Symmetry of Periodic Structures

In this section we will describe the two-dimensional space groups, or plane groups, which will serve as an introduction to the 230 space groups. The symmetry elements in the Euclidean plane are  

We identify the plane groups by the international symbols whose interpretation is quite straightforward  

The symbol for the crystal class may be derived from the symbol for a plane group by removing all references to translations (see Section 2.4.3 for the relation between a plane group and the crystal class)  

Note that the symbol for the plane group contains the complete information concerning the crystal system and the Bravais lattice. 

A set of points that is equivalent with respect to a symmetry group is called an orbit. The polyhedra of Fig. 2.17 represent orbits of point groups. The arrows in Fig. 2.29 represent the orbits of plane groups. For the majority of groups, there are several types of orbit that we refer to as general positions and special positions. We will illustrate this important point with the aid of the plane group p2mg (Fig, 2.30). 

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Space groups describe the symmetry of periodic structures. 

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