One- and two-dimensional lattices point groups

The strictly 1-dimensional pattern is of somewhat academic interest since only points lying along a line are permitted. The only symmetry element possible is an inversion point (symmetry centre) and accordingly the pattern is either a set of points repeating at regular intervals a, without symmetry, or pairs of points related by inversion points (Fig. 2.2(a)). Although the Inversion points (small black dots) were inserted only at the points of the lattice (that is, a distance a apart) further inversion points appear midway between the lattice points. This phenomenon, the 

We now come to the second point concerning plane patterns. An isolated object (for example, a polygon) can possess any kind of rotational symmetry but there is an important limitation on the types of rotational symmetry that a plane repeating pattern as a whole may possess. The possession of n-fold rotational symmetry would imply a pattern of -fold rotation axes normal to the plane (or strictly a pattern of -fold rotation points in the plane) since the pattern is a repeating one. In Fig. 2.4 let there be an axis of -fold rotation normal to the plane of the paper at /, and at Q one of the nearest other axes of -fold rotation. The rotation through Ivjn about Q transforms P into F and the same kind of rotation about P transforms Q into Q. It may happen that P and Q coincide, in which case n = 6. n all other cases PQ must be equal to, or an integral multiple of, PQ (since Q was chosen as one of the nearest axes), i.e. 4. The permissible values of n are therefore 1, 2, 3, 4, and 6. Since a 3-dimensional lattice may be regarded as built of plane nets the same restriction on kinds of symmetry applies to the 3-dimensional lattices, and hence to the symmetry of crystals. 

In Fig. 2.1 we Illustrated the most general form of the 2-dimensional lattice, but it is clear that if a pattern has 3-, 4-, or 6-fold symmetry the lattice also will be more symmetrical than that of Fig. 2.1. It is found that there are five, and only five, 2-dimensional lattices consistent with the permissible symmetry of 2-dimensional patterns (Fig. 2.5). 

We now have to find which combinations of symmetry elements are consistent with the five fundamental plane lattices. Let us consider the square lattice. We may 

Axial symmetry possible in plane patterns (see text). 

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