Wallpaper Group

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.

What this page calls pattern

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 $\text\,a.$ Another ''repetitive'' square has an $\text\,~a.~$ 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 ''C '' is a vertex of a repetitive pseudo‑ rhombus with thick stripes on its whole surface, called pseudo‑rhombus because of a concentric repetitive rhombus $\text\,~a,~$ 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 $\text\,~a.~$ Conversely, from elementary geometric tiles edge‑to‑edge, an artist like  M. C. Escher created attractive surfaces many times repeated. On image 2,  $a~\text$ 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.

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  $\mathit\text\mathit^~$ on image 1, 3 or 4, or a ± 120° rotation around a point like ''S '' 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 $\,\text\,\mathit\,\text\,\mathit\,^\;$ 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 ''p''1, 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:
Image:Wallpaper_group-p4m-2.jpg, Example A: Cloth, Tahiti
Image:Wallpaper_group-p4m-1.jpg, Example B: Ornamental painting, Nineveh, Assyria
Image:Wallpaper_group-p4g-2.jpg, Example C: Painted porcelain, China

Examples A and B have the same wallpaper group; it is called ''p''4''m'' in the IUCr notation and *442 in the orbifold notation. Example C has a different wallpaper group, called ''p''4''g'' 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 and then derived independently by George Pólya in 1924. 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 .

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:

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

However, example C is ''different''. It only has reflections in horizontal and vertical directions, ''not'' across diagonal axes. If 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.

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).
*Translations, denoted by ''T''_''v'', where ''v'' is a vector in R². This has the effect of shifting the plane applying displacement vector ''v''.
* Rotations, denoted by ''R''_''c'',''θ'', where ''c'' is a point in the plane (the centre of rotation), and ''θ'' is the angle of rotation.
* Reflections, or mirror isometries, denoted by ''F''_''L'', where ''L'' is a line in R². (''F'' is for "flip"). This has the effect of reflecting the plane in the line ''L'', called the reflection axis or the associated mirror.
*Glide reflections, denoted by ''G''_''L'',''d'', where ''L'' is a line in R² and ''d'' is a distance. This is a combination of a reflection in the line ''L'' and a translation along ''L'' by a distance ''d''.

The independent translations condition

The condition on linearly independent translations means that there exist linearly independent vectors ''v'' and ''w'' (in R²) 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

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, 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 ''p''31''m'', 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.

;Examples
* ''p''2 (''p''2): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
* ''p''4''gm'' (''p''4''mm''): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.
* ''c''2''mm'' (''c''2''mm''): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.
* ''p''31''m'' (''p''31''m''): Primitive cell, 3-fold rotation, mirror axis at 60°.

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

:

The remaining names are ''p''1, ''p''2, ''p''3, ''p''3''m''1, ''p''31''m'', ''p''4, and ''p''6.

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:
*#Digits before * denote centres of pure rotation (cyclic).
*#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.

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:
*A digit ''n'' without or before a * counts as .
*A digit ''n'' after a * counts as .
*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: + + = 2
*3*3: + 1 + = 2
*4*2: + 1 + = 2
*22×: + + 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 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.

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):

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 ''p''1 (o)

* Orbifold signature: o
* Coxeter notation (rectangular): infin;⁺,2,∞⁺or infin;sup>+× infin;sup>+
* Lattice: oblique
* Point group: C₁
* The group ''p''1 contains only translations; there are no rotations, reflections, or glide reflections.
;Examples of group ''p''1

Image:WallpaperP1.GIF, Computer generated

Image:Wallpaper_group-p1-3.jpg, Medieval wall diapering

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

Group ''p''2 (2222)

* Orbifold signature: 2222
* Coxeter notation (rectangular): infin;,2,∞sup>+
* Lattice: oblique
* Point group: C₂
* The group ''p''2 contains four rotation centres of order two (180°), but no reflections or glide reflections.

;Examples of group ''p''2

Image:WallpaperP2.GIF, Computer generated
Image:Wallpaper_group-p2-1.jpg, Cloth, Sandwich Islands (Hawaii)
Image:Wallpaper_group-p2-2.jpg, Mat on which an Egyptian king stood
Image:Wallpaper_group-p2-2 detail 2.jpg, Egyptian mat (detail)
Image:Wallpaper_group-p2-3.jpg, Ceiling of an Egyptian tomb
Image:Wallpaper_group-p2-4.jpg, Wire fence, U.S.

Group ''pm'' (**)

* Orbifold signature: **
* Coxeter notation: infin;,2,∞⁺or infin;⁺,2,∞* Lattice: rectangular
* Point group: D₁
* The group ''pm'' has no rotations. It has reflection axes, they are all parallel.

