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In minerals, the fundamental motifs are parts of clusters of three‐dimensional coordination polyhedra sufficient to establish the composition of the mineral. When these are repeated in three dimensions during mineral growth, they produce the long‐range order characteristic of crystalline substances (Figure 4.10). Like all fundamental units of pattern, these three‐dimensional motifs can be classified on the basis of their translation‐free symmetries.

Only 32 different three‐dimensional motif symmetries exist. These define 32 space point groups, each with unique space point group symmetry. In minerals, the 32 crystal classes – to one of which all minerals belong – correspond to the 32 space point group symmetries of the mineral's three‐dimensional motif. That the crystal classes were originally defined on the basis of the external symmetry of mineral crystals is another example of the fact that the external symmetry of minerals reflects the internal symmetry of their constituents. The 32 crystal classes belong to 6 (or 7) crystal systems, each with its own characteristic symmetry. Table 4.3 summarizes the crystal systems, the symmetries of the 32 space point groups or crystal classes and their names, which are based on general crystal forms. It is important to remember that a crystal cannot possess more symmetry than that of the motifs of which it is composed. However, it can possess less, depending on how the motifs are arranged and how the crystal developed during growth.