Awallpaper group (orplane symmetry group orplane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on thesymmetries in the pattern. Such patterns occur frequently inarchitecture anddecorative art, especially intextiles,tiles, andwallpaper.

The simplest wallpaper group, Groupp1, applies when there is no symmetry beyond simple translation of a pattern in two dimensions. The following patterns have more forms of symmetry, including some rotational and reflectional symmetries:

• 
  ExampleA: Cloth,Tahiti
• 
  ExampleB: Ornamental painting,Nineveh,Assyria
• 
  ExampleC: Paintedporcelain,China

ExamplesA andB have the same wallpaper group; it is calledp4m in theIUCr notation and*442 in theorbifold notation. ExampleC has a different wallpaper group, calledp4g or4*2 . The fact thatA andB have the same wallpaper group means that they have the same symmetries, regardless of the designs' superficial details; whereasC has a different set of symmetries.

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 simplerfrieze groups and the three-dimensionalspace groups.

Aproof that there are only 17 distinctgroups of such planar symmetries was first carried out byEvgraf Fedorov in 1891^[1] and then derived independently byGeorge Pólya in 1924.^[2] The proof that the list of wallpaper groups is complete came only after the much harder case ofspace groups had been done. The seventeen wallpaper groups are listed below; see§ The seventeen groups.

Asymmetry 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 betranslated (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 calledEuclidean plane isometries. For example:

• If oneshifts exampleB one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern isexactly the same as the starting pattern. This type of symmetry is called atranslation. ExamplesA andC are similar, except that the smallest possible shifts are in diagonal directions.
• If oneturns exampleB clockwise by 90°, around the centre of one of the squares, again one obtains exactly the same pattern. This is called arotation. ExamplesA andC also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation forC.
• One can alsoflip exampleB across a horizontal axis that runs across the middle of the image. This is called areflection. ExampleB also has reflections across a vertical axis, and across two diagonal axes. The same can be said forA.

However, exampleC isdifferent. It only has reflections in horizontal and vertical directions,not across diagonal axes. If one flips across a diagonal line, one doesnot 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 ofA andB is different from the wallpaper group ofC.

Another transformation is aglide reflection, a combination of reflection and translation parallel to the line of reflection.

Mathematically, a wallpaper group or plane crystallographic group is a type oftopologically discretegroup ofisometries of the Euclidean plane that contains twolinearly independenttranslations.

Two suchisometry groups are of the same type (of the same wallpaper group) if they arethe 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 noglide reflections, androtational symmetry is at most of order 2).

Unlike inthe three-dimensional case, one can equivalently restrict the affine transformations to those that preserveorientation.

It follows from theBieberbach conjecture that all wallpaper groups are different even as abstract groups (as opposed to e.g.frieze groups, of which two are isomorphic withZ).

2D patterns with double translational symmetry can be categorized according to theirsymmetry group type.

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

• Translations, denoted byT_v, wherev is avector inR². This has the effect of shifting the plane applyingdisplacement vectorv.
• Rotations, denoted byR_c,θ, wherec is a point in the plane (the centre of rotation), andθ is the angle of rotation.
• Reflections, ormirror isometries, denoted byF_L, whereL is a line inR². (F is for "flip"). This has the effect of reflecting the plane in the lineL, called thereflection axis or the associatedmirror.
• Glide reflections, denoted byG_L,d, whereL is a line inR² andd is a distance. This is a combination of a reflection in the lineL and a translation alongL by a distanced.

The condition on linearly independent translations means that there exist linearly independent vectorsv andw (inR²) such that the group contains bothT_v andT_w.

The purpose of this condition is to distinguish wallpaper groups fromfrieze groups, which possess a translation but not two linearly independent ones, and fromtwo-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves intwo 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 ofRⁿ withm linearly independent translations, wherem 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 translationT_v in the group, the vectorv has lengthat least ε (except of course in the case thatv 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 translationT_x for everyrational numberx, 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 thecrystallographic restriction theorem,^[3] and can be generalised to higher-dimensional cases.

Crystallography has 230space 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 ofCarl Hermann andCharles-Victor Mauguin. An example of a full wallpaper name in Hermann-Mauguin style (also calledIUCr notation) isp31m, with four letters or digits; more usual is a shortened name likecmm orpg.

