Wallpaper group

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A wallpaper is a mathematical object covering 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 composition of translations that leave a wallpaper unchanged.svg
A wallpaper is intrinsically linked to a commutative group of translations,
each element of which displaces it without changing anything,
in front of our motionless eye.
Wallpaper group-p4m-5.jpg
Example of an Egyptian design
with wallpaper group p4m
In a finite list are itemized all possible groups of isometries  under which a wallpaper can be invariant.

What this page calls pattern

A few patterns of Pythagorean tiling.svg
Image 1.
Examples of repetitive surfaces
on a Pythagorean tiling.
Pythagorean Squares is a wallpaper with reflection points.svg
Image 2.
The  minimal  area  of any of possible repetitive surfaces
  by  disregarding  the  colors
Such  Pythagorean tilings  can  be  seen  as  wallpapers  because  they are periodic.

Any periodic tiling can be seen as a wallpaper. More particularly, we can consider as a wallpaper a tiling by identical tiles edge‑to‑edge, necessarily periodic, and conceive from it a wallpaper by decorating in the same manner every tiling element, and eventually erase partly or entirely the boundaries between these tiles. Conversely, from every wallpaper we can construct such a tiling by identical tiles edge‑to‑edge, which bear each identical ornaments, the identical outlines of these tiles being not necessarily visible on the original wallpaper. Such repeated boundaries delineate a repetitive surface added here in dashed lines.

Such pseudo‑tilings connected to a given wallpaper are in infinite number. For example image 1 shows two models of repetitive squares in two different positions, which have Another repetitive square has an We could indefinitely conceive such repetitive squares larger and larger. An infinity of shapes of repetitive zones are possible for this Pythagorean tiling, in an infinity of positions on this wallpaper. For example in red on the bottom right‑hand corner of image 1, we could glide its repetitive parallelogram in one or another position. In common on the first two images: a repetitive square concentric with each small square tile, their common center being a point symmetry of the wallpaper.

Between identical tiles edge‑to‑edge, an edge is not necessarily a segment of right line. On the top left‑hand corner of image 3, point  is a vertex of a repetitive pseudo‑rhombus with thick stripes on its whole surface, called pseudo‑rhombus because of a concentric repetitive rhombus constructed from it by taking out a bit of surface somewhere to append it elsewhere, and keep the area unchanged. By the same process on image 3, a repetitive regular hexagon filled with vertical stripes is constructed from a rhombic repetitive zone Conversely, from elementary geometric tiles edge‑to‑edge, an artist like M. C. Escher created attractive surfaces many times repeated. On image 2,  the minimum area of a repetitive surface by disregarding colors, each repetitive zone in dashed lines consisting of five pieces in a certain arrangement, to be either a square or a hexagon, like in a proof of the Pythagorean theorem.

In the present article, a pattern is a repetitive parallelogram of minimal area in a determined position on the wallpaper. Image 1 shows two parallelogram‑shaped patterns — a square is a particular parallelogram —. Image 3 shows rhombic patterns — a rhombus is a particular parallelogram —.

On this page, all repetitive patterns (of minimal area) are constructed from two translations that generate the group of all translations under which the wallpaper is invariant. With the circle shaped symbol ⵔ of function composition, a pair like or  generates the group of all translations that transform the Pythagorean tiling into itself.

A few isometries and repetitive zones relative to a wallpaper V1.svg
Image 3.
In one or the other orientation, every rhombus
in dark dashed lines instances a same pattern,
because the rotation of center and ‑120° angle
leaves the wallpaper unchanged.
A few isometries and repetitive zones relative to a wallpaper V2.svg
Image 4.
The same wallpaper as previously by disregarding its colors.
Otherwise if the colors are considered, there is no longer
a center of rotation that leaves the wallpaper unchanged,
either at point  or  or H.
Is  considered  as  the  same  pattern  its  image
under an isometry keeping the wallpaper unchanged.

