10.1.6: Symmetry Operates on Everything

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10.16.png — Figure 10.16: A hexagonal prism terminated by pyramid faces

The crystal shown above in Figure 10.16 has a vertical 6-fold axis of symmetry. It has six identical prism faces and six identical pyramid faces on both the top and on bottom. It also has six identical edges around the prism faces and around the pyramid faces.

The crystal has six 2-fold rotation axes that pass through edges, and six other 2-fold axes that are perpendicular to the prism faces. The small insets in Figure 10.16 show one example of each, but five others are identical. Six vertical mirror planes pass through the edges of prism faces, and six other mirrors are perpendicular to the prism faces. The figure shows examples of each. The crystal also has a single horizontal mirror plane through its equator. So, symmetry operates on symmetry – there are six of everything (except the horizontal mirror because it is perpendicular to the 6-fold axis). This crystal also has an inversion center.

This crystal must also have six directions of identical atomic structure and, therefore, six directions with the same hardness, reflectivity, optical properties, and so on. The 6-fold rotation axis affects everything in a crystal, including faces, edges, corners, other symmetry elements, physical properties, optical properties, and atomic arrangement.