For wallpaper groups the full notation begins with eitherp orc, for aprimitive cell or aface-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 eitherm,g, or1, 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 (whenn > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or anm 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 crystalspace groups uses additional cell types.

Examples

• p2 (p2): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
• p4gm (p4gm): 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
Full	p1m1	p1g1	c1m1	p2mm	p2mg	p2gg	c2mm	p4mm	p4gm	p6mm

The remaining names arep1,p2,p3,p3m1,p31m,p4, andp6.

Orbifold notation for wallpaper groups, advocated byJohn 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, anorbifold, then describe that with a few symbols.

• A digit,n, indicates a centre ofn-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 symbolo denotes a handle on the orbifold.

The group denoted in crystallographic notation bycmm will, in Conway's notation, be2*22. The2 before the* says there is a 2-fold rotation centre with no mirror through it. The* itself says there is a mirror. The first2 after the* says there is a 2-fold rotation centre on a mirror. The final2 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 bypgg will be22×. There are two pure 2-fold rotation centres, and a glide reflection axis. Contrast this withpmg, Conway22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.

Coxeter'sbracket notation is also included, based on reflectionalCoxeter groups, and modified with plus superscripts accounting for rotations,improper rotations and translations.

An orbifold can be viewed as apolygon with face, edges, and vertices which can be unfolded to form a possibly infinite set of polygons which tile either thesphere, the plane or thehyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either aspherical symmetry group orHyperbolic symmetry group. The type of space the polygons tile can be found by calculating theEuler characteristic,χ = V − E + F, whereV is the number of corners (vertices),E is the number of edges andF 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 theorder of the symmetry group.

The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows:

• A digitn without or before a * counts as⁠n − 1/n⁠.
• A digitn after a * counts as⁠n − 1/2n⁠.
• Both * and × count as 1.
• The "no symmetry" o counts as 2.

For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2.

Examples

• 632:⁠5/6⁠ +⁠2/3⁠ +⁠1/2⁠ = 2
• 3*3:⁠2/3⁠ + 1 +⁠2/6⁠ = 2
• 4*2:⁠3/4⁠ + 1 +⁠1/4⁠ = 2
• 22×:⁠1/2⁠ +⁠1/2⁠ + 1 = 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 ishyperbolic; when positive,spherical orbad).

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

See alsothis 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.

Thebrown or yellow area indicates afundamental domain, i.e. the smallest part of the pattern that is repeated.

The diagrams on the right show the cell of thelattice corresponding to the smallest translations; those on the left sometimes show a larger area.

Cell structures forp1 by lattice type

Oblique			Hexagonal
Rectangular		Rhombic		Square

• Orbifold signature:o
• Coxeter notation (rectangular): [∞⁺,2,∞⁺] or [∞]⁺×[∞]⁺
• Lattice: oblique
• Point group: C₁
• The groupp1 contains only translations; there are no rotations, reflections, or glide reflections.

Examples of groupp1

• 
  Computer generated
• 
  Medieval walldiapering
• 
  21co-uniform tiling

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

Cell structures forp2 by lattice type

Oblique			Hexagonal
Rectangular		Rhombic		Square

• Orbifold signature:2222
• Coxeter notation (rectangular): [∞,2,∞]⁺
• Lattice: oblique
• Point group: C₂
• The groupp2 contains four rotation centres of order two (180°), but no reflections or glide reflections.

Examples of groupp2

• 
  Computer generated
• 
  Cloth,Sandwich Islands (Hawaii)
• 
  Ceiling of anEgyptiantomb
• 
  Wire fence, U.S.
• 
  15co-uniform tiling
• 
  Mat on which anEgyptian king stood
• 
  Egyptian mat (detail)

Cell structure forpm

Horizontal mirrors	Vertical mirrors

• Orbifold signature:**
• Coxeter notation: [∞,2,∞⁺] or [∞⁺,2,∞]
• Lattice: rectangular
• Point group: D₁
• The grouppm has no rotations. It has reflection axes, they are all parallel.

Examples of grouppm

(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 atomb atBiban el Moluk,Egypt
• 
  Egyptiantomb,Thebes
• 
  Ceiling of atomb at Gourna,Egypt. Reflection axis is diagonal
• 
  Indian metalwork at theGreat Exhibition in 1851. This is almostpm (ignoring short diagonal lines between ovals motifs, which make itp1)
• 
  6co-uniform tiling (slab)

Cell structures forpg

Horizontal glides	Vertical glides
Rectangular

• Orbifold signature:××
• Coxeter notation: [(∞,2)⁺,∞⁺] or [∞⁺,(2,∞)⁺]
• Lattice: rectangular
• Point group: D₁
• The grouppg contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.

