Isometric System, Crystallographic Axes, Symmetry and Forms

Crystallographic Axes: The crystallographic axes of the Isometric System are three in number, of equal lengths, and make right angles with each other. When properly orientated one axis in vertical and the other two are horizontal, one being parallel and the other perpendicular to the observer. Since the three axes are identical in character, they are interchangeable, and any one of them may serve as the vertical axis, etc. in giving the indices of a face of an isometric form, the order of the axes, etc.

Normal Class

Symmetry and Forms: They symmetry shown by the crystals of the Normal Class of the Isometric System is as follows. The three crystallographic axes are axes of tetragonal symmetry. These axes emerge in the middle of each of the octants formed by the intersection of the crystallographic axes. Further, there are six diagonal axes of binary symmetry, each of which bisects one of the angles between two of the crystallographic axes.

This class shows nine planes of symmetry, three of them being known as the axial planes, since each includes two crystallographic axes (see Fig. 29), and six being called diagonal planes, since each bisects the angles between two of the axial planes.

To summarize the symmetry of this section:

• 3 cruystallographic axes of tetragonal symmetry;
• 4 diagonal axes of trigonal symmetry;
• 6 diagonal axes of binary symmetry;
• 3 axial planes of symmetry;
• 6 diagonal planes of symmetry.

This symmetry, which is of the highest degree possible in solids with plane surfaces, defines the Normal Class of the Isometric System. Every crystal form and every combination of forms that belongs to this class must show its complete symmetry. It is important to remember that in this class the three crystallographic axes are of tetragonal symmetry, since this fact distinguishes the class from all others and by means of it the crystallographic axes can be easily located and a crystal properly orientated.