Geometric Solids

It consists of 16 vertices, 32 edges and 16 faces. In the case of a cube any polygon P consisting of vertices and edges of the cube separates the cube; i.e., there is a pair of points (and, of course, many pairs) of the cube, which do not belong to any polygonal line which is made up of points of the cube and which contains no points of the poly ' gon P. (It should be emphasized that the terms cube, tetrahedron, octahedron, etc. are used here to denote polyhedra and not solids bounded by certain polyhedra.) In the case of the polyhedron of the illustration the polygon ABCDA does not separate that polyhedron in the above sense ; however, any pair of polygons contained in the polyhedron, i.e., every vertex and every point of the sides of the polygon, are points which are either vertices or belong to edges and faces of the poly hedron, and having no points in common separates the polyhedron in the sense that a set S of polygons separates a polyhedron 7r if the polygons of S are contained in r and if there are two points of r such that any polygonal line which contains them and which is contained in it also contains a point of a polygon of S. For any polyhedron there exists a finite set T of polygons which have no points in common and such that the set T does not separate the polyhedron, while any set of non-intersecting polygons on the poly hedron, which contains more polygons than T, separates the poly hedron. The number of polygons in the set T is called the genus of the polyhedron. Thus it can be easily shown that the cube, tetrahedron, and the more common polyhedra are of genus zero, while the genus of the polyhedron of the above illustration is one. Speaking roughly, a polyhedron of genus p is not disconnected when it is cut along the circuits (polygons) of a certain set of p circuits, but it is disconnected when cut along any p+I circuits.

A most important theorem may now be stated. If v= the number of vertices, e= the number of edges, and f= the number of faces of a polyhedron of genus p, then v—e-l-f= 2 -2p. Thus in the case of the polyhedra of genus zero (such as the cube, octahedron, and icosahedron) v—e+f=2, while for the poly hedron illustrated above v—e+f= o. The number v—e+f is called the characteristic of the polyhedron. A remarkable fact in this connection is that, if two polyhedra have the same character istic (or genus), then a subdivision of the elements of each exists so that the polyhedra thus derived are isomorphic ; i.e., any vertex, edge and face of one is paired respectively with a vertex, edge and face of the other such that a pair of elements of the first polyhedron which are incident corresponds to a pair of elements of the second polyhedron, which also are incident, and conversely. Thus, if a vertex A' and an edge a' of the second polyhedron are paired with the vertex A and the edge a of the first polyhedron respectively, and if A is an end of a, then A' is an end of a'; and similarly if a vertex and a face or an edge and a face are incident.

The integer 2 minus the characteristic of a polyhedron is known as the connectivity number of the polyhedron and, as implied above, for a polyhedron in euclidean three-dimensional space it is either positive and even, or zero.

Classes of Polyhedra.

In the theory of polyhedra, considered merely as sets of objects, called vertices, edges and faces, which satisfy the relations of incidence and order of the definition of a polyhedron given above and which are not sets of points in any particular space, it is shown that there exists a polyhedron having any positive integer or zero as its connectivity number. In this theory two classes of polyhedra are distinguished. Any polyhedron of one of these classes has the property that there is at least one polygon on it which does not separate any region which is on the polyhedron and which contains the polygon. The term region is used here in a sense which, though analogous to, is a simple exten sion of, its meaning given above. All other polyhedra make up the other class. Polyhedra of the latter class are sometimes said to be two-sided; those of the former one-sided. The basis of this ter minology is suggested above in reference to the separation of euclidean space by polyhedra. Only two-sided polyhedra exist in euclidean space and any two-sided polyhedron is isomorphic with a polyhedron in euclidean space. A generalized polyhedron (see below) in euclidean space may be one-sided.

The terms non-orientable and orientable are also applied to polyhedra of the former and latter classes respectively, because either of two senses can be defined on an orientable polyhedron just as in the case of the euclidean plane or sphere, while that is impossible in the case of so-called non-orientable polyhedra. In the general theory of polyhedra, also called manifolds (q.v.) in that context, the above equation relating v, e, f and p holds for all orientable polyhedra, while the equation v —e-Ef =2—p is the corresponding equation for non-orientable polyhedra, the letters in both cases having analogous meanings. In the following, as in what preceded this digression, it is understood that the polyhedra are in euclidean three-dimensional space, and are therefore two sided or orientable.

Classification of Polyhedra.

It is suggested immediately that polyhedra may be classified according to genus but then the question of a finer classification arises. The goal is the complete description of the classes so that any two polyhedra of the same class and no two of different classes are isomorphic. Each of the classes of such a classification shall be called a type. It is obvious that the polyhedra of the same class have the same genus, but that the converse is not true. It is also obvious that polyhedra which are not isomorphic may form the boundary of the same solid. No such exhaustive classification has ever been completed, not even for the class of all polyhedra of genus zero.