page_title: Wallpaper group
section_title: Group p31m (3*3)
hierarchical_section_title: Wallpaper group / The seventeen groups / Group p31m (3*3)
context_page_description: A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles and tiles as well as wallpaper.
  The simplest wallpaper group, Group p1, applies when there is no symmetry other than the fact that a pattern repeats over regular intervals in two dimensions, as shown in the section on p1 below.
  Consider the following examples of patterns with more forms of symmetry:
  Examples A and B have the same wallpaper group; it is called p4m in the IUC notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.
  The number of symmetry groups depends on the number of dimensions in the patterns. Wallpaper groups apply to the two-dimensional case, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups.
context_section_description: Orbifold signature: 3*3
  Coxeter notation: [6,3⁺]
  Lattice: hexagonal
  Point group: D₃
  The group p31m has three different rotation centres of order three (120°), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre does not lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.
  Like for p3 and p3m1, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices can not be dark blue triangles.
  Examples of group p31m
caption: Example and diagram for p31m