Wallpaper group
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A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a twodimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.
Wallpaper groups are twodimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the threedimensional crystallographic groups (also called space groups).
Contents 
[edit] Introduction
Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.
Consider the following examples:

Examples A and B have the same wallpaper group; it is called p4m. Example C has a different wallpaper group, called p4g. 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.
A complete list of all seventeen possible wallpaper groups can be found below.
[edit] Symmetries of patterns
A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (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 which 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.
Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "bar code" in the centre of mass of each tile.
The types of transformations that are relevant here are called Euclidean plane isometries. For example:
 If we shift 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 pattern we started with. 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 we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain 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.
 We 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 we flip across a diagonal line, we do not get the same pattern back; what we do get is the original pattern shifted across by a certain distance. This is part of the reason that A and B have a different wallpaper group than C.
[edit] History
All 17 groups were used by Egyptian craftsmen, and used extensively in the Muslim world. A proof that there were only 17 possible patterns was first carried out by George Pólya in 1924.^{[1]}
[edit] 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 which 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 threedimensional case, we can equivalently restrict the affine transformations to those which 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.
[edit] 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 R^{2}. 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 R^{2}. (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 R^{2} 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.
[edit] The independent translations condition
The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R^{2}) such that the group contains both T_{v} and T_{w}.
The purpose of this condition is to distinguish wallpaper groups from frieze groups, which have only a single linearly independent translation, and from twodimensional 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. We 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.)
[edit] 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).
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, we 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 higherdimensional cases.
[edit] Notations for wallpaper groups
[edit] 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 we can use a similar notation for both kinds of groups, that of Carl Hermann and CharlesVictor Mauguin. An example of a full wallpaper name in HermannMauguin style 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 facecentred cell; these are explained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1fold (none), 2fold, 3fold, 4fold, or 6fold. 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 we choose that axis as 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 which 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. HermannMauguin notation for crystal space groups uses additional cell types.
Examples
 p2 (p211): Primitive cell, 2fold rotation symmetry, no mirrors or glide reflections.
 p4g (p4gm): Primitive cell, 4fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.
 cmm (c2mm): Centred cell, 2fold rotation, mirror axes both perpendicular and parallel to main axis.
 p31m (p31m): Primitive cell, 3fold rotation, mirror axis at 60°.
Here are all the names that differ in short and full notation.

Crystallographic short and full names Short p2 pm pg cm pmm pmg pgg cmm p4m p4g p6m Full p211 p1m1 p1g1 c1m1 p2mm p2mg p2gg c2mm p4mm p4gm p6mm
The remaining names are p1, p3, p3m1, p31m, p4, and p6.
[edit] Orbifold notation
Orbifold notation for wallpaper groups, introduced by John Horton Conway (Conway, 1992), is based not on crystallography, but on topology. We 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 nfold rotation. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.
 An asterisk, *, indicates a mirror. It interacts with the digits as follows:
 A cross, x, indicates a glide reflection. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so we do not notate them.
 The "no symmetry" symbol, o, stands alone, and indicates we have only lattice translations with no other symmetry.
Consider the group denoted in crystallographic notation by cmm; in Conway's notation, this will be 2*22. The 2 before the * says we have a 2fold rotation centre with no mirror through it. The * itself says we have a mirror. The first 2 after the * says we have a 2fold rotation centre on a mirror. The final 2 says we have an independent second 2fold rotation centre on a mirror, one which is not a duplicate of the first one under symmetries.
The group denoted by pgg will be 22x. We have two pure 2fold 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.

Conway and crystallographic correspondence Conway o xx *x ** 632 *632 Crystal. p1 pg cm pm p6 p6m Conway 333 *333 3*3 442 *442 4*2 Crystal. p3 p3m1 p31m p4 p4m p4g Conway 2222 22x 22* *2222 2*22 Crystal. p2 pgg pmg pmm cmm
[edit] Why there are exactly seventeen groups
An orbifold has a face, edges, and vertices; thus we can view it as a polygon. When we unfold it, that polygon tiles the plane, with each feature replicated infinitely by the action of the wallpaper symmetry group. Thus when Conway's orbifold notation mentions a feature, such as the 4fold rotation centre in 4*2, that feature unfolds into an infinite number of replicas across the plane. Hiding within this description is a key to the enumeration.
Consider a cube, with its corners, edges, and faces. We count 8 corners, 12 edges, and 6 faces. Alternately adding and subtracting, we note that 8 − 12 + 6 = 2. Now consider a tetrahedron. It has 4 corners, 6 edges, and 4 faces; and we note that 4 − 6 + 4 = 2. Let us explore further. For generality, use the term vertex instead of corner. Split a face with a new edge, causing one face to become two. Now we have 4 − 7 + 5 = 2. Next, split an edge with a new vertex, causing the one edge to become two. We have 5 − 8 + 5 = 2. This is not coincidence; it is a demonstration of the surface Euler characteristic, χ = V − E + F, and the beginning of a proof of its invariance.
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, we 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 before a * counts as (n−1)/n.
 A digit n after a * counts as (n−1)/2n.
 Both * and x 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 + 1/3 = 2
 4*2: 3/4 + 1 + 1/4 = 2
 22x: 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.
Incidentally, feature strings with other sums are not nonsense; they imply nonplanar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).
[edit] Guide to recognising wallpaper groups
To work out which wallpaper group corresponds to a given design, one may use the following table.
Least rotation 
Has reflection?  

