One of the most remarkable properties of a crystal is its symmetry, by which is meant the greater or less degree of regularity in the arrangement of its faces, edges, and vertices. A given substance may, according to circumstances, crystallize in a variety of forms or combinations of forms, but, with very few exceptions, all crystals of a given substance exhibit the same kind or grade of symmetry. There are three kinds of crystal symmetry, namely, in respect to a plane, a line or axis, and a point or center. A plane of symmetry divides a crystal into halves in such a way that for every point on one side of the plane there is a corresponding point directly opposite on the other side. Crystals may be cut into halves along various surfaces which are not symmetry planes. An axis of symmetry is a line about which a complete rotation (or in a few cases rotation combined with reflection) brings the crystal into the same relative position two, three, four or six times, these being called two, three, four, and sixfold axes of symmetry--no others being possible. A crystal has a center of symmetry when any line passing through it encounters corresponding points at equal distances from it on opposite sides. There are just 32 classes or combinations of the symmetry elements among crystals and just 232 definite crystal forms. Not only is it demonstrable that no more can exist, but actual experience with crystals of hundreds of species of minerals has never revealed any more. Obviously, then, symmetry furnishes us with a very scientific basis of classification of crystals, all of the 232 crystal forms constituting the 32 symmetry classes being in turn referable to seven fundamental crystal systems. To bring out the relations of the faces of a crystal and further aid in classification, prominent, straight lines or directions passing through the center of a crystal are chosen as crystallographic axes. Such axes may or may not coincide with symmetry axes. Basing our definitions upon both symmetry axes and crystallographic axes, the seven systems are as follows: 1. Isometric. There must be at least four threefold axes of symmetry, while the highest grade symmetry class of the five in the system includes three fourfold, four threefold, and six twofold axes of symmetry; nine planes of symmetry; and a center of symmetry. There are three interchangeable crystallographic axes at right angles to each other. [Illustration: Fig. 71.--Figures showing, a, crystal axes of Isometric system; b, points of emergence of the nine axes of symmetry in a cube of the Isometric system; c, nine planes of symmetry in a cubic crystal. (After Whitlock, New York State Museum.)] 2. Tetragonal. There must be one and only one fourfold symmetry axis, while the highest of its seven symmetry classes contains also four twofold axes of symmetry; five planes; and a center. Characterized by three crystallographic axes at right angles to each other, only two of them interchangeable.