The first virtual leaflets go over "Friezes", "Rosettes" and "Mosaics". They are the three possible classes to make a first distinction from the viewpoint of symmetry, among plane drawings.

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.

In mathematics the word "rosette" 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) contains only a finite number of transformations.

It can be proven that the only two possibilities for a rosette's symmetry group are cyclic groups (that are denoted with the symbol Cn and that contain n rotations) or dihedral (that are denoted with the symbol Dn and that contain n rotations and n reflections).

For any given integer number n, there is a corresponding cyclic group Cn and a corresponding dihedral group Dn.

cyclic groups		dihedral group
C1	C2	D1	D2
C3	C4	D3	D4
C5	C6	D5	D6
C7	C8	D7	D8
...	...	...	...

The word "frieze" in mathematics defines 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) contains translations, but only those which point in one direction and in fact are all multiples of a base translation.

This figure is thus unlimited (we can apply the same translation 2, 3, 1000 times and the figure will remain unchanged). When we say that a picture on a piece of paper, on a monument or on a computer screen is a "frieze" we are assuming with our imagination that the figure is extending beyond the limit of the page, the wall or the screen.

The symmetry groups of a given 'frieze' are seven and only seven. We list them here assigning to each of them a symbol and the name of some transformations. The symbol will be explained in the sequel. The names of transformations are "evocative", that is they refer to the transformations that best "characterize" the group (e.g. all groups in fact contain translations, thus it would be redundant to include them each time). The other transformations of the given group can be obtained by composing the basic transformations we indicate here.

	p111
	translations
	p112
	rotations
	p1a1
	glide-reflections (that is the composition of a translation and a reflection, whose axes is parallel to the translation vector)
	p1m1
	horizontal reflections (that is with axes parallel to the translation vector)
	pm11
	vertical reflections (that is with axes perpendicular to the translation vector)
	pma2
	vertical reflections and rotations
	pmm2
	vertical reflections and horizontal reflections

The symbol on the left indicates the type of group, it comes from crystallography; it is composed by 4 characters by means of the following rules:

• The first character is always p
• The second character can be 1 or m: (as in mirror). It is "m" when the symmetry group of the given figure contains vertical reflections:
  

  pm11	pma2	pmm2

  it is "1" in all other cases:
  

  p111	p112	p1a1	p1m1

• The third character can be "1" or "m" or "a". It is "m" if the symmetry group of the figure contains a horizontal reflection:
  

  p1m1	pmm2

  it is "a" if the symmetry group of the figure contains a glide-reflection with a horizontal line of reflection.
  

  p1a1	pma2

  it is "1" in all other cases
  

  p111	p112	pm11

• The fourth character can be 1 or 2: it is 2 if the symmetry group of the figure contains rotations of 180°
  

  p112	pma2	pmm2

  It is "1" in all other cases
  

  p111	p1a1	p1m1	pm11

You can find some interactive animations about friezes in Draw your own frieze and Recognize a frieze.