Wallpaper patterns

The word "wallpaper pattern" in mathematics indicates a plane figure whose symmetry group (that is the set of all those transformations of the plane that leave distances unchanged and map the figure onto itself) is discrete and contains some translations. These translations don't point in just one direction, as occurs for friezes, but in at least two different directions.

It is possible to prove that for wallpaper patterns there are 17 distinct symmetry groups (and seventeen only!).

Among them:

• Two groups contain rotations in 60° multiples (60°, 120°, 180°,240°, and 300°, and the identity):
  
  

  	632(p6) contains translations and rotations only (60° and multiple)
  	*632(p6m) contains reflections too.

  
  
  
• Three of them contain rotations in 90° multiples (90°, 180°, 270° and the identity):
  
  

  	442(p4) contains translations and rotations only (90° and multiples).
  	*442(p4m) contains reflections whose lines are pointing in four different directions.
  	4*2(p4g) contains reflections whose lines are pointing in two different directions.

  
  
  
• Three of them contain rotations in 120° multiples (120° and 240°, and the identity):
  
  

  	333(p3) contains translations and rotations only (in 120° multiples).
  	*333(p3m1) contains reflections too; all centres of rotation belong to an axis of symmetry of the figure.
  	3*3(p31m) contains reflections too; there are rotation centres that do not belong to an axis of symmetry of the figure.

  
  
  
• Five of them contain 180° rotations only (in addition to identity):
  
  

  	2222(p2) contains translations and rotations only (in 180&° multiples).
  	*2222(pmm) contains reflections whose lines point two different directions; all rotation centres belong to an axis of symmetry of the figure.
  	2*22(cmm) contains reflections whose lines point in two different directions; there are rotation centres that do not belong to an axis of symmetry of the figure.
  	22*(pmg) contains reflections whose lines point in one direction only.
  	22×(pgg) does not contain reflections; it contains glide-reflections (that is the composition of a translation and a reflection, whose axes is parallel to the translation vector).

  
  
  
• Four of them do not contain rotations (apart from the 360°, rotation [the identity] which belongs to the symmetry group of all figures):
  
  

  	o(p1) contains translations only.
  	**(pm) contains reflections; it does not contain glide-reflections, apart from the "mandatory ones", that result from the composition of the reflection in an axis of symmetry of the figure, with a parallel translation.
  	*×(cm) contains reflections too; it also contains glide-reflections, whose lines are parallel to axes of symmetry, but that in turn are not axes of symmetry of the figure.
  	××(pg) does not contain reflections; it contains glide-reflections.

In this description we have not indicated all the transformations that one can find in each group, nor have we written each characteristic and property, but we have provided enough to distinguish each one of the 17 groups.

You can find some interactive animations about wallpaper patterns in Draw your own wallpaper pattern and Recognize a wallpaper pattern.