Example of an Egyptian design with wallpaper group p4m

A wallpaper group (or plane symmetry group or plane crystallographic group) is 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. There are 17 possible distinct groups. Download high resolution version (1074x1063, 563 KB)Example of wallpaper group type p4m. ... Download high resolution version (1074x1063, 563 KB)Example of wallpaper group type p4m. ... Sphere symmetry group o. ... The Parthenon on top of the Acropolis, Athens, Greece Architecture (from Latin, architectura and ultimately from Greek, αρχιτεκτων, a master builder, from αρχι- chief, leader and τεκτων, builder, carpenter) is the art and science of designing buildings and structures. ... The decorative arts are traditionally defined as ornamental and functional works in ceramic, wood, glass, metal, or textile. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups). The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ... A crystallographic group is a topologically discrete subgroup of the group of isometries of some geometric space (typically, not necessarily a Euclidean space) with a compact fundamental domain. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...

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:

Example A: Cloth, Tahiti

Example B: Ornamental painting, Nineveh, Assyria

Example C: Painted porcelain, China

Download high resolution version (997x986, 432 KB)Example of wallpaper group type p4m. ... Download high resolution version (997x986, 432 KB)Example of wallpaper group type p4m. ... Map of French Polynesia Map of Tahiti and Moorea Tahiti is the largest island of French Polynesia, located in the archipelago of Society Islands in the southern Pacific Ocean at . ... Download high resolution version (1065x1057, 424 KB)Example of wallpaper group type p4m. ... Download high resolution version (1065x1057, 424 KB)Example of wallpaper group type p4m. ... , For other uses, see Nineveh (disambiguation). ... Relief from Assyrian capital of Dur Sharrukin, showing transport of Lebanese cedar (8th c. ... Download high resolution version (1054x1069, 435 KB)Example of wallpaper group type p4g. ... Download high resolution version (1054x1069, 435 KB)Example of wallpaper group type p4g. ... It has been suggested that Porcelain tile be merged into this article or section. ...

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.

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 first stripe "disappears" and a new stripe is "added" at the end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored. Sphere symmetry group o. ...

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, because 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: In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. ...

• 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 to C. Look up translate in Wiktionary, the free dictionary. ... A sphere rotating around its axis. ... Look up reflection in Wiktionary, the free dictionary. ...

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. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

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). In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In a group, the conjugate by g of h is ghg-1. ...

Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those which preserve orientation. The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...

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). A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ...

2D patterns with double translational symmetry can be categorized according to their symmetry group type. The symmetry group of an object (e. ...

Isometries of the Euclidean plane

Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information). In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. ...

• 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.

In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In Newtonian mechanics, displacement is one of two subtly different quantities measuring distance and direction. ... In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...

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 have only a single linearly independent translation, 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. A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ... The symmetry group of an object (e. ...

(It is possible to generalise this situation. We could for example study discrete groups of isometries of Rⁿ with m linearly independent translations, where m is any integer in the range 0 ≤ m ≤ n.)

The discreteness condition

The discreteness condition means that there is some positive real number ε, such that for every translation 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. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

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. The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ...

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 we 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 is p31m, with four letters or digits; more usual is a shortened name like cmm or pg. The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ... German professor of crystallography Carl Hermann (1898 June 17–1961 September 12) was an inventor (with Charles-Victor Mauguin) of an international standard notation for crystallographic groups. ... French professor of mineralogy Charles-Victor Mauguin (July 19, 1878 – April 25, 1958) was a founder of the International Union of Crystallography (IUCr), and inventor (with Carl Hermann) of an international standard notation for crystallographic groups. ...

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 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. In solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum volume cell corresponding to a single lattice point. ...

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. Hermann-Mauguin notation for crystal space groups uses additional cell types. The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...

Examples

• p2 (p211): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
• p4g (p4gm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.
• cmm (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	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.

Conway notation

Conway's 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. To meet Wikipedias quality standards, this article or section may require cleanup. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ...

• A digit, n, indicates a centre of n-fold 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:
  1. Digits before * are centres of pure rotation (cyclic).
  2. Digits after * are centres of rotation with mirrors through them (dihedral).

• 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 2-fold rotation centre with no mirror through it. The * itself says we have a mirror. The first 2 after the * says we have a 2-fold rotation centre on a mirror. The final 2 says we have an independent second 2-fold rotation centre on a mirror, one which is not a duplicate of the first one under symmetries. In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ... This article may be confusing for some readers, and should be edited to enhance clarity. ...

The group denoted by pgg will be 22x. We have 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.

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

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 4-fold 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. Look up Polygon in Wiktionary, the free dictionary. ...

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's 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. A cube [1] (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, facets or sides, with three meeting at each vertex. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

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. In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

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 non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical.) // List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ...

Guide to recognising 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.

Key to diagrams

Each group in the following list has two cell structure diagrams, which are interpreted as follows:

	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. Image File history File links Wallpaper_group_diagram_legend_rotation2. ... Image File history File links Wallpaper_group_diagram_legend_rotation3. ... Image File history File links Wallpaper_group_diagram_legend_rotation4. ... Image File history File links Wallpaper_group_diagram_legend_rotation6. ... Image File history File links Wallpaper_group_diagram_legend_reflection. ... Image File history File links Wallpaper_group_diagram_legend_glide_reflection. ...

