target="_blank" rel="nofollow" href="#ulink_e6b82dcf-2b66-5778-93cd-529416c5a7c5">Figure 4.2 (a) Two‐dimensional translation at right angles (t1 and t2) to generate a two‐dimensional mesh of motifs or nodes. (b) Two‐dimensional translation (t1 and t2) not at right angles to generate a two‐dimensional mesh or lattice. (c) Three‐dimensional translation (t1, t2, and t3) to generate a three‐dimensional space lattice.
Three‐dimensional translations are defined by three unit translation vectors (ta, tb, and tc or t1, t2, and t3, respectively). The translation in one direction is represented by the length and direction of ta or t1, the translation in the second direction is represented by tb or t2 and the translation in the third direction is represented by tc or t3. The result of any three‐dimensional translation is a space lattice. A space lattice is a three‐dimensional array of motifs or nodes in which every node has an environment similar to every other node in the array. Since crystalline substances such as minerals have long‐range, three‐dimensional order and since they may be thought of as motifs repeated in three dimensions, the resulting array of motifs is a crystal lattice. Figure 4.2c illustrates a space lattice produced by a three‐dimensional translation of nodes or motifs.
Figure 4.3 Five major types of rotational symmetry operations, viewed looking down rotational axes marked by blue symbols in the center of each circle: dot marks axis of onefold rotation (1), oval marks axis of twofold rotation (2), triangle marks axis of threefold rotation (3), square marks axis of fourfold rotation (4), and hexagon marks axis of sixfold rotation (6).
Rotation
Motifs can also be repeated by non‐translational symmetry operations. Many patterns can be repeated by rotation (n). Rotation (n) is a symmetry operation that involves the rotation of a pattern about an imaginary line or axis, called an axis of rotation (A), in such a way that every component of the pattern is perfectly repeated one or more times during a complete 360° rotation. The symbol “n” denotes the number of repetitions that occur during a complete rotation. Figure 4.3 uses triangle motifs to depict the major types of rotational symmetry (n) that occur in minerals and other crystals. The axis of rotation for each motif is perpendicular to the page. Table 4.1 summarizes the major types of rotational symmetry.
Table 4.1 Five common axes of rotational symmetry in minerals.
| Type | Symbolic notation | Description |
|---|---|---|
| Onefold axis of rotation | (1 or A1) | Any axis of rotation about which the motif is repeated only once during a 360° rotation (Figure 4.3 (1)) |
| Twofold axis of rotation | (2 or A2) | Motifs repeated every 180° or twice during a 360° rotation (Figure 4.3 (2)) |
| Threefold axis of rotation | (3 or A3) | Motifs repeated every 120° or three times during a complete rotation (Figure 4.3 (3)) |
| Fourfold axis of rotation | (4 or A4) | Motifs repeated every 90° or four times during a complete rotation (Figure 4.3 (4)) |
| Sixfold axis of rotation | (6 or A6) | Motifs repeated every 60° or six times during a complete rotation (Figure 4.3 (5)) |
Reflection
Reflection is as familiar to us as our own reflections in a mirror or that of a tree in a still body of water. It is also the basis for the concept of bilateral symmetry that characterizes many organisms (Figure 4.4). Yet it is a symmetry operation that is somewhat more difficult for most people to visualize than rotation. Reflection is a symmetry operation in which every component of a pattern is repeated by reflection across a plane called a mirror plane (m). Reflection occurs when each component is repeated by equidistant projection perpendicular to the mirror plane. Reflection retains all the components of the original motif but changes its “handedness”; the new motifs produced by reflection across a mirror plane are mirror images of each other (Figure 4.4). Symmetry operations that change the handedness of motifs are called enantiomorphic operations.
Figure 4.4 Two‐ and three‐dimensional motifs that illustrate the concept of reflection across a plane of mirror symmetry (m). (a) Mirror image of the letter “R”. (b) Bilateral symmetry of a butterfly; the two halves are nearly, but not quite, perfect mirror images of each other.
Source: Image from butterflywebsite.com. © Mikula Web Solutions.
One test for the existence of a mirror plane of symmetry is that all components of the motifs on one side of the plane are repeated at equal distances on the other side of the plane along projection lines perpendicular to the plane. If this is not true, the plane is not a plane of mirror symmetry.
Inversion
Inversion is perhaps the most difficult of the simple symmetry operations to visualize. Inversion involves the repetition of motifs by inversion through a point called a center of inversion (i). Inversion occurs when every component of a pattern is repeated by equidistant projection through a common point or center of inversion. The two “letters” in Figure 4.5 illustrates this enantiomorphic symmetry operation and shows the center through which inversion occurs. In some symbolic notations centers of inversion are symbolized by (c) rather than (i).