Possible groups linked to a pattern

A wallpaper remains on the whole unchanged under certain isometries, starting with certain translations that confer on the wallpaper a repetitive nature. One of the reasons to be unchanged under certain translations is that it covers the whole plane. No mathematical object in our minds is stuck onto a motionless wall! On the contrary an observer or his eye is motionless in front of a transformation, which glides or rotates or flips a wallpaper, eventually could distort it, but that would be out of our subject.

If an isometry leaves unchanged a given wallpaper, then the inverse isometry keeps it also unchanged, like translation  on image 1, 3 or 4, or a ± 120° rotation around a point like  on image 3 or 4. If they have both this property to leave unchanged a wallpaper, two isometries composed in one or the other order have then this same property to leave unchanged the wallpaper. To be exhaustive about the concepts of group and subgroups under the function composition, represented by the circle shaped symbol ⵔ, here is a traditional truism in mathematics:  everything remains itself under the identity transformation. This identity function can be called translation of zero vector or rotation of 360°.

A glide can be represented by one or several arrows if parallel and of same length and same sense, in same way a wallpaper can be represented either by a few patterns or by only one pattern, considered as a pseudo‑tile imagined repeated edge‑to‑edge with an infinite number of replicas. Image 3 shows two patterns with two different contents, and the one in dark dashed lines or one of its images under represents the same wallpaper on the following image 4, by disregarding the colors. Certainly a color is perceived subjectively whereas a wallpaper is an ideal object, however any color can be seen as a label that characterizes certain surfaces, we might think of a hexadecimal code of color as a label specific to certain zones. It may be added that a well‑known theorem deals with colors.

Groups are registered in the catalog by examining properties of a parallelogram, edge‑to‑edge with its replicas. For example its diagonals intersect at their common midpoints, center and symmetry point of any parallelogram, not necessarily symmetry point of its content. Other example, the midpoint of a full side shared by two patterns is the center of a new repetitive parallelogram formed by the two together, center which is not necessarily symmetry point of the content of this double parallelogram. Other possible symmetry point, two patterns symmetric one to the other with respect to their common vertex form together a new repetitive surface, the center of which is not necessarily symmetry point of its content.

Certain rotational symmetries are possible only for certain shapes of pattern. For example on image 2, a Pythagorean tiling is sometimes called pinwheel tilings because of its rotational symmetry of 90 degrees about the center of a tile, either small or large, or about the center of any replica of tile, of course. Also when two equilateral triangles form edge‑to‑edge a rhombic pattern, like on image 4 or 5 (future image 5), a rotational symmetry of 120 degrees about a vertex of a 120° angle, formed by two sides of pattern, is not always a symmetry point of the content of the regular hexagon formed by three patterns together sharing a vertex, because they does not always contain the same motif.

First examples of groups

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:

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.

Symmetries of patterns

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.

A glide reflection will map a set of left and right footprints into each other Krok 6.png
A glide reflection will map a set of left and right footprints into each other

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, 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

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

The independent translations condition

The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R2) such that the group contains both Tv and Tw.

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 Rn 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 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.

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.

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 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.

  • p2 (p2): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
  • p4gm (p4mm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.
  • c2mm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.
  • p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.

Here are all the names that differ in short and full notation.

Crystallographic short and full names
Short pm pg cm pmm pmg pgg cmm p4m p4g p6m

The remaining names are p1 , p2 , p3 , p3m1 , p31m , p4 , and p6 .

Orbifold notation

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.

  • A digit, n, indicates a centre of n-fold rotation corresponding to a cone point on the orbifold. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.
  • An asterisk, *, indicates a mirror symmetry corresponding to a boundary of the orbifold. It interacts with the digits as follows:
    1. Digits before * denote centres of pure rotation (cyclic).
    2. Digits after * denote centres of rotation with mirrors through them, corresponding to "corners" on the boundary of the orbifold (dihedral).
  • A cross, ×, occurs when a glide reflection is present and indicates a crosscap on the orbifold. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so need no notation.
  • The "no symmetry" symbol, o, stands alone, and indicates there are only lattice translations with no other symmetry. The orbifold with this symbol is a torus; in general the symbol o denotes a handle on the orbifold.