Examples of grouppg

• 
  Computer generated
• 
  Mat withherringbone pattern on whichEgyptian king stood
• 
  Egyptian mat (detail)
• 
  Pavement withherringbone pattern inSalzburg. Glide reflection axis runs northeast–southwest
• 
  One of the colorings of thesnub square tiling; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than justpg, then it isp4g (see there for this image with equally colored triangles)^[5]
• 
  6co-uniform tiling made only ofpentagons

Without the details inside the zigzag bands the mat ispmg; with the details but without the distinction between brown and black it ispgg.

Ignoring the wavy borders of the tiles, the pavement ispgg.

Cell structure forcm

Horizontal mirrors	Vertical mirrors
Rhombic

• Orbifold signature:*×
• Coxeter notation: [∞⁺,2⁺,∞] or [∞,2⁺,∞⁺]
• Lattice: rhombic
• Point group: D₁
• The groupcm contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis isnot a reflection axis; it is halfway between two adjacent parallel reflection axes.
• This group applies for symmetrically staggered rows (i.e. there is a shift per row of half the translation distance inside the rows) of identical objects, which have a symmetry axis perpendicular to the rows.

Examples of groupcm

• 
  Computer generated
• 
  Dress ofAmun, fromAbu Simbel,Egypt
• 
  Dado fromBiban el Moluk,Egypt
• 
  Bronze vessel inNimroud,Assyria
• 
  Spandrels ofarches, theAlhambra,Spain
• 
  Soffit of arch, theAlhambra,Spain
• 
  Persiantapestry
• 
  Indian metalwork at theGreat Exhibition in 1851
• 
  Dress of a figure in atomb atBiban el Moluk,Egypt
• 
  6co-uniform tiling with hexagonal cells
• 
  Textile pattern:houndstooth

Cell structure forpmm

rectangular	square

• Orbifold signature:*2222
• Coxeter notation (rectangular): [∞,2,∞] or [∞]×[∞]
• Coxeter notation (square): [4,1⁺,4] or [1⁺,4,4,1⁺]
• Lattice: rectangular
• Point group: D₂
• The grouppmm has reflections in two perpendicular directions, and four rotation centres of order two (180°) located at the intersections of the reflection axes.

Examples of grouppmm

• 
  2D image of latticefence, U.S. (in 3D there is additional symmetry)
• 
  Mummy case stored inThe Louvre
• 
  Mummy case stored inThe Louvre. Would be typep4m except for the mismatched coloring
• 
  8co-uniform tiling with all non-slab planigons

Cell structures forpmg

Horizontal mirrors	Vertical mirrors

• Orbifold signature:22*
• Coxeter notation: [(∞,2)⁺,∞] or [∞,(2,∞)⁺]
• Lattice: rectangular
• Point group: D₂
• The grouppmg has two rotation centres of order two (180°), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes.

Examples of grouppmg

• 
  Computer generated
• 
  Cloth,Sandwich Islands (Hawaii)
• 
  Ceiling ofEgyptiantomb
• 
  Floor tiling inPrague, theCzech Republic
• 
  Bowl fromKerma
• 
  4co-uniform tiling
• 
  Pentagon packing

Cell structures forpgg by lattice type

Rectangular	Square

• Orbifold signature:22×
• Coxeter notation (rectangular): [((∞,2)⁺,(∞,2)⁺)]
• Coxeter notation (square): [4⁺,4⁺]
• Lattice: rectangular
• Point group: D₂
• The grouppgg contains two rotation centres of order two (180°), and glide reflections in two perpendicular directions. The centres of rotation are not located on the glide reflection axes. There are no reflections.

