Stereographic projection diagrams

A stereogram is to be interpreted as a schematic diagram of an object viewed down the principal axis. Horizontal mirror planes and equatorial rotation axes lie in the plane of the diagram. Points in space are designated by x if they lie above the plane of the diagram and by o if they lie below it. 

Problem 4-4. Construct stereographic projection diagrams for the symmetry groups of HOCl, cyclopropane, allene, and benzene. 

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In drawing stereographic projection diagrams, it is conventional to indicate improper axes, by an open polygon at the center, and a proper axis, C n, by a filled polygon (see Figure 4.1c). 

The symmetry operations for the rotation groups can most easily be found by means of the stereographic projection diagrams. For the groupC4t, this takes the form of Figure VII-1. The square in the center represents the 4-fold axis. Starting with point 1, where the + indicates... 

Fig. VII-2. Stereographic projection diagrams for the simple point groups.

In the character tables, the operations of the various groups are collected into classes. For example, the group C3, has three classes the class of Ej the class of the two 3-fold axes, and the class of the three planes of symmetry. This may readily be verified from the definition of a class and the stereographic projection diagrams. 

All thirty-two crystal classes in stereographic projection are tabulated in Fig. 9.1.8. For each crystal class, two diagrams showing the equipoints and symmetry elements are displayed side by side. Note that the thicker lines indicate mirror planes, and the meaning of the graphic symbols of the symmetry elements are given in Table 9.3.2. 

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