Formal Definition and Discussion
Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations.
Two such isometry groups are of the same type (of the same wallpaper group) if they are the same up to an affine transformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry (this is only the case if there are no mirrors and no glide reflections, and rotational symmetry is at most of order 2).
Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation.
It follows from the Bieberbach theorem that all wallpaper groups are different even as abstract groups (as opposed to e.g. Frieze groups, of which two are isomorphic with Z).
2D patterns with double translational symmetry can be categorized according to their symmetry group type.
Read more about this topic: Wallpaper Group
Famous quotes containing the words formal, definition and/or discussion:
“There must be a profound recognition that parents are the first teachers and that education begins before formal schooling and is deeply rooted in the values, traditions, and norms of family and culture.”
—Sara Lawrence Lightfoot (20th century)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“Power is action; the electoral principle is discussion. No political action is possible when discussion is permanently established.”
—Honoré De Balzac (1799–1850)