Symmetry Group S( P ) Symmetries in the Group p111 Only translation p1a1 Translation and glide-reflection p1m1 Translation and horizontal reflection pm11 Translation and vertical reflection p112 Translation and rotation pma2 Translation, vertical reflection, glide- reflection, and rotation pmm2 Translation, vertical reflection, and rotation Since there are fifteen pattern types and seven symmetry groups, obviously there are going to be multiple pattern types to a certain symmetry group. The difference will come in the induced groups. Observe the figures below to see an example of each of the fifteen strip patterns and an explanation to which symmetry group and the induced group goes with each pattern type. PS1 p111 c1 7
PS2 p1a1 c1 PS3 p1m1 c1 PS4 p1m1 d1 PS5 pm11 c1 PS6 pm11 d1 PS7 p112 c1 8
PS8 p112 d1 PS9 pma2 c1 PS10 pma2 c2 PS11 pma2 d1 PS12 pmm2 c1 9
PS13 pmm2 d1 PS14 pmm2 d1 PS15 pmm2 d2 Notice that in the case of PS13 and PS14 they both have an induced group of d1. A motif transitive subgroup is a subgroup of the symmetry group that maps any motif onto any other copy of the motif in the pattern. These two pattern types differ because they have different motif-transitive proper subgroups. I will not elaborate on this any further, but you can refer to Grünbaum and Shephard for further details. Pattern Types of Periodic Patterns There are fifty-one different pattern types within the periodic patterns labeled PP1 through PP51. A periodic pattern is a pattern that classifies further the seventeen wallpaper groups. The wallpaper groups are as follows taken from Tilings and Patterns (Grünbaum and Shephard, 1987). Symmetry Groups Description p1 Only translation pg Two glide-reflections pm Two reflections cm One glide-reflection, one reflection p2 Four rotations of period two pgg Two glide-reflections, two rotations of period 2 10
pmg Two glide-reflections, one reflection, two rotations of period two pmm Four reflections, four rotations of period two cmm Two glide-reflections, two reflections, four rotations of period two p3 Three rotations of period three p31m One glide-reflection, one reflection, two rotations of period three p3m1 One glide-reflection, one reflection, three rotations of period three p4 One rotation of period two, two rotations of period four p4g Two glide-reflections, one reflection, one rotation of period two, one rotation of period four p4m One glide-reflection, three reflections, one rotation of period two, one rotation of period four p6 One rotation of period two, one rotation of period four, one rotation of period six p6m Two glide-reflections, two reflections, one rotation of period two, one rotation of period four, one rotation of period six Furthermore, the wallpaper symmetry groups come from an induced group of either cn, where 1 n or dn, where 1 n . In a couple of cases, the symmetry group and induced group of two pattern types are the same, so one must look further to the motif-transitive subgroup to distinguish a difference in pattern types. See an example of each of the fifty- one different periodic pattern types along with their symmetry group and induced group below.
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- Fall '08
- Number Theory, Symmetry group, symmetry groups, Grünbaum, pattern types