Glide reflection

This remarkable five five-cell pattern is called a glider because after two iteration steps it produces a pattern that is both diagonally displaced by one site and reflected in a diagonal line - i.e., patterns separated by two steps are related by a glide reflection. In this way, the original pattern is reproduced in a diagonally displaced position every four iterations. 

With class 6 we encounter a new type of translational symmetry, the glide reflection. This is a combination of reflection and translation by one half the unit translation. The glide line is represented by a broken (dashed) line, whereas the mirror is represented by a solid line. 

Finally, in class 7 we have four types of symmetry operation (1) simple (unit) translation (2) transverse reflection (3) twofold rotation and (4) glide reflections. As in class 5, not all of these symmetry operations are independent. If we begin with class 1 and introduce explicitly only the glide reflection and one transverse reflection, all the other operations will arise as products of these. Again, this is analogous to the way point groups behave. 

Thus far we have addressed the symmetry of crystalline arrays only in terms of the proper rotations and the rotation-inversion operations (the latter including simple inversion, as 1, and reflection, as 2) that occur in point symmetries, along with the lattice translation operations. However, for a complete discussion of symmetry in crystalline solids, we require two more types of operation in which translation is combined with either reflection or rotation. These are, respectively, glide-reflections (or, as commonly called, glides) and screw-rotations. 

Glide-Reflection. This operation is referred to a symmetry element called a glide plane. We have already employed a glide line (its 2D equivalent) in developing the 2D space groups. 

Why must a glide-reflection operation entail a translation of i the repeat distance and no other fraction of it ... 

A glide reflection is one in which S is a reflection in the glide plane followed by a translation w, not necessarily parallel to the reflection plane. The three possible types of translation w are described in Table 16.3. 

The product of three reflections in separate planes is an improper isometry. We again distinguish two cases (1) Two reflections in parallel planes combined with a third reflection in a plane perpendicular to the other two results is a translation-reflection or glide reflection (2) two reflections in intersecting planes... 

The strips may be divided into unit cells or eventually reduced unit cells (see Fig. 1), when the space group of the strip contains operations involving glide-reflections or screw-rotations, i.e. a combination of an improper reflection or two-fold rotation followed by a non-primitive translation of half a unit cell, which by themselves do not leave the lattice invariant. Each cell will contain w sites, w x L = 2N. 

Along with the primitive translations, and glide-reflections when appropriate, there axe other symmetry operations belonging to the space group. In bipartite systems, it is relevant to classify any symmetry operation according to whether it leaves each sublattice invariant or transforms one into each other (see, for instance, Fig. 2). 

Figure 2 (a) The polyphenantrene polymer with its glide-reflection plane. Here the A and B sublattices are invariant under this symmetry operation, (b) Polyaceacene polymer with its glide-reflection plane. Notice that here the A and B sublattices transform one into each other under this symmetry operation. 

For a given strip, it may happen that under a symmetry operation one configuration of LR-SPO D transforms into a configuration of different LR-SPO, D. That happens in particular for symmetry operations that transform one sublattice into each other, since then the arrows representing the SP will change their direction. Sometimes such a symmetry operation may be the one step translation (or the glide reflection when appropriate), T- For instance, when 6 0, A and B sublattices always must transform one into each other under T,... 

All the above examples applied to point groups. Antisymmetry and color symmetry, of course, may be introduced in space-group symmetries as well as examples illustrate in Figures 8-32, 8-37, and 9-46 (in the discussion of space groups). If we look only at the close-up of the tower in Figure 4-14b, it also has tranlational antisymmetry, specifically anti-glide-reflection symmetry together with similarity symmetry (these symmetries will be discussed in Chapter 8). 

Figure 8-1. Planar decoration with two-dimensional space group after Budden [4], (a) The decoration (b) Symmetry elements of the pattern (c) Some of the glide reflection planes and their effects in the pattern.

Figure 8-2. Top Canon illustrating simple repetition Bottom repetition combined with glide reflection.

Among the projected symmetry elements in Figure 9-22c, there are some which are derived from the generating elements. This is the case, for example, for vertical glide-reflection planes with elementary translations all and bll (represented by broken lines), translations (dot-dash lines), vertical screw axes 2, and 42, and symmetry centers (small hollow circles, some of which lie above the plane by 1/4 of the elementary translation). 

Fixed-Point-Free Motions. These include translations, screw rotations, and glide reflections. Because the primitive translation vector, Eq. 1.2, joins any two lattice points, an equivalent statement is that Eq. 1.2 represents the operation of translational symmetry bringing one lattice point into coincidence with another. However, we must choose the basis vectors (a, b, c) so as to include all lattice points, thus defining a... 

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