;Examples of group ''pm''

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

Image:WallpaperPM.gif, Computer generated
Image:Wallpaper_group-pm-3.jpg, Dress of a figure in a tomb at Biban el Moluk, Egypt
Image:Wallpaper_group-pm-4.jpg, Egyptian tomb, Thebes
Image:Wallpaper_group-pm-1.jpg, Ceiling of a tomb at Gourna, Egypt. Reflection axis is diagonal
Image:Wallpaper_group-pm-5.jpg, Indian metalwork at the Great Exhibition in 1851. This is almost ''pm'' (ignoring short diagonal lines between ovals motifs, which make it ''p''1)

Group ''pg'' (××)

* Orbifold signature: ××
* Coxeter notation: ∞,2)⁺,∞⁺or infin;⁺,(2,∞)⁺* Lattice: rectangular
* Point group: D₁
* The group ''pg'' contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.

;Examples of group ''pg''

Image:WallpaperPG.GIF, Computer generated
Image:Wallpaper_group-pg-1.jpg, Mat with herringbone pattern on which Egyptian king stood
Image:Wallpaper_group-pg-1 detail.jpg, Egyptian mat (detail)
Image:Wallpaper_group-pg-2.jpg, Pavement with herringbone pattern in Salzburg. Glide reflection axis runs northeast–southwest
Image:Tile 33434.svg, 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 ''p''4''g'' (see there for this image with equally colored triangles)If one thinks of the squares as the background, then one can see a simple patterns of rows of rhombuses.

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'' (*×)

* Orbifold signature: *×
* Coxeter notation: infin;⁺,2⁺,∞or infin;,2⁺,∞⁺* Lattice: rhombic
* Point group: D₁
* The group ''cm'' contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis is ''not'' 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 group ''cm''

Image:WallpaperCM.GIF, Computer generated
Image:Wallpaper_group-cm-1.jpg, Dress of Amun, from Abu Simbel, Egypt
Image:Wallpaper_group-cm-2.jpg, Dado from Biban el Moluk, Egypt
Image:Wallpaper_group-cm-3.jpg, Bronze vessel in Nimroud, Assyria
Image:Wallpaper_group-cm-4.jpg, Spandrils of arches, the Alhambra, Spain
Image:Wallpaper_group-cm-5.jpg, Soffitt of arch, the Alhambra, Spain
Image:Wallpaper_group-cm-6.jpg, Persian tapestry
Image:Wallpaper_group-cm-7.jpg, Indian metalwork at the Great Exhibition in 1851
Image:Wallpaper_group-pm-2.jpg, Dress of a figure in a tomb at Biban el Moluk, Egypt

Group ''pmm'' (*2222)

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

;Examples of group ''pmm''

Image:Wallpaper_group-pmm-1.jpg, 2D image of lattice fence, U.S. (in 3D there is additional symmetry)
Image:Wallpaper_group-pmm-2.jpg, Mummy case stored in The Louvre
Image:Wallpaper_group-pmm-4.jpg, Mummy case stored in The Louvre. Would be type ''p''4''m'' except for the mismatched coloring

Group ''pmg'' (22*)

* Orbifold signature: 22*
* Coxeter notation: ∞,2)⁺,∞or infin;,(2,∞)⁺* Lattice: rectangular
* Point group: D₂
* The group ''pmg'' 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 group ''pmg''

Image:WallpaperPMG.GIF, Computer generated
Image:Wallpaper_group-pmg-1.jpg, Cloth, Sandwich Islands (Hawaii)
Image:Wallpaper_group-pmg-2.jpg, Ceiling of Egyptian tomb
Image:Wallpaper_group-pmg-3.jpg, Floor tiling in Prague, the Czech Republic
Image:Wallpaper_group-pmg-4.jpg, Bowl from Kerma
Image:2-d pentagon packing.svg, Pentagon packing

Group ''pgg'' (22×)

* Orbifold signature: 22×
* Coxeter notation (rectangular): (∞,2)⁺,(∞,2)⁺)* Coxeter notation (square): ⁺,4⁺* Lattice: rectangular
* Point group: D₂
* The group ''pgg'' 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 group ''pgg''

Image:WallpaperPGG.GIF, Computer generated
Image:Wallpaper_group-pgg-1.jpg, Bronze vessel in Nimroud, Assyria
Image:Wallpaper_group-pgg-2.jpg, Pavement in Budapest, Hungary

Group ''cmm'' (2*22)

* Orbifold signature: 2*22
* Coxeter notation (rhombic): infin;,2⁺,∞* Coxeter notation (square): 4,4,2⁺)* Lattice: rhombic
* Point group: D₂
* The group ''cmm'' has reflections in two perpendicular directions, and a rotation of order two (180°) whose centre is ''not'' on a reflection axis. It also has two rotations whose centres ''are'' on a reflection axis.
*This group is frequently seen in everyday life, since the most common arrangement of bricks 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 group ''cmm''

