Symmetry Plane, Symmetry Axis
and Classes of Crystals

Crystals are grouped together into different classes according to the symmetry which they show. The symmetry of crystals is of three kinds, namely: 1. Symmetry in respect to a plane; 2. Symmetry in respect to a line; 3. Symmetry in respect to a point.

Symmetry Plane
A symmetry plane is an imaginary plane which divides a crystal into halves, each of which is the mirror image of the other. Figure 5 will illustrate the character of such a plane. The shaded portion of the figure shows the position of the one plane of symmetry that a crystal of this sort possesses. For each face, edge or point on one side of the plane there is a corresponding face, edge or point in a similar position on the other side of the plane.

Symmetry Axis
Asymmetry axis is a imaginary line through a crystal about which the crystal may be revolved as upon an axis and repeat itself in appearance two or more times during the revolution. IN Fig. 6 the line C-C’ is an axis of symmetry, for when the crystal represented is revolved upon it, it will have, after a revolution of 180º, the same appearance as at first; or in other words, similar planes, edges, etc., will appear in the places of the corresponding planes and edges of the original position. Point A’ will occupy the original position of A, B’ that of B, etc. since the crystal is repeated twice in appearance during a complete revolution, this axis is said to be one of binary symmetry, we have axes of trigonal (threefold), tetragonal (fourfold) and hexagonal (sixfold) symmetry. 

Center of Symmetry
A crystal has a center of symmetry if an imaginary line is passed from some point on its surface through its center, and a similar point is found on the line at an equal distance beyond center, C, of the crystal, the distance AC and A’C being equal.

Symmetry Classes of Crystals
With the basic assumption that crystal forms were the outward expression of an internal regular structure, crystallographers early attempted to develop theoretical structure that could account for the various kinds of symmetry shown by crystals. In this way fourteen different fundamental parallelepipeds were derived which could act as unit cells in the building up of the crystal structure. While these fourteen different units could account for the more important of the symmetry classes of crystals they failed in certain cases. To explain these it was necessary to assume that the unit cells were grouped in respect to each other in some other way than the usual rectilinear arrangement. By the assumption that the cells had been rotated or moved in respect to each other it was possible to derive some sixty five different point systems from the original fourteen unit cells. Even them there were still some symmetry classes that remained unexplained. These have been accounted for by the further assumption that the various point systems might interpenetrate each other. In this way a total of two hundred and thirty different arrangements of the unit cells was possible. All these, however, fall into thirty-two distinct symmetry classes and from theoretical considerations it has been shown that these thirty-two classes may be further grouped into six systems, the classes of each system having certain close relations to each other. These systems are known as the Isometric, Tetragonal, Hexagonal, Orthorhombic, Monoclinic and Triclinic Systems. All crystals will be found to belong to one or the other of these systems. As stated above, there are thirty-two possible subdivisions of these six systems, but the majority of them are only of theoretical interest, since practically all know species can be placed in one or the other of some ten or twelve classes.