rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole polyhedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Dual pairs of polyhedra have their configuration matrices rotated 180 degrees from each other. Classification The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question—one that requires an explicit construction. Geometric proof The following geometric argument is very similar to the one given by Euclid in the Elements: Topological proof A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that V − E + F = 2, and the fact that pF = 2E = qV, where p stands for the number of edges of each face and q for the number of edges meeting at each vertex. Combining these equations one obtains the equation Simple algebraic manipulation then gives Since E is strictly positive we must have Using the fact that p and q must both be at least 3, one can easily see that there are only five possibilities for {p, q}: {3, 3}, {4, 3}, {3, 4}, {5, 3}, {3, 5}. Geometric properties Angles There are a number of angles associated with each Platonic solid. The dihedral angle is the interior angle between any two face planes. The dihedral angle, θ, of the solid {p,q} is given by the formula This is sometimes more conveniently expressed in terms of the tangent by The quantity h (called the Coxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively. The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2. The defect, δ, at any vertex of the Platonic solids {p,q} is By a theorem of Descartes, this is equal to 4 divided by the number of vertices (i.e. the total defect at all vertices is 4). The three-dimensional analog of a plane angle is a solid angle. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by This follows from the