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 solids (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 (Sloane's 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 (Sloane's A063722; in the order tetrahedron, octahedron = cube, dodecahedron = icosahedron).

solid	Schläfli symbol	Wythoff symbol	C&R symbol				group
cube		3 2 4		8	12	6
dodecahedron		3 2 5		20	30	12
icosahedron		5 2 3		12	30	20
octahedron		4 2 3		6	12	8
tetrahedron		3 2 3		4	6	4

The duals of Platonic solids are other Platonic solids and, in fact, the dual of the tetrahedron is another tetrahedron. Let be the inradius of the dual polyhedron (corresponding to the insphere, which touches the faces of the dual solid), be the midradius of both the polyhedron and its dual (corresponding to the midsphere, which touches the edges of both the polyhedron and its duals), the circumradius (corresponding to the circumsphere of the solid which touches the vertices of the solid) of the Platonic solid, and the edge length of the solid. Since the circumsphere and insphere are dual to each other, they obey the relationship

(1)

(Cundy and Rollett 1989, Table II following p. 144). In addition,

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

The following two tables give the analytic and numerical values of these distances for Platonic solids with unit side length.

solid
cube
dodecahedron
icosahedron
octahedron
tetrahedron

solid
cube	0.5	0.70711	0.86603
dodecahedron	1.11352	1.30902	1.40126
icosahedron	0.75576	0.80902	0.95106
octahedron	0.40825	0.5	0.70711
tetrahedron	0.20412	0.35355	0.61237

Finally, let be the area of a single face, be the volume of the solid, and the polyhedron edges be of unit length on a side. The following table summarizes these quantities for the Platonic solids.

solid
cube	1	1
dodecahedron
icosahedron
octahedron
tetrahedron

The following table gives the dihedral angles and angles subtended by an edge from the center for the Platonic solids (Cundy and Rollett 1989, Table II following p. 144).

solid	(rad)	()		()
cube		90.000		70.529
dodecahedron		116.565		41.810
icosahedron		138.190		63.435
octahedron		109.471		90.000
tetrahedron		70.529		109.471

The number of polyhedron edges meeting at a polyhedron vertex is . The Schläfli symbol can be used to specify a Platonic solid. For the solid whose faces are -gons (denoted ), with touching at each polyhedron vertex, the symbol is . Given and , the number of polyhedron vertices, polyhedron edges, and faces are given by

			(8)
			(9)
			(10)

The plots above show scaled duals of the Platonic solid embedded in a cumulated form of the original solid, where the scaling is chosen so that the dual vertices lie at the incenters of the original faces (Wenninger 1983, pp. 8-9).
Since the Platonic solids are convex, the convex hull of each Platonic solid is the solid itself. Minimal surfaces for Platonic solid frames are illustrated in Isenberg (1992, pp. 82-83).
Wolfram