# Classification of wallpaper groups

From Groupprops

## 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 |
---|---|---|---|---|---|

None | Trivial | Yes | No | Yes | |

Glide reflection | Two-element subgroup generated by reflection | No | No | No | |

Reflection | Two-element subgroup generated by reflection | Yes | No | No | |

Reflection and glide reflection | Klein four-group generated by two reflections | No | No | No | |

Rotation by | Two-element subgroup generated by -rotation | Yes | Yes | Yes | |

Reflection and rotation by | Klein four-group generated by two reflections | No | No | No | |

Reflection, rotation by with rotation center off reflection axis | Klein four-group generated by two reflections | No | No | No | |

Two orthogonal reflections | Klein four-group generated by two reflections | Yes | Yes | No | |

Rotation by , glide reflection | Klein four-group generated by two reflections | No | No | No | |

Rotation by | Four-element subgroup generated by -rotation | Yes | No | Yes | |

Rotation by , Reflection | Dihedral group of order eight | No | No | No | |

Rotation by | Three-element subgroup generated by rotation | Yes | No | Yes | |

Rotation by , Reflection | Dihedral group of order six | Yes | No | No | |

Rotation by , Reflection, with rotation center off reflection axis | Dihedral group of order six | No | No | No | |

Rotation by | Six-element subgroup generated by rotation | Yes | No | ||

Rotation by , Reflection | Dihedral group:D12 | Yes | Yes | No |

## Facts used

- 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 . Note that the case of and gives equivalent lattices.