The development of natural faces on a crystal is far from being haphazard, but is governed by the condition that the angles between similar faces, whether on the same crystal or on different crystals, are equal, however varying may be the shapes and the relative sizes of the faces (Fig. 1), and by the tendency of the faces bounding the crystal to fall into series with parallel edges, such series being termed zones. Each species has a characteristic type of crystallization, which may be referred to one of the following six systems:— 1. _Cubic._—Crystals in this system can be referred to three edges, which are mutually at right angles, and in which the directional characters are identical in value. These principal edges are known as axes. Some typical forms are the cube (Fig. 2), characteristic of fluor; the octahedron (Fig. 3), characteristic of diamond and spinel; the dodecahedron (Fig. 4), characteristic of garnet; and the triakisoctahedron, or three-faced octahedron (Fig. 5). [Illustration: FIG. 2.—Cube.] [Illustration: FIG. 3.—Octahedron.] [Illustration: FIG. 4.—Dodecahedron.] All crystals belonging to this system are singly refractive. 2. _Tetragonal._—Such crystals can be referred to three axes, which are mutually at right angles, but in only two of them are the directional characters identical. A typical form is a four-sided prism, _mm_, of square section, terminated by four triangular faces, _p_ (Fig. 6), the usual shape of crystals of zircon and idocrase. [Illustration: FIG. 5.—Triakisoctahedron, or Three-faced Octahedron.] [Illustration: FIG. 6.—Tetragonal Crystal.] Crystals belonging to this system are doubly refractive and uniaxial, _i.e._ they have one direction of single refraction (cf. p. 45), which is parallel to the unequal edge of the three mentioned above. [Illustration: FIG. 7.—Two alternative sets of Axes in the Hexagonal System.] 3. _Hexagonal._—Such crystals can be referred alternatively either to a set of three axes, _X_, _Y_, _Z_ (Fig. 7), which lie in a plane perpendicular to a fourth, _H_, and are mutually inclined at angles of 60°, or to a set of three, _a_, _b_, _c_, which are not at right angles as in the cubic system, but in which the directional characters are identical. The fourth axis in the first arrangement is equally inclined to each in the second set of axes. Many important species crystallize in this system—corundum (sapphire, ruby), beryl (emerald, aquamarine), tourmaline, quartz, and phenakite. The crystals usually display a six-sided prism, terminated by a single face, _c_, perpendicular to the edge of the prism _m_ (Fig. 8), _e.g._ emerald, or by six or twelve inclined faces, _p_ (Fig. 9), _e.g._ quartz, crystals of which are so constant in form as to be the most familiar in the Mineral Kingdom. Tourmaline crystals (Fig. 10) are peculiar because of the fact that often one end is obviously to the eye flatter than the other. [Illustration: FIGS. 8-10.—Hexagonal Crystals.]