The notion of symmetry furnishes another basis for the investi gation of properties of regularity of polyhedra. It is a funda mental concept in the definition of an important class of poly hedral solids known as crystals. A configuration is called sym metrical if it admits of a congruence transformation, in the broad sense, into itself, which is not the identity. In this scheme of classification, two configurations are in the same class if the group of congruence transformations which transform one of the configurations into itself is the same as the group of con gruence transformations which transform the other into itself.
A Crystal.—A crystal is a solid whose boundary is a polyhedron which has the following properties : In the first place its vertices, edges and faces are respectively on points, lines and planes of a rational space net. A rational space net is the set S of all points, lines and planes such that there are five points, P1, P2, and no four of which are in a plane, such as the vertices of a tetrahedron and a point interior thereto, with the property that if a denotes any element (point, line or plane) of S, then there exists a finite sequence of elements of S, P1, P2, P3, P4, P5, a,, a,, . a„, a, such that the first five elements are the five points mentioned above and any other element of the sequence is uniquely determined by being common to or containing preceding elements of the sequence. Secondly, the polyhedron is trans formed into itself by the transformations of a finite group of congruence transformations, in the broad sense, which transform a rational space net into itself. Such a group is known as a crystallo graphic group. There exist thirty-two crystallographic groups which thus determine the same number of classes of crystals.
by requiring an even number, not necessarily two, of sides inci dent with each vertex.) A regular generalized polygon is defined accordingly ; e.g., any regular generalized polygon, sometimes called a star polygon, with n vertices is obtained by joining in order every k th vertex of a regular simple polygon of n vertices, where k is any integer between I and n/2.
In an analogous way the notion of a generalized polyhedron is defined ; i.e., by relinquishing the condition that two elements have no points in common, and at the same time allowing a face to be determined by a generalized plane polygon. The generalized polygon is not the boundary of the face in the ordinary sense. For example, the sides of each pentagon which bounds a face of a regular dodecahedron when prolonged form a star pentagon, and all such star pentagons determine the faces of the great do decahedron. The latter generalized polyhedron is regular in a sense which is an obvious extension of the meaning of that term as applied to ordinary polyhedra. There are just four such regu lar generalized polyhedra and detailed data concerning them are given in an accompanying table. The other definitions involving the idea of regularity as applied to ordinary polyhedra evidently admit of like extension so as to apply to generalized polyhedra, and many interesting types of generalized polyhedra accordingly present themselves.
Explanation of Tables.—The symbols of the tables have the fol lowing meanings : f denotes the number of faces of the poly hedron, v the number of vertices, and e the number of edges; of the f faces, m2 are bounded by polygons of m sides, and of these faces are incident to each vertex. Similarly, in the case of the numbers n, and s, si, s2. a is the number of times that the projection of the generalized polyhedron from its centre on to a sphere about that centre covers the sphere; and are respectively the analogous numbers for a polygon which determines a face of the polyhedron, and for a polyhedral angle of the polyhedron ; p is the genus of the polyhedron (the genus of all polyhedra in the first table is, of course, zero). Each polyhedron of a bracketed pair is dual to the other; the tetra hedron is self-dual.