A most useful general method of thus classifying polyhedra is based on the problem of the determination of all types of poly hedra of 1+1 faces or v+i vertices, assuming that all the types of polyhedra of f faces or v vertices, respectively, are known. For example, any type of convex polyhedron (for definition see below) of faces can be obtained by cutting with a plane all the faces of a certain convex polyhedron of f faces, which are incident with the same vertex. In the case of the special convex polyhedra which have only vertices which are incident with three and only three edges, this process of "cutting off the corners" results in a polyhedron of the same speciality and gives a comparatively com plete result. If (f) denotes the number of types of convex poly hedra which have f faces and all of whose vertices are incident with exactly three edges, then (4) = i, V (5) = i, V/ (6) = 2, (7) = V/ (8) = 14, (9) =5o, and (I0) = Very little, indeed, is known of the general behaviour of the function ,p (f). If the polyhedra also satisfy the special condition that each polyhedron contains a face which is adjacent to all of the remaining faces, then a recursion formula for the value of (f) is known, but even in this very special case it is sufficiently complicated. A similar degree of completeness exists in the theory of the convex polyhedra, i.e., the so-called duals of those just considered, i.e., those which have only triangular faces. A poly hedron is said to be the dual (see DUALITY) or reciprocal of an other if each vertex of the former is paired with a unique face of the second, so that thereby every face of the second is paired with just one vertex of the first, and if each edge of the first is paired with a unique edge of the second in a reciprocal way, and if a pair of incident elements of the second polyhedron corresponds to a pair of incident elements of the first.
Another general procedure in studying different types of poly hedra consists of making simple replacements of some of the ele ments of a polyhedron by new elements. For example, a face may be replaced by the two faces and the edge, which result from the division of the original face by a line which cuts its boundary in exactly two points, and different replacements are made according as the dividing line passes through no vertex, one vertex or two vertices incident with the face. For some such replacements the type of the polyhedron is not changed, while for others the type of the resulting polyhedron is different from but simply relatea to that of the original one. Many important theorems result from the consideration of such replacements of elements.
connected polyhedron of genus zero. Such a polyhedron will be said to be convex-like, because with regard only to incidence rela tions of its elements it does not differ from a convex polyhedron. The precise fact is given in the fundamental theorem that every convex-like polyhedron is of the same type as that of a certain convex polyhedron. As an important preliminary theorem here we have that every convex-like polyhedron is derivable from a tetrahedron by simple replacements of elements such as referred to above.
A convex polyhedron, already referred to several times, is de fined in several equivalent ways. One definition requires that no line which is not in the plane of any face of the polyhedron con tains more than two points of the elements of the polyhedron, and another is that, if a is any face of the polyhedron, then all of the elements of the polyhedron except a and the vertices and edges incident with a are on the same side of the plane which contains a. Again, a polyhedron which forms the boundary of a convex solid is a convex polyhedron, where by a convex solid is meant a solid which is such that, if two ends of a segment are points of the solid, then every point of the segment is a point of the solid. It is worth noting that these definitions require that a convex polyhedron be contained in a three-dimensional space, while the concept of a convex-like polyhedron is independent of any sur rounding space.
Some further results of the foregoing will now be noted. If v, e and f respectively denote the number of vertices, of edges and of faces of a polyhedron then, since at least three edges are incident with each vertex and also with each face, 3v< 2e and 2e; and if the polyhedron is of genus zero, then v—ed-f = 2. Conversely, it is proved that, if v, e and f are positive integers which satisfy these three conditions, the two inequalities and the equality, then there exists a polyhedron, in particular a convex polyhedron, with v vertices, e edges and f faces. Further, it follows easily from the latter equality that every polyhedron of genus zero and, therefore, in particular every convex polyhedron (for it is shown easily that the genus of a convex polyhedron is zero) has at least one face that is bounded by a polyhedron of either 3, 4 or 5 sides, and at least one vertex which is incident with 3, 4 or 5 edges, and has at least either one triangular face or one trilinear vertex, i.e., a vertex incident with exactly three edges. Also, there exist polyhedra and, in particular, convex polyhedra having any number of edges, if that number is an integer greater than five and different from seven. There is no polyhedron having exactly seven edges. Finally a fact that is remarkable (even in view of the greater richness of the theory of polyhedra in euclidean three-dimen sional space as compared to the theory of polygons in a plane) is that there exist polyhedral solids, evidently not convex, in euclidean three-dimensional space, which are not decomposable into a finite number of tetrahedral regions which have no points which are not on their boundaries in common, and whose vertices are vertices of the polyhedral solid. That an analogous decomposition of a polygonal region into triangular regions always exists is a theorem of euclidean plane geometry.