Examples of grouppgg

• 
  Computer generated
• 
  Bronze vessel inNimroud,Assyria
• 
  Pavement inBudapest,Hungary
• 
  4co-uniform tiling (strictly trihexagonal)

Cell structures forcmm by lattice type

Rhombic	Square

• Orbifold signature:2*22
• Coxeter notation (rhombic): [∞,2⁺,∞]
• Coxeter notation (square): [(4,4,2⁺)]
• Lattice: rhombic
• Point group: D₂
• The groupcmm has reflections in two perpendicular directions, and a rotation of order two (180°) whose centre isnot on a reflection axis. It also has two rotations whose centresare on a reflection axis.
• This group is frequently seen in everyday life, since the most common arrangement ofbricks in a brick building (running bond) utilises this group (see example below).

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:

• symmetrically staggered rows of identical doubly symmetric objects
• a checkerboard pattern of two alternating rectangular tiles, of which each, by itself, is doubly symmetric
• a checkerboard pattern of alternatingly a 2-fold rotationally symmetric rectangular tile and its mirror image

Examples of groupcmm

• 
  Computer generated
• 
  Elongated triangular tiling
• 
  Suburbanbrick wall usingrunning bond arrangement, U.S.
• 
  Ceiling ofEgyptiantomb. Ignoring colors, this would bep4g
• 
  Egyptian
• 
  Persiantapestry
• 
  Egyptiantomb
• 
  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
• 
  3co-uniform tiling (Krötenheerdt)

• Orbifold signature:442
• Coxeter notation: [4,4]⁺
• Lattice: square
• Point group: C₄
• The groupp4 has two rotation centres of order four (90°), and one rotation centre of order two (180°). It has no reflections or glide reflections.

Examples of groupp4

Ap4 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 acheckerboard pattern of two such tiles, a factor√2 smaller and rotated 45°.

• 
  Computer generated
• 
  Ceiling ofEgyptiantomb; ignoring colors this isp4, otherwisep2
• 
  Ceiling ofEgyptiantomb
• 
  Overlaid patterns
• 
  Frieze, theAlhambra,Spain. Requires close inspection to see why there are no reflections
• 
  Viennese cane
• 
  Renaissance earthenware
• 
  Pythagorean tiling
• 
  Generated from a photograph
• 
  4co-uniform tiling

• Orbifold signature:*442
• Coxeter notation: [4,4]
• Lattice: square
• Point group: D₄
• The groupp4m has two rotation centres of order four (90°), and reflections in four distinct directions (horizontal, vertical, and diagonals). It has additional glide reflections whose axes are not reflection axes; rotations of order two (180°) are centred at the intersection of the glide reflection axes. All rotation centres lie on reflection axes.

This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes. Also it corresponds to acheckerboard pattern of two of such squares.

Examples of groupp4m

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

• 
  Computer generated
• 
  Square tiling
• 
  Tetrakis square tiling; ignoring colors, this isp4m, otherwisecmm
• 
  Truncated square tiling (ignoring color also, with smaller translations)
• 
  Ornamental painting,Nineveh,Assyria
• 
  Storm drain, U.S.
• 
  Egyptianmummy case
• 
  Persianglazed tile
• 
  Compact packing of two sizes of circle
• 
  4co-uniform tiling (Krötenheerdt)

Examples displayed with the smallest translations diagonal:

• 
  checkerboard
• 
  Cloth,Otaheite (Tahiti)
• 
  Egyptiantomb
• 
  Cathedral ofBourges
• 
  Dish fromTurkey,Ottoman period

• Orbifold signature:4*2
• Coxeter notation: [4⁺,4]
• Lattice: square
• Point group: D₄
• The groupp4g has two centres of rotation of order four (90°), which are each other's mirror image, but it has reflections in only two directions, which are perpendicular. There are rotations of order two (180°) whose centres are located at the intersections of reflection axes. It has glide reflections axes parallel to the reflection axes, in between them, and also at an angle of 45° with these.

Ap4g pattern can be looked upon as acheckerboard 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 groupp4m (with diagonal translation cells).

Examples of groupp4g

• 
  Bathroomlinoleum, U.S.
• 
  Paintedporcelain, China
• 
  Fly screen, U.S.
• 
  Painting, China
• 
  one of the colorings of thesnub square tiling (see also atpg)
• 
  4co-uniform tiling (fractalization of snub square tiling)

• Orbifold signature:333
• Coxeter notation: [(3,3,3)]⁺ or [3^[3]]⁺
• Lattice: hexagonal
• Point group: C₃
• The groupp3 has three different rotation centres of order three (120°), but no reflections or glide reflections.