The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern which is repeated. In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...

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. See lattice for other meanings of this term, both within and without mathematics. ...

The seventeen groups

Group p1

Cell structure for p1

Cell structure for o

• Orbifold notation: o.
• The group p1 contains only translations; there are no rotations, reflections, or glide reflections.

Examples of group p1 Cell diagram for wallpaper group type p1. ... Image File history File links Wallpaper_group_diagram_p1. ... Image File history File links Wallpaper_group_diagram_p1. ...

Computer generated

Tomb, Thebes, Egypt. Easily confused with type p2.

Persian tapestry

Mediæval wall diapering

The two translations (cell sides) can each have different lengths, and can form any angle. Example of wallpaper plane class p1 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1093x1047, 367 KB)Example of wallpaper group type p1. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... For the ancient capital of Boeotia, see Thebes, Greece. ... Download high resolution version (1043x1095, 329 KB)Example of wallpaper group type p1. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... This article is about tapestry the textile. ... Download high resolution version (698x923, 235 KB)Example of wallpaper group type p1. ... The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... In Heraldry, Diapering (from an old term for damask) is a technique in which those who draw achievements of arms attempt to decorate large areas of flat colour by drawing crosshatches or arabesques. ...

Group p2

Cell structure for p2

Cell structure for 2222

• Orbifold notation: 2222.
• The group p2 contains four rotation centres of order two (180°), but no reflections or glide reflections.

Examples of group p2 Cell diagram for wallpaper group type p2. ... Image File history File links Wallpaper_group_diagram_p2. ... Image File history File links Wallpaper_group_diagram_p2. ...

Computer generated	Cloth, Sandwich Islands (Hawaii)	Mat on which Egyptian king stood	Egyptian mat (detail)
Ceiling of Egyptian tomb	Wire fence, U.S.

Example of wallpaper plane class p2 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1018x980, 381 KB)Example of wallpaper group type p2. ... The Sandwich Islands was the name given to Hawaii by Captain James Cook on his discovery of the islands on January 18, 1778. ... This article or section does not cite its references or sources. ... Download high resolution version (1070x1066, 639 KB)Example of wallpaper group type p2. ... Image File history File links Wallpaper_group-p2-2_detail_2. ... Download high resolution version (1085x1075, 534 KB)Example of wallpaper group type p2. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (966x882, 298 KB)Example of wallpaper group type p2. ... A fence is a freestanding structure designed to restrict or prevent movement across a boundary. ... For other uses, see United States (disambiguation) and US (disambiguation). ...

Group pm

Cell structure for pm

Cell structure for **

• Orbifold notation: **.
• The group pm has no rotations. It has reflection axes, they are all parallel.

Examples of group pm Cell diagram for wallpaper group type pm. ... Image File history File links Wallpaper_group_diagram_pm. ... Image File history File links Wallpaper_group_diagram_pm. ...

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

Computer generated	Dress of a figure in a tomb at Biban el Moluk, Egypt	Egyptian tomb, Thebes	Ceiling of a tomb at Gourna, Egypt. Reflection axis is diagonal.
Indian metalwork at the Great Exhibition in 1851. The plane group is almost pm (ignoring short diagonal lines between ovals motifs).

Example of wallpaper plane class pm File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1033x994, 320 KB)Example of wallpaper group type pm. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Valley of the Kings The Valley of the Kings, or Wadi el-Muluk (وادي الملوك) in Arabic, is a valley in Egypt where tombs were built for the Pharaohs of the New Kingdom, the Eighteenth through Twentieth Dynasties. ... Download high resolution version (1078x1075, 353 KB)Example of wallpaper group type pm. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... For the ancient capital of Boeotia, see Thebes, Greece. ... Download high resolution version (1082x1076, 468 KB)Example of wallpaper group type pm. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (1064x1074, 475 KB)Example of wallpaper group type pm. ... The Great Exhibition: Paxtons Crystal Palace enclosed full-grown trees in Hyde Park. ...

Group pg

Cell structure for pg

Cell structure for xx

• 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 Cell diagram for wallpaper group type pg. ... Image File history File links Wallpaper_group_diagram_pg. ... Image File history File links Wallpaper_group_diagram_pg. ...

Computer generated

Mat on which Egyptian king stood

Egyptian mat (detail)

Pavement in Salzburg. Glide reflection axis runs northeast-southwest.

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. Example of wallpaper plane class pg File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1091x1064, 653 KB)Example of wallpaper group type pg. ... Image File history File links Wallpaper_group-pg-1_detail. ... Image File history File links Download high resolution version (2568x1944, 1021 KB) Summary Computer-enhanced photograph of pavement near the Nonnberg Abbey, Salzburg, Austria, illustrating wallpaper group pg. Licensing I, the creator of this work, hereby release it into the public domain. ... Flag of Salzburg Salzburg (population 145,000 in 2005) is a city in western Austria and the capital of the federal state of Salzburg (population 520,000 in 2003). ...