Image:WallpaperCMM.GIF, Computer generated
File:1-uniform_n8.svg, Elongated triangular tiling
Image:Wallpaper_group-cmm-1.jpg, Suburban brick wall using running bond arrangement, U.S.
Image:Wallpaper_group-cmm-2.jpg, Ceiling of Egyptian tomb. Ignoring colors, this would be ''p''4''g''
Image:Wallpaper_group-cmm-3.jpg, Egyptian
Image:Wallpaper_group-cmm-4.jpg, Persian tapestry
Image:Wallpaper_group-cmm-5.jpg, Egyptian tomb
Image:Wallpaper_group-cmm-6.jpg, Turkish dish
Image:2-d dense packing r1.svg, A compact packing of two sizes of circle
Image:2-d dense packing r3.svg, Another compact packing of two sizes of circle
Image:2-d dense packing r7.svg, Another compact packing of two sizes of circle

Group ''p''4 (442)

* Orbifold signature: 442
* Coxeter notation: ,4sup>+
* Lattice: square
* Point group: C₄
* The group ''p''4 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 group ''p''4

A ''p''4 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 smaller and rotated 45°.

Image:WallpaperP4.GIF, Computer generated
Image:Wallpaper_group-p4-1.jpg, Ceiling of Egyptian tomb; ignoring colors this is ''p''4, otherwise ''p''2
Image:Wallpaper_group-p4-2.jpg, Ceiling of Egyptian tomb
Image:A_wallpaper_pattern_Overlaid_patterns.svg, Overlaid patterns
Image:Wallpaper_group-p4-3.jpg, Frieze, the Alhambra, Spain. Requires close inspection to see why there are no reflections
Image:Wallpaper_group-p4-4.jpg, Viennese cane
Image:Wallpaper_group-p4-5.jpg, Renaissance earthenware
File:A tri-colored Pythagorean tiling View 4.svg, Pythagorean tiling
File:Lizard p4 p4.png, Generated from a photograph

Group ''p''4''m'' (*442)

* Orbifold signature: *442
* Coxeter notation: ,4* Lattice: square
* Point group: D₄
* The group ''p''4''m'' 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 a checkerboard pattern of two of such squares.

;Examples of group ''p''4''m''

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

Image:WallpaperP4M.GIF, Computer generated
Image:1-uniform_n5.svg, Square tiling
Image:Tile V488.svg, Tetrakis square tiling; ignoring colors, this is ''p''4''m'', otherwise ''c''2''m''
Image:Tile 488.svg, Truncated square tiling (ignoring color also, with smaller translations)
Image:Wallpaper_group-p4m-1.jpg, Ornamental painting, Nineveh, Assyria
Image:Wallpaper_group-p4m-3.jpg, Storm drain, U.S.
Image:Wallpaper_group-p4m-5.jpg, Egyptian mummy case
Image:Wallpaper_group-p4m-6.jpg, Persian glazed tile
Image:2-d dense packing r4.svg, Compact packing of two sizes of circle

Examples displayed with the smallest translations diagonal:

Image:Tile 4,4.svg, checkerboard
Image:Wallpaper_group-p4m-2.jpg, Cloth, Otaheite ( Tahiti)
Image:Wallpaper_group-p4m-4.jpg, Egyptian tomb
Image:Wallpaper_group-p4m-7.jpg, Cathedral of Bourges
Image:Wallpaper_group-p4m-8.jpg, Dish from Turkey, Ottoman period

Group ''p''4''g'' (4*2)

* Orbifold signature: 4*2
* Coxeter notation: ⁺,4* Lattice: square
* Point group: D₄
* The group ''p''4''g'' 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.
A ''p''4''g'' 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 ''p''4''m'' (with diagonal translation cells).

;Examples of group ''p''4''g''

Image:Wallpaper_group-p4g-1.jpg, Bathroom linoleum, U.S.
Image:Wallpaper_group-p4g-2.jpg, Painted porcelain, China
Image:Wallpaper_group-p4g-3.jpg, Fly screen, U.S.
Image:Wallpaper_group-p4g-4.jpg, Painting, China
File:Uniform tiling 44-h01.png, one of the colorings of the snub square tiling (see also at ''pg'')

Group ''p''3 (333)

* Orbifold signature: 333
* Coxeter notation: 3,3,3)sup>+ or ^[3/sup>.html"_;"title=".html"_;"title="[3">[3/sup>">.html"_;"title="[3">^[3/sup>sup>+
*_Lattice:_hexagonal
*_Point_group:_C₃
*_The_group_''p''3_has_three_different_rotation_centres_of_order_three_(120°),_but_no_reflections_or_glide_reflections.

Imagine_a_tessellation_

A_tessellation_or_tiling_is_the_covering_of_a__surface,_often_a__plane,_using_one_or_more__geometric_shapes,_called_''tiles'',_with_no_overlaps_and_no_gaps._In_mathematics,_tessellation_can_be_generalized_to_higher_dimensions_and_a_variety_of_...