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, Coxeter and crystallographic correspondence
Crystallographic p1 pg cm pm p6 p6m
Crystallographic p3 p3m1 p31m p4 p4m p4g
Crystallographic p2 pgg pmg pmm cmm

Why there are exactly seventeen groups

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 ).

Guide to recognizing wallpaper groups

To work out which wallpaper group corresponds to a given design, one may use the following table. [4]

Size of smallest
Has reflection?
360° / 6 p6m (*632) p6 (632)
360° / 4Has mirrors at 45°? p4 (442)
Yes: p4m (*442) No: p4g (4*2)
360° / 3Has rot. centre off mirrors? p3 (333)
Yes: p31m (3*3) No: p3m1 (*333)
360° / 2Has perpendicular reflections?Has glide reflection?
Has rot. centre off mirrors? pmg (22*) Yes: pgg (22×) No: p2 (2222)
Yes: cmm (2*22) No: pmm (*2222)
noneHas glide axis off mirrors?Has glide reflection?
Yes: cm (*×) No: pm (**) Yes: pg (××) No: p1 (o)

See also this overview with diagrams.

The seventeen groups

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):

Wallpaper group diagram legend rotation2.svg a centre of rotation of order two (180°).
Wallpaper group diagram legend rotation3.svg a centre of rotation of order three (120°).
Wallpaper group diagram legend rotation4.svg a centre of rotation of order four (90°).
Wallpaper group diagram legend rotation6.svg a centre of rotation of order six (60°).
Wallpaper group diagram legend reflection.svg an axis of reflection.
Wallpaper group diagram legend glide reflection.svg 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.

Group p1 (o)

Example and diagram for p1 SymBlend p1.svg
Example and diagram for p1
Cell structures for p1 by lattice type
Wallpaper group diagram p1.svg
Wallpaper group diagram p1 half.svg
Wallpaper group diagram p1 rect.svg
Wallpaper group diagram p1 rhombic.svg
Wallpaper group diagram p1 square.svg
Examples of group p1

The two translations (cell sides) can each have different lengths, and can form any angle.

Group p2 (2222)

Example and diagram for p2 SymBlend p2.svg
Example and diagram for p2
Cell structures for p2 by lattice type
Wallpaper group diagram p2.svg
Wallpaper group diagram p2 half.svg
Wallpaper group diagram p2 rect.svg
Wallpaper group diagram p2 rhombic.svg
Wallpaper group diagram p2 square.svg
Examples of group p2

Group pm (**)

Example and diagram for pm SymBlend pm.svg
Example and diagram for pm
Cell structure for pm
Wallpaper group diagram pm.svg
Horizontal mirrors
Wallpaper group diagram pm rotated.svg
Vertical mirrors
Examples of group pm

(The first three have a vertical symmetry axis, and the last two each have a different diagonal one.)

Group pg (××)

Example and diagram for pg SymBlend pg.svg
Example and diagram for pg
Cell structures for pg
Wallpaper group diagram pg.svg
Horizontal glides
Wallpaper group diagram pg rotated.svg
Vertical glides
Examples of group pg

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.

