A wallpaper is a drawing that covers a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane crystallographic group) is in 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, especially in textiles, tessellations and tiles as well as wallpaper.
A union of an infinite number of separate surfaces can make up a given wallpaper. An infinite number of shapes and positions on the wallpaper are possible for a repetitive surface. For instance on the left, two adjacent squares of different sizes of the Pythagorean tiling form together a repetitive surface you can imagine, the union of which makes up the infinite Pythagorean tiling.
In the present article, a “pattern” is a repetitive parallelogram of minimal area in a determined position on the wallpaper. The image shows two shapes of repetitive parallelograms of minimal area denoted by a — a square is a particular parallelogram —, and possible positions of these surfaces. In the upper right‑hand corner, the repetitive square has a more interesting position on the wallpaper because of its point symmetry with respect to the center of a small tiling element.
In the present article, all repetitive patterns (of minimal area) are constructed from two translations that generate the group of all translations under which the wallpaper is invariant. On the image, the circle shaped symbol ⵔ represents the function composition, and a pair like or generates the group of all translations that transform the Pythagorean tiling into itself. Perhaps we should say that in the mathematical tradition the observer or his eye is motionless, when a transformation moves or transforms a geometric object.
The simplest wallpaper group, Group p1, applies when there is no symmetry other than the fact that a pattern repeats over regular intervals in two dimensions, as shown in the section on p1 below.
The following examples are patterns with more forms of symmetry:
Example A: Cloth, Tahiti
Examples A and B have the same wallpaper group; it is called p4m in the IUCr notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2 . 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.
The number of symmetry groups depends on the number of dimensions in the patterns. Wallpaper groups apply to the two-dimensional case, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.
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.^{[2]} The proof that the list of wallpaper groups is complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups.
A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (in other words, 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 that 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.
The types of transformations that are relevant here are called Euclidean plane isometries. For example:
However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If one flips across a diagonal line, one does not get the same pattern back, but the original pattern shifted across by a certain distance. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C.
Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection.
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, one 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.
Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).
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. One 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 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, 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 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,^{[3]} and can be generalised to higher-dimensional cases.
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 one 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 (also called IUCr notation) is p31m, with four letters or digits; more usual is a shortened name like cmm or pg.
For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell; these are explained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis that is the main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180°/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.
A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells that are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types.
Here are all the names that differ in short and full notation.
Short | pm | pg | cm | pmm | pmg | pgg | cmm | p4m | p4g | p6m |
---|---|---|---|---|---|---|---|---|---|---|
Full | p1m1 | p1g1 | c1m1 | p2mm | p2mg | p2gg | c2mm | p4mm | p4gm | p6mm |
The remaining names are p1, p2, p3, p3m1, p31m, p4, and p6.
Orbifold notation for wallpaper groups, advocated by John Horton Conway (Conway, 1992) (Conway 2008), is based not on crystallography, but on topology. One can fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols.
The group denoted in crystallographic notation by cmm will, in Conway's notation, be 2*22. The 2 before the * says there is a 2-fold rotation centre with no mirror through it. The * itself says there is a mirror. The first 2 after the * says there is a 2-fold rotation centre on a mirror. The final 2 says there is an independent second 2-fold rotation centre on a mirror, one that is not a duplicate of the first one under symmetries.
The group denoted by pgg will be 22×. There are two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.
Coxeter's bracket notation is also included, based on reflectional Coxeter groups, and modified with plus superscripts accounting for rotations, improper rotations and translations.
Conway | o | ×× | *× | ** | 632 | *632 |
---|---|---|---|---|---|---|
Coxeter | [∞^{+},2,∞^{+}] | [(∞,2)^{+},∞^{+}] | [∞,2^{+},∞^{+}] | [∞,2,∞^{+}] | [6,3]^{+} | [6,3] |
Crystallographic | p1 | pg | cm | pm | p6 | p6m |
Conway | 333 | *333 | 3*3 | 442 | *442 | 4*2 |
---|---|---|---|---|---|---|
Coxeter | [3^{[3]}]^{+} | [3^{[3]}] | [3^{+},6] | [4,4]^{+} | [4,4] | [4^{+},4] |
Crystallographic | p3 | p3m1 | p31m | p4 | p4m | p4g |
Conway | 2222 | 22× | 22* | *2222 | 2*22 |
---|---|---|---|---|---|
Coxeter | [∞,2,∞]^{+} | [((∞,2)^{+},(∞,2)^{+})] | [(∞,2)^{+},∞] | [∞,2,∞] | [∞,2^{+},∞] |
Crystallographic | p2 | pgg | pmg | pmm | cmm |
An orbifold can be viewed as a polygon with face, edges, and vertices which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a spherical symmetry group or Hyperbolic symmetry group. The type of space the polygons tile can be found by calculating the Euler characteristic, χ = V − E + F, where V is the number of corners (vertices), E is the number of edges and F is the number of faces. If the Euler characteristic is positive then the orbifold has an elliptic (spherical) structure; if it is zero then it has a parabolic structure, i.e. a wallpaper group; and if it is negative it will have a hyperbolic structure. When the full set of possible orbifolds is enumerated it is found that only 17 have Euler characteristic 0.