Yes  No  
360° / 6  p6m  p6  
360° / 4 

p4  
360° / 3 

p3  
360° / 2 



none 


See also this overview with diagrams.
[edit] The seventeen groups
Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows:
On the righthand 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 which 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.
[edit] Group p1
 Orbifold notation: o.
 The group p1 contains only translations; there are no rotations, reflections, or glide reflections.
Examples of group p1
The two translations (cell sides) can each have different lengths, and can form any angle.
[edit] Group p2
 Orbifold notation: 2222.
 The group p2 contains four rotation centres of order two (180°), but no reflections or glide reflections.
Examples of group p2



[edit] Group pm
 Orbifold notation: **.
 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.)



[edit] Group pg
 Orbifold notation: xx.
 The group pg contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.
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.
[edit] Group cm
 Orbifold notation: *x.
 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






[edit] Group pmm
 Orbifold notation: *2222.
 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



[edit] Group pmg
 Orbifold notation: 22*.
 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




[edit] Group pgg
 Orbifold notation: 22x.
 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
[edit] Group cmm
 Orbifold notation: 2*22.
 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 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 2fold rotationally symmetric rectangular tile and its mirror image
Examples of group cmm



[edit] Group p4
 Orbifold notation: 442.
 The group p4 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 p4


[edit] Group p4m
 Orbifold notation: *442.
 The group p4m 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 alternating 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 (like on a checkerboard):

[edit] Group p4g
 Orbifold notation: 4*2.
 The group p4g 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.
In p4g there is a checkerboard pattern of 4fold rotational tiles and their mirror image, or looking at it differently (by shifting half a tile) a checkerboard pattern of horizontally and vertically symmetric tiles and their 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).
Note that the diagram on the left represents in area twice the smallest square that is repeated by translation.
Examples of group p4g



[edit] Group p3
 Orbifold notation: 333.
 The group p3 has three different rotation centres of order three (120°), but no reflections or glide reflections.
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. 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 hexagons of regular shape and equal size, with the sides corresponding to the smallest translations. 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. For a given image, nine of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the centres can be each of three selections of the red triangles, or of the blue or the green.
Examples of group p3


[edit] Group p3m1
 Orbifold notation: *333.
 The group p3m1 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 for p3, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, and both are symmetric, but the two are not equal, and not each other's mirror image. For a given image, three of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the vertices can be the red, the dark blue or the green triangles.
Examples of group p3m1





[edit] Group p31m
 Orbifold notation: 3*3.
 The group p31m 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 does not 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 for p3 and p3m1, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices can not be dark blue triangles.
Examples of group p31m

[edit] Group p6
 Orbifold notation: 632.
 The group p6 has one rotation centre of order six (60°); it has also 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 no reflections or glide reflections.
Examples of group p6

[edit] Group p6m
 Orbifold notation: *632.
 The group p6m has one rotation centre of order six (60°); it has also 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.
Examples of group p6m


[edit] Lattice types
There are five lattice types, 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 cell consists of two equilateral triangles (hexagonal lattice, itself p6m).
 In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself p4m).
 In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself pmm), therefore the diagrams show a rectangle, but a special case is that it actually is a square.
 In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself cmm); a special case is that it actually is a square.
 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 lattice, itself p2), therefore the diagrams show a parallelogram, but special cases are that it actually is a rectangle, rhombus, or square.
[edit] 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:
 6 for p2
 5 for pmm, pmg, pgg, and cmm
 4 for the rest
However, within each wallpaper group, all symmetry groups are algebraically isomorphic.
Some symmetry group isomorphisms:
 p1: Z^{2}
 pm: Z × D_{∞}
 pmm: D_{∞} × D_{∞}
[edit] Dependence of wallpaper groups on transformations
 The wallpaper group of a pattern is invariant under isometries and uniform scaling (similarity transformations).
 Translational symmetry is preserved under arbitrary bijective affine transformations.
 Rotational symmetry of order two ditto; this means also that 4 and 6fold rotation centres at least keep 2fold 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 changes p6m, p4g, and p3m1 into cmm, p3m1 into cm, and p4m, depending on direction of expansion/contraction, into pmm or cmm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from p6m to p4m.
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 4fold 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.
[edit] Web demo and software
There exist several software graphic tools that 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.
 Tess, a nagware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.
 Kali, free graphical symmetry editor available online and for download.
 Inkscape, a free vector 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.
 SymmetryWorks is a commercial plugin for Adobe Illustrator, supports all 17 groups.
 Arabeske is a free standalone tool, supports a subset of wallpaper groups.
[edit] See also
Wikimedia Commons has media related to: Wallpaper group diagrams 
 List of planar symmetry groups (summary of this page)
 Tessellation
 Point group
 Crystallography
 Symmetry groups in one dimension
 M. C. Escher
 Aperiodic tiling
[edit] References
 ^ Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 0691023743
 The Grammar of Ornament (1856), by Owen Jones. Many of the images in this article are from this book; it contains many more.
 J. 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 Series 165. Cambridge University Press, Cambridge. pp. 438–447
 Grünbaum, Branko; Shephard, G. C. (1987): Tilings and Patterns. New York: Freeman. ISBN 0716711931.
 Pattern Design, Lewis F. Day
[edit] External links
 The 17 plane symmetry groups by David E. Joyce
 Introduction to wallpaper patterns by Chaim GoodmanStrauss and Heidi Burgiel
 Description by Silvio Levy
 Example tiling for each group, with dynamic demos of properties
 Overview with example tiling for each group
 Tiling plane and fancy by Steve Edwards
 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.