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A wallpaper group (or plane symmetry group or plane crystallographic 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. There are 17 possible distinct groups.

Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups).

Introduction

Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.

Consider the following examples:

Examples A and B have the same wallpaper group; it is called p4m . Example C has a different wallpaper group, called p4g . 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.

A complete list of all seventeen possible wallpaper groups can be found below.

Symmetries of patterns

A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns which repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry — when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.

Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "bar code" in the centre of mass of each tile.

The types of transformations that are relevant here are called Euclidean plane isometries. For example:

  • If we shift example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the pattern we started with. This type of symmetry is called a translation . Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.
  • If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain exactly the same pattern. This is called a rotation . Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C .
  • We can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection . Example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A .

However, example C is different . It only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a diagonal line, we do not get the same pattern back; what we do get is the original pattern shifted across by a certain distance. This is part of the reason that A and B have a different wallpaper group than C .

History

All 17 groups were used by Egyptian craftsmen, and used extensively in the Muslim world. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891 and then derived independently by George Pólya in 1924.

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 which 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 which 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.

Isometries of the Euclidean plane

Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).

  • Translations , denoted by T v , where v is a vector in R 2 . This has the effect of shifting the plane applying displacement vector v .
  • Rotations , denoted by R c,θ , where c is a point in the plane (the centre of rotation), and θ is the angle of rotation.
  • Reflections , or mirror isometries , denoted by F L , where L is a line in R 2 . ( F is for "flip"). This has the effect of reflecting the plane in the line L , called the reflection axis or the associated mirror .
  • Glide reflections , denoted by G L , d , where L is a line in R 2 and d is a distance. This is a combination of a reflection in the line L and a translation along L by a distance d .

The independent translations condition

The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R 2 ) such that the group contains both T v and T w .

The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups which only repeat along a single axis.

(It is possible to generalise this situation. We could for example study discrete groups of isometries of R n with m linearly independent translations, where m is any integer in the range 0 ≤ m ≤ n .)

The discreteness condition

The discreteness condition means that there is some positive real number ε, such that for every translation T v in the group, the vector v has length at least ε (except of course in the case that v is the zero vector).

The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, we might have for example a group containing the translation T x for every rational number x , which would not correspond to any reasonable wallpaper pattern.

One important and nontrivial consequence of the discreteness condition in combination with the independent translations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180°, 120°, 90°, or 60°. This fact is known as the crystallographic restriction theorem, and can be generalised to higher-dimensional cases.

Notations for wallpaper groups

Crystallographic notation

Crystallography has 230 space groups to distinguish, far more than the 17 wallpaper groups, but many of the symmetries in the groups are the same. Thus we can use a similar notation for both kinds of groups, that of Carl Hermann and Charles-Victor Mauguin. An example of a full wallpaper name in Hermann-Mauguin style is

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