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. Reversing the process, one can assign numbers to the features of the orbifold, but fractions, rather than whole numbers. Because the orbifold itself is a quotient of the full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Euler characteristic by the order of the symmetry group.
The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows:
For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2.
Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2.
Feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).
To work out which wallpaper group corresponds to a given design, one may use the following table.^{[4]}
Size of smallest rotation |
Has reflection? | |||||
---|---|---|---|---|---|---|
Yes | No | |||||
360° / 6 | p6m (*632) | p6 (632) | ||||
360° / 4 | Has mirrors at 45°? | p4 (442) | ||||
Yes: p4m (*442) | No: p4g (4*2) | |||||
360° / 3 | Has rot. centre off mirrors? | p3 (333) | ||||
Yes: p31m (3*3) | No: p3m1 (*333) | |||||
360° / 2 | Has perpendicular reflections? | Has glide reflection? | ||||
Yes | No | |||||
Has rot. centre off mirrors? | pmg (22*) | Yes: pgg (22×) | No: p2 (2222) | |||
Yes: cmm (2*22) | No: pmm (*2222) | |||||
none | Has glide axis off mirrors? | Has glide reflection? | ||||
Yes: cm (*×) | No: pm (**) | Yes: pg (××) | No: p1 (o) |
See also this overview with diagrams.
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):
a centre of rotation of order two (180°). | |
a centre of rotation of order three (120°). | |
a centre of rotation of order four (90°). | |
a centre of rotation of order six (60°). | |
an axis of reflection. | |
an axis of glide reflection. |
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.
The diagrams on the right show the cell of the lattice corresponding to the smallest translations; those on the left sometimes show a larger area.
Oblique |
Hexagonal | ||||
---|---|---|---|---|---|
Rectangular |
Rhombic |
Square |
Computer generated
The two translations (cell sides) can each have different lengths, and can form any angle.
Oblique |
Hexagonal | ||||
---|---|---|---|---|---|
Rectangular |
Rhombic |
Square |
Computer generated
Cloth, Sandwich Islands (Hawaii)
Mat on which an Egyptian king stood
Egyptian mat (detail)
Wire fence, U.S.
Horizontal mirrors |
Vertical mirrors |
---|
(The first three have a vertical symmetry axis, and the last two each have a different diagonal one.)
Computer generated
Dress of a figure in a tomb at Biban el Moluk, Egypt
Indian metalwork at the Great Exhibition in 1851. This is almost pm (ignoring short diagonal lines between ovals motifs, which make it p1)
Horizontal glides |
Vertical glides |
---|---|
Rectangular |
Computer generated
Mat with herringbone pattern on which Egyptian king stood
Egyptian mat (detail)
Pavement with herringbone pattern in Salzburg. Glide reflection axis runs northeast–southwest
One of the colorings of the snub square tiling; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than just pg, then it is p4g (see there for this image with equally colored triangles)^{[5]}
Without the details inside the zigzag bands the mat is pmg; with the details but without the distinction between brown and black it is pgg.
Ignoring the wavy borders of the tiles, the pavement is pgg.
Horizontal mirrors |
Vertical mirrors |
---|---|
Rhombic |
Computer generated
Dress of Amun, from Abu Simbel, Egypt
Dado from Biban el Moluk, Egypt
Indian metalwork at the Great Exhibition in 1851
Dress of a figure in a tomb at Biban el Moluk, Egypt
rectangular |
square |
---|
2D image of lattice fence, U.S. (in 3D there is additional symmetry)
Mummy case stored in The Louvre
Mummy case stored in The Louvre. Would be type p4m except for the mismatched coloring
Horizontal mirrors |
Vertical mirrors |
---|
Computer generated
Cloth, Sandwich Islands (Hawaii)
Floor tiling in Prague, the Czech Republic
Bowl from Kerma
Pentagon packing
Rectangular |
Square |
---|
Computer generated
Rhombic |
Square |
---|
The rotational symmetry of order 2 with centres of rotation at the centres of the sides of the rhombus is a consequence of the other properties.
