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC. In this work, Plato equated the tetrahedron with the "element" fire, the cube with earth, the icosahedron with water, the octahedron with air, and the dodecahedron with the stuff of which the constellations and heavens were made (Cromwell 1997). Predating Plato, the neolithic people of Scotland developed the five solids a thousand years earlier. The stone models are kept in the Ashmolean Museum in Oxford (Atiyah and Sutcliffe 2003).

Schläfli (1852) proved that there are exactly six regular bodies with Platonic properties (i.e., regular polytopes) in four dimensions, three in five dimensions, and three in all higher dimensions. However, his work (which contained no illustrations) remained practically unknown until it was partially published in English by Cayley (Schläfli 1858, 1860). Other mathematicians such as Stringham subsequently discovered similar results independently in 1880 and Schläfli's work was published posthumously in its entirety in 1901.

If is a polyhedron with congruent (convex) regular polygonal faces, then Cromwell (1997, pp. 77-78) shows that the following statements are equivalent.

1. The vertices of all lie on a sphere.

2. All the dihedral angles are equal.

3. All the vertex figures are regular polygons.

4. All the solid angles are equivalent.

5. All the vertices are surrounded by the same number of faces.

Let (sometimes denoted ) be the number of polyhedron vertices, (or ) the number of graph edges, and (or ) the number of faces. The following table gives the Schläfli symbol, Wythoff symbol, and C&R symbol, the number of vertices , edges , and faces , and the point groups for the Platonic solids (Wenninger 1989). The ordered number of faces for the Platonic solids are 4, 6, 8, 12, 20 (OEIS A053016; in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron), which is also the ordered number of vertices (in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron). The ordered number of edges are 6, 12, 12, 30, 30 (OEIS A063722; in the order tetrahedron, octahedron = cube, dodecahedron = icosahedron).