Figure 1.4 Dense filling of 2D space by regular geometrical figures.

      Finally, we obtain:

(1.5)

It follows from Eq. (1.5) that there is a very limited set of regular figures (with 2 < n ≤ 6) useful for producing periodic patterns, which fill the 2D space with no voids (i.e. providing integer numbers, p). These are hexagons (n = 6, p = 3, ϕ = 60°), squares (n = 4, p = 4, ϕ = 90°), and triangles (n = 3, m = 6, ϕ = 120°). Based on the value of central angle, ϕ, these regular figures possess the sixfold, fourfold, and threefold rotation axes, respectively. Since they are related to regular geometrical figures, these rotation axes are called high-symmetry elements. Regarding the twofold axis, it fits the symmetry of the parallelogram, which also can be used for filling the 2D space without voids but does not represent a regular geometrical figure. For this reason, the twofold rotation axis is classified as a low symmetry element (together with inversion center, , and mirror plane, m). It also comes out from Eq. (1.5), that regular figures with fivefold rotation axis (n = 5), as well as with rotation axes higher than n > 6, are incompatible with translational symmetry, i.e. cannot be used for producing periodic patterns without voids since parameter, p, is not an integer number.

In the absence of the long-range translational symmetry, however, as in quasicrystals, one can find additional rotation axes, e.g. fivefold ( = 72°), as for 2D construction shown in Figure 1.3 or for icosahedral symmetry in 3D. The latter can be found in two Platonic bodies: regular icosahedrons and dodecahedrons. Regular dodecahedron has 12 pentagonal faces and 20 vertices, in each of them three faces meet (Figure 1.5). Therefore, the fivefold axes are normal to the pentagonal faces. In contrast, regular icosahedron has 20 triangular faces and 12 vertices, in each of them five faces meet (Figure 1.6). Therefore, the fivefold axes connect the body center and each vertex. Note that regular pentagon (plane figure) has central angle 72° and is characterized by the so-called golden ratio τ (the ratio between the pentagon diagonal, d_p, and pentagon edge, a_p, see Figure 1.7):

(1.6)

Figure 1.5 Dodecahedron sculpted by 12 pentagonal faces.

Figure 1.6 Icosahedron sculpted by 20 triangular faces.

Figure 1.7 Regular pentagon with edges equal a_p and diagonals equal d_p. The ratio, , is called the golden ratio (Eq. (1.6)).

which is of great importance to the quasicrystal diffraction conditions (described later in this chapter).

Permitted combinations of local symmetry elements (totally 32 in regular crystals) are called point groups. A set of different crystals, possessing the same point group symmetry, form certain crystal class. Point group symmetry is responsible for anisotropy of physical properties in crystals, as explained in more detail further in this chapter.

Figure 1.8 Unit cells of the following side-centered Bravais lattices: A-type (a), B-type (b), C-type (c). Translation vectors, a₁, a₂, a₃, are indicated by dashed arrows.

Figure 1.9 Unit cells of the following centered Bravais lattices: (a) face-centered (F-type) and (b) body-centered (I-type). Translation vectors, a₁, a₂, a₃, are indicated by dashed arrows.

Bravais lattices defined by Eq. (1.1) are primitive (P) since they effectively contain only one atom per unit cell. However, in some symmetry

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