The pattern corresponds to each of the following:
Computer generated
one of the 8 semi-regular tessellations
Suburban brick wall using running bond arrangement, U.S.
Turkish dish
A compact packing of two sizes of circle
Another compact packing of two sizes of circle
Another compact packing of two sizes of circle
A p4 pattern can be looked upon as a repetition in rows and columns of equal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a checkerboard pattern of two such tiles, a factor √2 smaller and rotated 45°.
Computer generated
Overlaid patterns
Viennese cane
Renaissance earthenware
Generated from a photograph
This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes. Also it corresponds to a checkerboard pattern of two of such squares.
Examples displayed with the smallest translations horizontal and vertical (like in the diagram):
Computer generated
one of the 3 regular tessellations
Demiregular tiling with triangles; ignoring colors, this is p4m, otherwise c2m
one of the 8 semi-regular tessellations (ignoring color also, with smaller translations)
Storm drain, U.S.
Compact packing of two sizes of circle
Examples displayed with the smallest translations diagonal:
checkerboard
Cathedral of Bourges
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. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90° rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells).
Bathroom linoleum, U.S.
Painted porcelain, China
Fly screen, U.S.
Painting, China
one of the colorings of the snub square tiling (see also at pg)
Imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, but the two are not equal, not each other's mirror image, and not both symmetric (if the two are equal it is p6, if they are each other's mirror image it is p31m, if they are both symmetric it is p3m1; if two of the three apply then the third also, and it is p6m). For a given image, three of these tessellations are possible, each with rotation centres as vertices, i.e. for any tessellation two shifts are possible. In terms of the image: the vertices can be the red, the blue or the green triangles.
Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by √3. Then this wallpaper group corresponds to the case that all hexagons are equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror image symmetry (if they have rotational symmetry of order six it is p6, if they are symmetric with respect to the main diagonals it is p31m, if they are symmetric with respect to lines perpendicular to the sides it is p3m1; if two of the three apply then the third also, it is p6m). For a given image, three of these tessellations are possible, each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of the hexagons can be the red, the blue or the green triangles.
Computer generated
one of the 8 semi-regular tessellations (ignoring the colors: p6); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image
Like for p3, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, and both are symmetric, but the two are not equal, and not each other's mirror image. For a given image, three of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the vertices can be the red, the dark blue or the green triangles.
one of the 3 regular tessellations (ignoring colors: p6m)
another regular tessellation (ignoring colors: p6m)
one of the 8 semi-regular tessellations (ignoring colors: p6m)
Persian ornament
Painting, China (see detailed image)
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. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices can not be dark blue triangles.
Painting, China
Compact packing of two sizes of circle
A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with C_{3} symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with C_{6} symmetry (with the edges of the tiles not necessarily part of the pattern).
Computer generated
Persian ornament
A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with D_{3} symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D_{6} symmetry (with the edges of the tiles not necessarily part of the pattern). Thus the simplest examples are a triangular lattice with or without connecting lines, and a hexagonal tiling with one color for outlining the hexagons and one for the background.
Computer generated
one of the 8 semi-regular tessellations
another semi-regular tessellation
another semi-regular tessellation
Compact packing of two sizes of circle
Another compact packing of two sizes of circle
There are five lattice types or Bravais lattices, corresponding to the five possible wallpaper groups of the lattice itself. The wallpaper group of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.
The actual symmetry group should be distinguished from the wallpaper group. Wallpaper groups are collections of symmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetry groups, in the sense of actual groups of isometries. 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.
The numbers of degrees of freedom are:
However, within each wallpaper group, all symmetry groups are algebraically isomorphic.
Some symmetry group isomorphisms:
Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously for some patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction produces a pattern with 4-fold symmetry) is not counted as a form of extra symmetry.
Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change.
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.
Several software graphic tools will let you create 2D patterns using wallpaper symmetry groups. Usually you can edit the original tile and its copies in the entire pattern are updated automatically.
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