Classification of wallpaper groups

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Statement

This article completely classifies all the space groups in two dimensions, also called the wallpaper groups. The classification up to affine space type is the same as the classification up to crystallographic space type, and there are a total of seventeen types.

The seventeen types are as follows:

IUC name Description of non-translation generators Point group Splits Full automorphism group Orientation-preserving
p1 None Trivial Yes No Yes
pg Glide reflection Two-element subgroup generated by reflection No No No
pm Reflection Two-element subgroup generated by reflection Yes No No
cm Reflection and glide reflection Klein four-group generated by two reflections No No No
p2 Rotation by \pi Two-element subgroup generated by \pi-rotation Yes Yes Yes
pmg Reflection and rotation by \pi Klein four-group generated by two reflections No No No
cmm Reflection, rotation by \pi with rotation center off reflection axis Klein four-group generated by two reflections No No No
pmm Two orthogonal reflections Klein four-group generated by two reflections Yes Yes No
pgg Rotation by \pi, glide reflection Klein four-group generated by two reflections No No No
p4 Rotation by \pi/2 Four-element subgroup generated by \pi/2-rotation Yes No Yes
p4g Rotation by \pi/2, Reflection Dihedral group of order eight No No No
p3 Rotation by 2\pi/3 Three-element subgroup generated by rotation Yes No Yes
p3m1 Rotation by 2\pi/3, Reflection Dihedral group of order six Yes No No
p31m Rotation by 2\pi/3, Reflection, with rotation center off reflection axis Dihedral group of order six No No No
p6 Rotation by \pi/3 Six-element subgroup generated by rotation Yes No
p6m Rotation by \pi/3, Reflection Dihedral group:D12 Yes Yes No

Facts used

  1. Crystallographic restriction: This is the main lemma used in the classification, and it states that if a lattice possesses a nontrivial rotational symmetry, then it is spanned by two shortest vectors of equal length at an angle of \pi/2, \pi/3, 2\pi/3. Note that the case of \pi/3 and 2\pi/3 gives equivalent lattices.