Periodic Pattern

Wallpaper Groups or Crystallographic Lattices - with Poster

The periodic pattern can be characterized according to their symmetry properties. Depending on the amount of differnt symmetry operations which transform the pattern into itself, they belong to one of the 17 different Wallpaper Groups. You can see a direct comparison of the 17 symmetry groups for a certain pattern on a separate poster.

I started with the calculation of symmetric patterns for my lecture on "group theory in solid state physics", as the Wallpaper Groups are identical to Crystallographic Lattices in two dimensions. In contrast, the Crystallographic Lattices of minerals describe the symmetry properties of periodic systems in three dimensions, where a lot more different types exist.

The images below were obtained by random use of different coordinate transformations and coloring schemes and final symmetrisation according to one of the 17 wallpaper groups. The random numbers, which select the different transformations are stored, so that each pattern generation can be repeated with different resolutions or other settings. The calculations were performed with the numeric language Matlab.

1920x1200, 791.62 KB	1920x1200, 296.67 KB	1920x1200, 811.33 KB	1920x1200, 509.21 KB	1920x1200, 1.27 MB	1920x1200, 132.33 KB
1920x1200, 552.39 KB	1920x1200, 1.45 MB	1920x1200, 767.91 KB	1920x1200, 484.38 KB	1920x1200, 1.32 MB	1920x1200, 1.62 MB
1920x1200, 1.68 MB	1920x1200, 1.52 MB	1920x1200, 651.96 KB	1920x1200, 1.76 MB	1920x1200, 880.86 KB	1920x1200, 1.35 MB
1920x1200, 2.09 MB	1920x1200, 512.64 KB	1920x1200, 913.65 KB	1920x1200, 1.18 MB	1920x1200, 196.41 KB	1920x1200, 1.46 MB
1920x1200, 493.61 KB	1920x1200, 1.64 MB	1920x1200, 689.97 KB	1920x1200, 522.41 KB

See the poster on the 17 different symmetry groups.

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e-mail: Gerhard.Brunthaler@jku.at