Imagine atessellation 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 isp6, if they are each other's mirror image it isp31m, if they are both symmetric it isp3m1; if two of the three apply then the third also, and it isp6m). 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 isp6, if they are symmetric with respect to the main diagonals it isp31m, if they are symmetric with respect to lines perpendicular to the sides it isp3m1; if two of the three apply then the third also, it isp6m). 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 groupp3

• 
  Computer generated
• 
  Snub trihexagonal tiling (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
• 
  Street pavement inZakopane,Poland
• 
  Tiling based off a similar tiling in theAlhambra,Spain
• 
  6co-uniform tiling, each rotation point surrounded by a 3-fold cluster

• Orbifold signature:*333
• Coxeter notation: [(3,3,3)] or [3^[3]]
• Lattice: hexagonal
• Point group: D₃
• The groupp3m1 has three different rotation centres of order three (120°). It has reflections in the three sides of an equilateral triangle. The centre of every rotation lies on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

Like forp3, 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 groupp3m1

• 
  Triangular tiling (ignoring colors:p6m)
• 
  Hexagonal tiling (ignoring colors:p6m)
• 
  Truncated hexagonal tiling (ignoring colors:p6m)
• 
  Persianglazed tile (ignoring colors:p6m)
• 
  Persian ornament
• 
  Painting,China (see detailed image)
• 
  6co-uniform tiling (smallest one containing 3 different 3-clusters)

• Orbifold signature:3*3
• Coxeter notation: [6,3⁺]
• Lattice: hexagonal
• Point group: D₃
• The groupp31m has three different rotation centres of order three (120°), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre doesnot lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

Like forp3 andp3m1, 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 groupp31m

• 
  Persianglazed tile
• 
  Paintedporcelain,China
• 
  Painting,China
• 
  Compact packing of two sizes of circle
• 
  4co-uniform tiling

• Orbifold signature:632
• Coxeter notation: [6,3]⁺
• Lattice: hexagonal
• Point group: C₆
• The groupp6 has one rotation centre of order six (60°); two rotation centres of order three (120°), which are each other's images under a rotation of 60°; and three rotation centres of order two (180°) which are also each other's images under a rotation of 60°. It has no reflections or glide reflections.

A pattern with this symmetry can be looked upon as atessellation of the plane with equal triangular tiles withC₃ 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 groupp6

• 
  Computer generated
• 
  Regular polygons
• 
  Wall panelling, theAlhambra,Spain
• 
  Persian ornament
• 
  Floret pentagonal tiling
• 
  7co-uniform tiling with horizontal and 60° translations

• Orbifold signature:*632
• Coxeter notation: [6,3]
• Lattice: hexagonal
• Point group: D₆
• The groupp6m has one rotation centre of order six (60°); it has two rotation centres of order three, which only differ by a rotation of 60° (or, equivalently, 180°), and three of order two, which only differ by a rotation of 60°. It has also reflections in six distinct directions. There are additional glide reflections in six distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

A pattern with this symmetry can be looked upon as atessellation of the plane with equal triangular tiles withD₃ 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 atriangular lattice with or without connecting lines, and ahexagonal tiling with one color for outlining the hexagons and one for the background.

Examples of groupp6m

• 
  Computer generated
• 
  Trihexagonal tiling
• 
  Small rhombitrihexagonal tiling
• 
  Great rhombitrihexagonal tiling
• 
  Persianglazed tile
• 
  Bronze vessel inNimroud,Assyria
• 
  Byzantinemarble pavement,Rome
• 
  Paintedporcelain,China
• 
  Paintedporcelain,China
• 
  Compact packing of two sizes of circle
• 
  Another compact packing of two sizes of circle
• 
  7co-uniform tiling
• 
  King's dress,Khorsabad,Assyria; this is almostp6m (ignoring inner parts of flowers, which make itcmm)

There are fivelattice types orBravais 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.

• In the 5 cases of rotational symmetry of order 3 or 6, the unit cell consists of two equilateral triangles (hexagonal lattice, itselfp6m). They form a rhombus with angles 60° and 120°.
• In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itselfp4m).
• In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itselfpmm). It may also be interpreted as a centered rhombic lattice. Special cases: square.
• In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itselfcmm). It may also be interpreted as a centered rectangular lattice. Special cases: square, hexagonal unit cell.
• In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell is in general a parallelogram (parallelogrammatic or oblique lattice, itselfp2). Special cases: rectangle, square, rhombus, hexagonal unit cell.

