Abstract Algebra Theory and Applications - Computer Science ...

10.2 SYMMETRY 185Table 10.1. The 17 wallpaper groupsNotation andReflectionsSpace Groups Point Group Lattice Type or Glide Reflections?p1 Z 1 parallelogram nonep2 Z 2 parallelogram nonep3 Z 3 hexagonal nonep4 Z 4 square nonep6 Z 6 hexagonal nonepm D 1 rectangular reflectionspg D 1 rectangular glide reflectionscm D 1 rhombic bothpmm D 2 rectangular reflectionspmg D 2 rectangular glide reflectionspgg D 2 rectangular bothc2mm D 2 rhombic bothp3m1, p31m D 3 hexagonal bothp4m, p4g D 4 square bothp6m D 6 hexagonal both(Figure 10.8). The wallpaper groups can now be classified according to thetypes of reflections that occur in each group: these are ordinarily reflections,glide reflections, both, or none.Theorem 10.8 There are exactly 17 wallpaper groups.p4mp4gFigure 10.9. The wallpaper groups p4m and p4gThe 17 wallpaper groups are listed in Table 10.1. The groups p3m1 andp31m can be distinguished by whether or not all of their threefold centerslie on the reflection axes: those of p3m1 must, whereas those of p31m maynot. Similarly, the fourfold centers of p4m must lie on the reflection axeswhereas those of p4g need not (Figure 10.9). The complete proof of this