simple Cartesian coordinates of the vertices are given below. The Greek letter φ is used to represent the golden ratio ≈ 1.6180. The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign (central symmetry), or, in the other cases, by exchanging two coordinates (reflection with respect to any of the three diagonal planes). These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or , one of two sets of 4 vertices in dual positions, as h{4,3} or . Both tetrahedral positions make the compound stellated octahedron. The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t{3,4} or , also called a snub octahedron, as s{3,4} or , and seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the compound of five cubes. Combinatorial properties A convex polyhedron is a Platonic solid if and only if all its faces are congruent convex regular polygons, none of its faces intersect except at their edges, and the same number of faces meet at each of its vertices. Each Platonic solid can therefore be denoted by a symbol {p, q} where p is the number of edges (or, equivalently, vertices) of each face, and q is the number of faces (or, equivalently, edges) that meet at each vertex. The symbol {p, q}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below. All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have: The other relationship between these values is given by Euler's formula: This can be proved in many ways. Together these three relationships completely determine V, E, and F: Swapping p and q interchanges F and V while leaving E unchanged. For a geometric interpretation of this property, see . As a configuration The elements of a polyhedron can be expressed in a configuration matrix. The