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As discussed previously, any motif can be represented by a point called a node. Points or nodes can be translated some distance in one direction by a unit translation vector t_a or t₁ to produce a line of nodes or motifs. Nodes can also be translated some distances in two directions t_a and t_b or t₁ and t₂ to produce a two‐dimensional array of points called a plane mesh or plane net. Simple translation of nodes in two directions produces five basic types of two‐dimensional patterns (Figure 4.9). The smallest units of such meshes, which contain the unit translation vectors and at least one node, are called unit meshes (unit nets). Unit meshes contain all the information necessary to produce the larger two‐dimensional pattern. They contain only translation symmetry information. The five basic types of unit mesh are classified on the basis of (1) the unit translation vector lengths (equal or unequal), the angles between them (90°, 60°, and 120° or none of these) and (2) whether they have nodes only at the corners (primitive = p) or have an additional node in the center (c) of the mesh.

Figure 4.8 The 10 plane point groups defined by rotational and reflection symmetry.

Source: Klein and Hurlbut (1985). © John Wiley & Sons.

Square unit meshes (Figure 4.9a) are primitive and have equal unit translation vectors at 90° angles to each other (p, t_a = t_b, γ = 90°). Primitive rectangular unit meshes (Figure 4.9b) differ in that, although the unit translation vectors intersect at right angles, they are of unequal lengths (p, t_a ≠ t_b, γ = 90°). Diamond unit meshes have equal unit translation vectors that intersect at angles other than 60°, 90° or 120°. Diamond lattices can be produced and represented by primitive diamond unit meshes (p, t_a = t_b, γ ≠ 60°, 90° or 120°). They can also be produced by the translation of centered rectangular unit meshes (Figure 4.9c) in which the two unit mesh sides are unequal, the angle between them is 90°, and there is a second node in the center of the mesh (c, t_a ≠ t_b, γ = 90°). In a centered rectangular mesh there is a total content of two nodes = two motifs. If one looks closely, one may see evidence for glide reflection in the centered rectangular mesh and/or the larger diamond lattice. The hexagonal unit mesh (Figure 4.9d) is a special form of the primitive diamond mesh because, although the unit translation vectors are equal, the angles between them are 60° and 120° (p, t_a = t_b, γ = 120°). Rotation through 120^o generates three such unit meshes which combine to produce a larger pattern with hexagonal symmetry. Oblique unit meshes (Figure 4.9e) are primitive and are characterized by unequal unit translation vectors that intersect at angles that are not 90°, 60° or 120° (p, t_a ≠ t_b, γ ≠ 90°, 60° or 120°) and produce the least regular, least symmetrical two‐dimensional lattices. The arrays of nodes on planes within minerals always correspond to one of these basic patterns.