Wallpaper group

Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and wallpaper.

A proof that there are only 17 distinct groups of such planar symmetries was first carried out by Evgraf Fedorov in 1891[1] and then derived independently by George Pólya in 1924.

Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely.

Unlike in the three-dimensional case, one can equivalently restrict the affine transformations to those that preserve orientation.

One could for example study discrete groups of isometries of Rn with m linearly independent translations, where m is any integer in the range 0 ≤ m ≤ n.) The discreteness condition means that there is some positive real number ε, such that for every translation Tv in the group, the vector v has length at least ε (except of course in the case that v is the zero vector, but the independent translations condition prevents this, since any set that contains the zero vector is linearly dependent by definition and thus disallowed).

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, one might have for example a group containing the translation Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern.

The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.

All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice.

One can fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols.

Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.

When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, and polygon faces, which must be consistent with the Euler characteristic.

Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2.

(When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).

Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows (it is the shape that is significant, not the colour): On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated) differently.

The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern that is repeated.

This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes.

Examples displayed with the smallest translations horizontal and vertical (like in the diagram): Examples displayed with the smallest translations diagonal: A p4g pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image.

Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells).

Imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations.

Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by √3.

Like for p3, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations.

Like for p3 and p3m1, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations.

These depend, apart from the wallpaper group, on a number of parameters for the translation vectors, the orientation and position of the reflection axes and rotation centers.

If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change.

Usually you can edit the original tile and its copies in the entire pattern are updated automatically.