[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.
3. Trigonal. Characterized by one and only one threefold symmetry axis, the highest of the five classes having also three twofold axes; four planes; and a center. Crystallographic axes as for hexagonal.
4. Hexagonal. One and only one sixfold axis of symmetry must be present, but the highest of the seven classes also has six twofold axes; seven planes; and a center. Characterized by four crystallographic axes, one vertical and three interchangeable horizontal axes making angles of 60 degrees with each other.
5. Orthorhombic. There must be no axis of symmetry higher than a twofold and three prominent directions (i.e., parallel to important faces) at right angles to each other, the highest grade of the three classes having three twofold axes; three planes; and a center. There are three noninterchangeable crystallographic axes at right angles.
6. Monoclinic. There is no axis of symmetry higher than a twofold and only two prominent directions at right angles to each other, the highest of the three classes having one twofold axis; one plane; and a center. There are three noninterchangeable crystallographic axes, only two of which are at right angles.
7. Triclinic. There is no axis of symmetry of any kind, and there are no prominent directions at right angles. One of the two classes has a center of symmetry only, and the other no symmetry at all. Characterized by three noninterchangeable crystallographic axes, none at right angles.
A fact which should be strongly emphasized is that crystals only, of all the objects of nature, can be definitely referred to the above seven systems comprising the 32 classes of symmetry, and 232 crystal forms. Since there are about 1,000 mineral species and only 232 fundamental forms, it necessarily follows that two or more species may crystallize in the same form within a class, so that it is not always possible to tell the species of mineral merely by its crystal form. It is, however, a remarkable fact that, where two or more substances crystallize in the same class (i.e., show the same grade of symmetry) each substance almost invariably exhibits "crystal habit" which is a pronounced tendency to crystallize in certain relatively few forms or combinations of forms out of the many possibilities. It is clear, then, that grade of symmetry combined with "habit" are of great practical value in determining crystallized minerals, because, on the basis of symmetry, a crystal is referred to a certain definite symmetry class in which only a limited number of substances crystallize, and then, by its characteristic "habit," the particular substance can be told.