Figure 4.1 Representative mineral crystals: representative mineral crystals: (a) pyrite.

      Source: Courtesy of Doug Moore.

      (b) quartz.

      Source: Robert Lavinsky. iRocks.com—CCBYSA 3.0, https://commons.wikimedia.org/wiki/File:Quartz‐153455.jpg; last accessed 09/24/2020.

      Mineralogists have developed language to describe the symmetry of crystals and the crystal faces that enclose them. Familiarizing students with the concepts and terminology of crystal symmetry and crystal faces is one of the primary goals of this chapter. A second goal of this chapter is to build connections between crystal chemistry (Chapters 2 and 3) and crystallography. This involves an explanation of the relationships between chemical composition, coordination polyhedra and the crystal structures, crystal faces, and crystal forms that develop as crystals grow.

       Motifs and nodes

      When minerals begin to form, atoms or ions bond together, so that partial or complete coordination polyhedra develop (Chapter 2). Because the ions on the corners and edges of coordination polyhedra have unsatisfied electrostatic charges, they tend to bond to additional ions available in the environment as the mineral grows. Eventually, a small cluster of coordination polyhedra is formed that contains all the coordination polyhedra characteristic of the mineral and its chemical composition. In any mineral, we can recognize a small cluster of coordination polyhedra that contains the mineral's fundamental composition and unit of pattern or motif. As the mineral continues to grow, additional clusters of the same pattern of coordination polyhedra are added to form a mineral crystal with a three‐dimensional geometric pattern – a long‐range, three‐dimensional crystal structure. Clusters of coordination polyhedra are added, often one atom or ion at a time, as (1) the crystal nucleates, (2) it becomes a microscopic crystal, and, if growth continues, (3) it becomes a macroscopic crystal. Growth continues in this manner until the environmental conditions that promote growth change and growth ceases.

Long‐range, geometric arrangements of atoms and/or ions in crystals are produced when a fundamental array of atoms, a unit of pattern or motif, is repeated in three dimensions to produce the crystal structure. A motif is the smallest unit of pattern that, when repeated by a set of symmetry operations, will generate the long‐range pattern characteristic of the crystal. In minerals, the motif is composed of one or more coordination polyhedra. In wallpaper, it is a basic set of design elements that are repeated to produce a two‐dimensional pattern, whereas in a brick wall the fundamental motif is that of a single brick that is repeated in space to form the three‐dimensional structure. The repetition of these fundamental units of pattern by a set of rules called symmetry operations can produce a two‐ or three‐dimensional pattern with long‐range order. When several different motifs could be repeated by a similar set of symmetry operations, we may wish to emphasize the general rules by which different motifs may be repeated to produce a particular type of long‐range order. In such cases, it is useful to represent motifs by using a point. A point used to represent any motif is called a node. The pattern or array of atoms about every node must be the same throughout the pattern the nodes represent.

4.2 SYMMETRY OPERATIONS

      4.2.1 Simple symmetry operations

      Symmetry operations may be simple or compound. Simple symmetry operations produce repetition of a unit of pattern or motif using a single type of operation. Compound symmetry operations involve motif repetition using a combination of two symmetry operations. Simple symmetry operations, discussed in this section, include (1) translation, by specific distances in specified directions, (2) rotation, about a specified set of axes, (3) reflection, across a mirror plane, and (4) inversion, through a point called a center. These operations are discussed below and provide useful insights into the geometry of crystal structures and the three‐dimensional properties of such crystals.

       Translation

The symmetry operation called translation involves the periodic repetition of nodes or motifs by systematic linear translation. Two‐dimensional translation of basic design elements generates a row of similar elements (Figure 4.2a). The translation is defined by the unit translation vector (t), a specific length, and direction of systematic displacement by which the pattern is repeated. Motifs other than triangles, any motif, could be translated by the same unit translation vector to produce a one‐dimensional pattern. In minerals, the motifs are clusters of atoms or coordination polyhedra that are repeated by translation.

      Two‐dimensional translations are defined by two unit translation vectors (t_aand t_b ort₁and t₂, respectively).