Geometric Solids

Metrical Properties of Polyhedra.

The preceding exposi tion has been concerned largely with non-metrical properties. In succeeding paragraphs metrical properties of polyhedra are also considered. These properties being based on the intuitive notions of size and shape were the chief objects of study in the very early development of the subject. The chief topics here are the congruence of polyhedra and the properties of polyhedra which have elements or groups of elements which are congruent respec tively. Two configurations are said to be congruent in the narrow sense if, and only if, one can be transformed into the other by a rigid motion. Two configurations are said to be congruent in the broad sense if, and only if, one can be transformed into the other by a succession or product of a finite number of rigid mo tions and planar symmetries ; i.e., reflections in a plane. An equivalent set of definitions is the following : Two configurations are congruent in the broad sense if, and only if, the points of one are in one-one correspondence with the points of the other such that the distances between pairs of corresponding points are equal. If sense as well as distance is preserved by the correspond ence, then and only then are the two configurations congruent in the narrow sense. The terms of these definitions have precise abstract meanings but in order to save space they are omitted here ; their ordinary meanings should give a sufficient under standing.

An outstanding theorem on the congruence of polyhedra is that if two convex polyhedra are isomorphic such that corresponding faces are congruent, then the polyhedra are congruent in the broad sense. The theorem is obviously not true in the case of polyhedra which are not convex, and an interesting theory concerning the infinitesimal and finite deformations of non-convex polyhedra, which preserve length, presents itself.

Regular and Semi-regular Polyhedra.—The question of the con gruence of polyhedra suggests the consideration of a single poly hedron whose elements satisfy certain relations of congruence. At the extreme of speciality there are the so-called regular poly hedra. Because of their very conspicuous regularity they are among the oldest objects of study in the theory of polyhedra. A polyhedron is a regular polyhedron if, and only if, the bound ary of each face is a regular polygon which is congruent to the boundary of every other face and each polyhedral angle of the polyhedron is regular, i.e., having congruent face angles and dihedral angles resp., and congruent to every other polyhedral angle of the polyhedron. There are exactly five types of regular polyhedra; further details of these are given in the table below. A generalization of the regular polyhedra are the Archimedian polyhedra; all the faces of such a polyhedron are bounded by regular polygons, not all congruent to each other, and all the polyhedral angles are convex and congruent to each other. There are thirteen types of such polyhedra besides the semi-regular prisms and prismoids.

A prism is a polyhedron of which two faces, called its bases, lie in parallel planes and each of the remaining faces is bounded by a parallelogram and is adjacent to each base. A prismoid is a

polyhedron which also has two faces, its bases, which are in parallel planes, while each of its other faces is triangular and incident with a vertex of one base and a side of the other. If every face of a prism or prismoid is regular, i.e., bounded by a regular polygon, and if the bases of the prismoid are congruent, then the prism or prismoid are said to be semi-regular. The Archimedian polyhedra and their reciprocals form the class of the so-called semi-regular polyhedra. The semi-regular polyhedra can be obtained by such simple operations as taking plane sections of and combining regular polyhedra, and by dualizing. For example, the cuboctahedron is obtained by cutting off the vertices of the cube or octahedron by planes passing through the mid points of the edges ; the icosidodecahedron by a similar treat ment of the dodecahedron or icosahedron. The reciprocals of these two Archimedian polyhedra are respectively the rhombic dodecahedron and the rhombic triacontahedron, which have in all twelve and thirty faces respectively, each face being a rhombus. For further details of semi-regular polyhedra, see table p. 945.

The idea of semi-regular polyhedra leads to the later generaliza tion of the notion of regularity, which is contained in the idea of a polyhedron transformable into itself by a congruence transfor mation, in the broad sense, such that any polyhedral angle of the polyhedron is transformed thereby into any other. Re ciprocally there is defined a polyhedron which is transformable into itself by a congruence transformation so that any face of the polyhedron is transformed into any other. Polyhedra of the former kind have a circumscribing sphere while those of the latter have an inscribed sphere. The mere requirement of the congruence of all of the polyhedral angles or of all of the faces of a polyhedron does not imply the existence of a circumscribing or an inscribed sphere respectively. Archimedian polyhedra are polyhedra of the former kind which have their faces bounded by regular polygons which are not all congruent to each other. A complete classification of the convex polyhedra satisfying this more general definition of regularity has been made, according to the kind of faces and the various relations of symmetry which are satisfied, and the number of distinctive types is finite. When the requirement of convexity is removed the complexity of the problem of classification is very great, and no classification of any satisfying degree of completeness has been made. In con formity with the present viewpoint a regular polyhedron may be defined as a polyhedron which is transformable into itself by a congruence transformation, in the broad sense, so that any vertex is transformed into any other vertex, any edge incident with the former vertex into any edge incident with the latter vertex, and any face incident with the former edge into any face incident with the latter edge.