Fig. 18.—Measured Crystal of Anatase.
But there is one system of symmetry, the highest possible, and which has already been referred to as the cubic system, which combines in itself all but one (the hexagonal axis) of the elements of symmetry. Indeed, not only does it possess a tetragonal, a trigonal, and a digonal axis of symmetry, but also ten other symmetry axes; for these three automatically involve altogether the presence of no less than three tetragonal, four trigonal, and six digonal axes of symmetry, together with nine planes of symmetry, twenty-two elements of symmetry being thus present in all.
The perfections of the cube, the simple lines of which are illustrated in Fig. 20, as the expression of the highest kind of symmetry, with angles all right angles and sides and edges all equal, were so fully appreciated by the geometrical minds of the ancient Greek philosophers, imbued with the innate love of symmetry characteristic of their nation, that to them the cube became the emblem of perfection. We are reminded of this interesting fact in the Book of Revelation, which, in describing in its inimitable language the wonders of the Holy City, speaks of it as “lying foursquare,” and attributes to it the properties of the cube, that “The length and the breadth and the height of it are equal.”
Fig. 19.—Crystal of Calcite.
Fig. 20.—The Cube.
Fig. 21.—The Hexakis Octahedron.
The full symmetry of the cubic system is not realised, however, by a study of the cube alone; we only appreciate it when we come to examine the general form of the cubic system, that which is produced by starting with a face oblique to all three axes, and with different amounts of obliquity to each, and seeing how many repetitions of the face the symmetry demands. The presence of such a face involves as a matter of fact, when all the elements of symmetry are satisfied, the presence also of no less than forty-seven others, symmetrically situated, the forty-eight-sided figure produced being the hexakis octahedron shown in Fig. 21, and which is occasionally actually found developed in nature as the diamond. All diamonds do not by any means exhibit this form so wonderfully rich in faces, but diamonds are from time to time found which do show all the forty-eight faces well developed.