GEOMETRIC SOLIDS IN GENERAL The boundary of a solid may consist, of course, of several poly hedra ; e.g., a cube and one or more smaller cubes inside of the former. Some of these polyhedra may have points in common. The great diversity of such disconnected boundaries makes classi fying them difficult ; a rough classification is according to the num ber of pieces into which the boundary falls. As remarked earlier, the boundary of a solid may be a closed surface or consist of a number of such, or it may be a set of points of great irregularity. Solids with such irregular boundaries can be obtained by limiting processes in which certain constructions are repeated an infinite number of times.
The study of the general properties of solids of the class of solids whose boundaries are closed surfaces is made along lines suggested by the theory of polyhedral solids, a sub-class of the for mer; for a closed surface can be transformed continuously into a polyhedron. This property may be taken as the defining property of a closed surface. The concept of genus carries over directly, when simple continuous curves are used in place of the polygonal lines and polygons of the definition of that term given above.
Simple examples of closed surfaces are the sphere, ellipsoid, the surface of a cone, of a cylinder, and of an anchor ring, i.e., a torus. With regard only to the primitive properties of continuity and connectivity, the first four closed surfaces are not different ; the torus, however, is different from all of the other four. The genus of each of the first four is zero, that of the torus, one ; it is impossible to transform continuously the torus into any of the others, but any one of the first four can be continuously trans formed into any other. The similarity between the torus and the polyhedron illustrated above should be noted; the genus of each is unity and either is continuously transformable into the other. When projective or metric properties are regarded, distinctions between any two of the examples cited immediately appear. (See