Group cm (*×)

Example and diagram for cm SymBlend cm.svg
Example and diagram for cm
Cell structure for cm
Wallpaper group diagram cm.svg
Horizontal mirrors
Wallpaper group diagram cm rotated.svg
Vertical mirrors
Examples of group cm

Group pmm (*2222)

Example and diagram for pmm SymBlend pmm.svg
Example and diagram for pmm
Cell structure for pmm
Wallpaper group diagram pmm.svg
Wallpaper group diagram pmm square.svg
Examples of group pmm

Group pmg (22*)

Example and diagram for pmg SymBlend pmg.svg
Example and diagram for pmg
Cell structures for pmg
Wallpaper group diagram pmg.svg
Horizontal mirrors
Wallpaper group diagram pmg rotated.svg
Vertical mirrors
Examples of group pmg

Group pgg (22×)

Example and diagram for pgg SymBlend pgg.svg
Example and diagram for pgg
Cell structures for pgg by lattice type
Wallpaper group diagram pgg.svg
Wallpaper group diagram pgg square.svg
Examples of group pgg

Group cmm (2*22)

Example and diagram for cmm SymBlend cmm.svg
Example and diagram for cmm
Cell structures for cmm by lattice type
Wallpaper group diagram cmm.svg
Wallpaper group diagram cmm square.svg

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:

Examples of group cmm

Group p4 (442)

Example and diagram for p4 SymBlend p4.svg
Example and diagram for p4
Cell structure for p4 Wallpaper group diagram p4 square.svg
Cell structure for p4
Examples of group p4

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°.

Group p4m (*442)

Example and diagram for p4m SymBlend p4m.svg
Example and diagram for p4m
Cell structure for p4m Wallpaper group diagram p4m square.svg
Cell structure for p4m

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 of group p4m

Examples displayed with the smallest translations horizontal and vertical (like in the diagram):

Examples displayed with the smallest translations diagonal:

Group p4g (4*2)

Example and diagram for p4g SymBlend p4g.svg
Example and diagram for p4g
Cell structure for p4g Wallpaper group diagram p4g square.svg
Cell structure for p4g

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).

Examples of group p4g

Group p3 (333)

Example and diagram for p3 SymBlend p3.svg
Example and diagram for p3
Cell structure for p3 Wallpaper group diagram p3.svg
Cell structure 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, 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.

Examples of group p3

Group p3m1 (*333)

Example and diagram for p3m1 SymBlend p3m1.svg
Example and diagram for p3m1
Cell structure for p3m1 Wallpaper group diagram p3m1.svg
Cell structure for p3m1

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 blue or the green triangles.

Examples of group p3m1

Group p31m (3*3)

Example and diagram for p31m SymBlend p31m.svg
Example and diagram for p31m
Cell structure for p31m Wallpaper group diagram p31m.svg
Cell structure for p31m

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 must be the red triangles, not the blue triangles.

Examples of group p31m

Group p6 (632)

Example and diagram for p6 SymBlend p6.svg
Example and diagram for p6
Cell structure for p6 Wallpaper group diagram p6.svg
Cell structure for p6

A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with C3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with C6 symmetry (with the edges of the tiles not necessarily part of the pattern).

Examples of group p6

Group p6m (*632)

Example and diagram for p6m SymBlend p6m.svg
Example and diagram for p6m
Cell structure for p6m Wallpaper group diagram p6m.svg
Cell structure for p6m

A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with D3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D6 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.

Examples of group p6m

Lattice types

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.

Symmetry groups

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:

Dependence of wallpaper groups on transformations

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.

Web demo and software

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.

See also


  1. E. Fedorov (1891) "Симметрія на плоскости" (Simmetrija na ploskosti, Symmetry in the plane), Записки Императорского С.-Петербургского минералогического общества (Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society), series 2, 28 : 345–390 (in Russian).
  2. Pólya, George (November 1924). "Über die Analogie der Kristallsymmetrie in der Ebene" [On the analog of crystal symmetry in the plane]. Zeitschrift für Kristallographie (in German). 60 (1–6): 278–282. doi:10.1524/zkri.1924.60.1.278. S2CID   102174323.
  3. Klarreich, Erica (5 March 2013). "How to Make Impossible Wallpaper". Quanta Magazine. Retrieved 2021-04-07.
  4. Radaelli, Paulo G. Symmetry in Crystallography. Oxford University Press.
  5. If one thinks of the squares as the background, then one can see a simple patterns of rows of rhombuses.

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