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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). The translation in one direction is represented by the length and direction of t_a or t₁. Translation in the second direction is represented by the length and direction of t_b or t₂. The pattern generated depends on the length of the two unit translation vectors and the angles between their directions. The result of any two‐dimensional translation is a plane lattice or plane mesh. A plane lattice is a two‐dimensional array of motifs or nodes in which every node has an environment similar to every other node in the array (Figure 4.2a, b).

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 (t_a, t_b, and t_c or t₁, t₂, and t₃, respectively). The translation in one direction is represented by the length and direction of t_a or t₁, the translation in the second direction is represented by t_b or t₂ and the translation in the third direction is represented by t_c 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).