Wallpaper Groups
The wallpaper groups are the 17 possible plane symmetry groups. They are commonly represented using Hermann-Mauguin-like symbols or in orbifold notation (Zwillinger 1995, p. 260).
| orbifold notation | Hermann-Mauguin symbol |
| o | p1 |
| 2222 | p2 |
| ** | pm |
| xx | pg |
| *2222 | pmm |
| 22* | pmg |
| 22x | pgg |
| x* | cm |
| 2*22 | cmm |
| 442 | p4 |
| *442 | p4m |
| 4*2 | p4g |
| 333 | p3 |
| *333 | p3ml |
| 3*3 | p3lm |
| 632 | p6 |
| *632 | p6m |
Patterns created with Artlandia SymmetryWorks for each of these groups are illustrated above.
Beautiful patterns can be created by repeating geometric and artistic motifs according to the symmetry of the wallpaper groups, as exemplified in works by M. C. Escher and in the patterns created by I. Bakshee in the Wolfram Language using Artlandia, illustrated above.
For a description of the symmetry elements present in each space group, see Coxeter (1969, p. 413).
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