Geometric Solids

Polyhedral Solids.—In the study of solids aid is obtained from the study of a special class, polyhedral solids, i.e., solids bounded by configurations formed by polyhedra. The theory of polyhedra suggests means of classifying more general solids and also methods of attack in the solution of problems of more general solids. Furthermore, the fact that any solid can be approximated to as closely as desired by a polyhedral solid adds to the impor tance of the theory of such solids to a study of solids in general. A precise formulation of the fact just alluded to is the following : If S denotes any solid, then there exists a sequence of polyhedral solids, P1, P2, P3, • • Pn . . ., such that every point of the solid P, is a point of the interior of S and also of the interior of the solid and every point of the interior of S is a point of the interior a polyhedral solid P. of the sequence. In particular, there exist such sequences of polyhedral solids such that each polyhedral solid is composed of a finite number of cubes such that no two of the cubes have interior points in common and every cube has a face in common with another one of the cubes. An other method of approximating to a solid by means of polyhedral solids is that in which the vertices of the approximating polyhedral solids are points of the boundary of the given solid, or, as it is usually stated, the polyhedron which bounds the polyhedral solid is inscribed in the boundary of the given solid. However, the existence of sequences of approximating polyhedral solids of the latter kind has been established only in the case of special solids.

Definitions.

According to the definition of polyhedral solids, it is immediately evident that the theory of polyhedra is of much importance in the study of those solids. Preparatory to the consideration of the properties of polyhedra the following definitions are made. A segment is the set of all points of a (straight) line which are between two points of that line, and each of the latter points is called an end of the segment. A simple polygon is a finite set of points (vertices of the polygon) and seg ments (sides of the polygon) such that (a) each point of the set is the end of two and only two segments of the set, (b) the ends of each segment of the set are points of the set, (c) no two ele ments (i.e., points or segments) of the set have a point in common, and finally (d) no proper subset of the set of points and segments satisfies (a), (b) and (c). It is a theorem of euclidean plane geometry that every simple polygon in a plane r is the planar boundary of a unique connected region of finite extent which is contained in r. The planar boundary of a set of points in a plane and a connected region in a plane are the two-dimensional analogues of the boundary of a solid and the interior of a solid respectively as defined above, circles of the plane replacing the spheres in the definitions referred to. Saying that a set of points

is of finite extent means that it is contained in the interior of a sphere. The planar region thus determined by a simple plane polygon is called a polygonal region.

A finite set of points, segments and polygonal regions is called a polyhedron, if and only if (a) every segment of the set is to side of the boundaries of two and only two polygonal regions of the set, and every side of the boundary of a polygonal region of the set is a segment of the set, (b) every point of the set is an end of at least one segment of the set, and the ends of any seg ment of the set are points of the set, (c) if P is any point of the set, the set o of all polygonal regions of the set whose boundaries have P as a vertex form a cycle, i.e., every segment of the set which has P as an end is a side of the boundaries of two polygonal regions of the set a and no proper subset of a has that property, (d) no two elements (i.e., points, segments or polygonal regions) of the set have a point in common, and (e) there is no proper subset of the set of points, segments, and polygonal regions which satisfies (a), (b), (c) and (d). By omitting some of the condi tions of this definition, more general configurations which are also called polyhedra are defined thereby, and the set just defined is called a simple or an ordinary polyhedron. Just so there are polygons which are not simple and which are defined by less re strictive definitions than that given above. In what follows, the words polygon and polyhedron signify respectively, simple polygon and simple polyhedron. Despite the qualifying term simple, the class of simple polyhedra has a most interesting and important theory which still leaves much to be done.

Non-metrical Properties.—That every polyhedron is the boundary of a unique solid of finite extent and also the boundary of a unique solid not of finite extent is a theorem of euclidean three-dimensional geometry which requires a proof of considerable length. This theorem is analogous to the theorem of euclidean two-dimensional geometry concerning the separation of a plane by a polygon contained in it ; but at this point the spatial theory becomes much richer. Considering merely the internal connection of the elements of a polygon, one is very much like the other, but in the case of polyhedra that is not so. Besides polyhedra such as the tetrahedron, cube octahedron, etc., there are those which can be obtained by cutting one or more square holes out of the limited solid bounded by a cube, tetrahedron, etc. Between a polyhedron thus obtained and a cube or tetrahedron there is a fundamental difference. The diagram illustrates one of these _ polyhedra of more complicated structure.