The actualsymmetry 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 ofdegrees of freedom are:

• 6 forp2
• 5 forpmm,pmg,pgg, andcmm
• 4 for the rest.

However, within each wallpaper group, all symmetry groups are algebraically isomorphic.

Some symmetry group isomorphisms:

• p1:Z²
• pm:Z ×D_∞
• pmm: D_∞ × D_∞.

• The wallpaper group of a pattern is invariant under isometries and uniformscaling (similarity transformations).
• Translational symmetry is preserved under arbitrary bijectiveaffine transformations.
• Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry.
• Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection. It changesp6m,p4g, andp3m1 intocmm,p3m1 intocm, andp4m, depending on direction of expansion/contraction, intopmm orcmm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction fromp6m top4m.

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.

• MadPattern, a free set of Adobe Illustrator templates that support the 17 wallpaper groups
• Tess, ashareware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.
• Wallpaper Symmetry is a free online JavaScript drawing tool supporting the 17 groups. Themain page has an explanation of the wallpaper groups, as well as drawing tools and explanations for the otherplanar symmetry groups as well.
• TALES GAME, a free software designed for educational purposes which includes the tessellation function.
• KaliArchived 2018-12-16 at theWayback Machine, online graphical symmetry editorJava applet (not supported by default in browsers).
• KaliArchived 2020-11-21 at theWayback Machine, free downloadable Kali for Windows and Mac Classic.
• Inkscape, afreevector graphics editor, supports all 17 groups plus arbitrary scales, shifts, rotates, and color changes per row or per column, optionally randomized to a given degree. (See[1])
• SymmetryWorks is a commercial plugin forAdobe Illustrator, supports all 17 groups.
• EscherSketch is a free online JavaScript drawing tool supporting the 17 groups.
• Repper is a commercial online drawing tool supporting the 17 groups plus a number of non-periodic tilings

Wikimedia Commons has media related toWallpaper group diagrams.

• List of planar symmetry groups (summary of this page)
• Aperiodic tiling
• Crystallography
• Layer group
• Mathematics and art
• M. C. Escher
• Point group
• Symmetry groups in one dimension
• Tessellation

• The Grammar of Ornament (1856), byOwen Jones. Many of the images in this article are from this book; it contains many more.
• John H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.),Groups, Combinatorics and Geometry, Proceedings of the L.M.S. Durham Symposium, July 5–15, Durham, UK, 1990; London Math. Soc. Lecture Notes Series165. Cambridge University Press, Cambridge. pp. 438–447
• John H. Conway, Heidi Burgiel andChaim Goodman-Strauss (2008):The Symmetries of Things. Worcester MA: A.K. Peters.ISBN 1-56881-220-5.
• Branko Grünbaum and G. C. Shephard (1987):Tilings and patterns. New York: Freeman.ISBN 0-7167-1193-1.
• Pattern Design, Lewis F. Day

• International Tables for Crystallography Volume A: Space-group symmetry by the International Union of Crystallography
• The 17 plane symmetry groupsArchived 2018-11-20 at theWayback Machine by David E. Joyce
• Introduction to wallpaper patterns by Chaim Goodman-Strauss and Heidi Burgiel
• Description by Silvio Levy
• Example tiling for each group, with dynamic demos of properties
• Overview with example tiling for each group, by Brian Sanderson
• Escher Web Sketch, a java applet with interactive tools for drawing in all 17 plane symmetry groups
• Burak, a Java applet for drawing symmetry groups.Archived 2009-02-18 at theWayback Machine
• A JavaScript app for drawing wallpaper patterns
• Circle-Pattern on Roman Mosaics in Greece
• Seventeen Kinds of Wallpaper PatternsArchived 2017-10-12 at theWayback Machine the 17 symmetries found in traditional Japanese patterns.
• Baloglou, George (2002)."An elementary, purely geometrical classification of the 17 planar crystallographic groups (wallpaper patterns)". Archived fromthe original on 2018-08-07. Retrieved2018-07-22.

• Crystallography
• Discrete groups
• Euclidean symmetries
• Ornaments

• CS1 German-language sources (de)
• Articles with short description
• Short description is different from Wikidata
• Webarchive template wayback links
• Commons category link is on Wikidata