Categories: "symmetry"
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 glidereflections (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 glidereflections, 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 glidereflections, 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 glidereflections.
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  
glidereflections (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
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
p1a1 pma2
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
p111 p1a1 p1m1 pm11
You can find some interactive animations about friezes in Draw your own frieze and Recognize a frieze.