SOLIDS, GEOMETRIC. The term solid is not widely used in geometry as a precise generic term, but it commonly appears in the names of certain special classes of geometric configurations.
A geometric configuration is any set of points in space which, unless the contrary is stated, is understood to be the space of euclidean three-dimensional geometry. We may set down that any geometric configuration is a geometric solid, or simply a solid, if every point of the configuration is the centre of a sphere whose interior contains only points of the configuration, and if the con figuration is not composed of two configurations which have no points in common and which are such that every point of each configuration is the centre of a sphere whose interior contains only points of that configuration. It will be seen readily that according to this definition the interior of a sphere is a solid, while the con figuration consisting of that interior and the sphere, i.e., the spherical surface, is not a solid. It is easy to modify the given definition so as to permit calling the latter configuration a solid. This requires making precise the idea of the boundary of a configuration. A boundary point of a configuration is a point such that every sphere having it as centre contains points of the configuration and also points which do not belong to the configura tion. A boundary point of a configuration need not belong to the configuration. The set of all boundary points of a configuration is called the boundary of the configuration. We shall now define a solid as any configuration consisting of a solid according to the original definition and the boundary of the latter configuration, and shall refer to the latter configuration as the interior of the solid thus newly defined. There are simple solids such as those commonly referred to as the sphere, cube, pyramid, ellipsoid, cylinder and the less simple ones, such as the torus or anchor ring and a sphere with a number of holes bored through it, and so on to solids of great complexity; e.g., the abstract analogue of natural objects, such as a sponge or the steel framework of a building. The task of making a complete detailed classification of solids is thus an enormous one. For certain restricted, though important,
classes of solids the problem of classification has been solved to such an extent that the associated theory is an interesting and important chapter of mathematics. An example of such a class is the class of polyhedra, which is taken up in some detail below.
A useful procedure in attacking the general problem of classi fication is the study of the boundaries of solids. Although several solids may have the same boundary the ambiguity thus arising in the determination of a solid by its boundary is easily settled in the case of a great class of solids; viz., all those which have been studied in a systematic way. Thus a classification of the bound aries of solids determines a classification of solids. The bound aries of solids may be extremely complicated configurations, but in the case of the solids of the large class just referred to the boundaries have a speciality which is sufficient for a quite com plete classification and theory. The solids referred to are those whose boundaries are closed surfaces. A sphere, a torus, a poly hedron are examples of closed surfaces, and a more complicated example of such a surface is the surface of the solid which results from boring any number (finite or infinite) of holes, properly located, through a sphere. A precise definition of the concept of closed surface will not be given now and an ordinary intuitive counterpart of that concept will serve provisionally.
In studying the properties of solids, it is helpful to divide those properties into metric and non-metric properties. A general property of the latter kind involves merely the way in which parts of the solid are attached to each other, i.e., the connection of the solid ; while a metric property refers to size and shape, thus in volving a comparison with other solids in regard to such relations as congruence and similarity. A special kind of non-metric prop erty, sometimes called descriptive, involves such notions as linear ity, parallelism and convexity. The non-metrical properties are the more primitive and simple, but metrical relations (such as congruence) are also fundamental.