Ignoring the wavy borders of the tiles, the pavement is pgg.

Group cm

Cell structure for cm

Cell structure for *x

• 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 groups 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. Cell diagram for wallpaper group type cm. ... Image File history File links Wallpaper_group_diagram_cm. ... Image File history File links Wallpaper_group_diagram_cm. ...

Examples of group cm

Computer generated	Dress of Amun, from Abu Simbel, Egypt	Dado from Biban el Moluk, Egypt	Bronze vessel in Nimroud, Assyria
Spandrils of arches, the Alhambra, Spain	Soffitt of arch, the Alhambra, Spain	Persian tapestry	Indian metalwork at the Great Exhibition in 1851
Dress of a figure in a tomb at Biban el Moluk, Egypt

Example of wallpaper class cm File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1088x1054, 394 KB)Example of wallpaper group type cm. ... Amun (also spelt Amon, Amoun, Amen, and rarely Imenand, and spelt in Greek as Ammon, and Hammon) was the name of a deity, in Egyptian mythology, who gradually rose to become one of the most important deities, before fading into obscurity. ... Model showing the relative positions of the Abu Simbel temples before and after relocation Categories: Ancient Egypt stubs | Wonders of the World ... Download high resolution version (1078x1066, 383 KB)Example of wallpaper group type cm. ... The lower part of a wall, below the dado rail and above the skirting board. ... Valley of the Kings The Valley of the Kings, or Wadi el-Muluk (وادي الملوك) in Arabic, is a valley in Egypt where tombs were built for the Pharaohs of the New Kingdom, the Eighteenth through Twentieth Dynasties. ... Download high resolution version (1050x1052, 517 KB)Example of wallpaper group type cm. ... ... Nimrud is a ancient Assyrian city located south of Nineveh on the river Tigris. ... Relief from Assyrian capital of Dur Sharrukin, showing transport of Lebanese cedar (8th c. ... Download high resolution version (1123x1729, 800 KB)Example of wallpaper group type cm. ... In the masonry arch, a spandrel or spandril (formerly splaundrel, a word of unknown origin) is the space between two arches. ... Isometric view of a typical arch An arch is a curved structure capable of spanning a space while supporting significant weight (e. ... View of the Alhambra from the Mirador St Nicolas in the Albaycin of Granada. ... Download high resolution version (1626x1523, 1212 KB)Example of wallpaper group type cm. ... Soffit (from Fr. ... View of the Alhambra from the Mirador St Nicolas in the Albaycin of Granada. ... Download high resolution version (1132x1282, 396 KB)Example of wallpaper group type cm. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... This article is about tapestry the textile. ... Download high resolution version (1067x1064, 417 KB)Example of wallpaper group type cm. ... The Great Exhibition: Paxtons Crystal Palace enclosed full-grown trees in Hyde Park. ... Download high resolution version (1080x946, 290 KB)Example of wallpaper group type pm. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Valley of the Kings The Valley of the Kings, or Wadi el-Muluk (وادي الملوك) in Arabic, is a valley in Egypt where tombs were built for the Pharaohs of the New Kingdom, the Eighteenth through Twentieth Dynasties. ...

Group pmm

Cell structure for pmm

Cell structure for *2222

• 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 Cell diagram for wallpaper group type pmm. ... Image File history File links Wallpaper_group_diagram_pmm. ... Image File history File links Wallpaper_group_diagram_pmm. ...

Computer generated	2D image of lattice fence, U.S. (in 3D there is additional symmetry)	Mummy case stored in The Louvre	Ceiling of Egyptian tomb (almost cmm)
Mummy case stored in The Louvre. Would be type p4 except for the mismatched coloring.

Example of wallpaper plane class pmm File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1029x951, 380 KB)Example of wallpaper group type pmm. ... A fence is a freestanding structure designed to restrict or prevent movement across a boundary. ... For other uses, see United States (disambiguation) and US (disambiguation). ... Download high resolution version (1076x1069, 562 KB)Example of wallpaper group type pmm. ... A mummy is a corpse whose skin and dried flesh have been preserved by either intentional or accidental exposure to chemicals, extreme cold or dryness, or airlessness. ... The main courtyard of the Louvre. ... Download high resolution version (1086x1064, 450 KB)Example of wallpaper group type pmm. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (1070x1068, 490 KB)Example of wallpaper group type pmm. ... A mummy is a corpse whose skin and dried flesh have been preserved by either intentional or accidental exposure to chemicals, extreme cold or dryness, or airlessness. ... The main courtyard of the Louvre. ...

Group pmg

Cell structure for pmg

Cell structure for 22*

• 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 Cell diagram for wallpaper group type pmg. ... Image File history File links Wallpaper_group_diagram_pmg. ... Image File history File links Wallpaper_group_diagram_pmg. ...

Computer generated	Cloth, Sandwich Islands (Hawaii)	Ceiling of Egyptian tomb	Floor tiling in Prague, the Czech Republic
Bowl from Kerma

Example of wallpaper plane class pmg File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (993x1011, 499 KB)Example of wallpaper group type pmg. ... The Sandwich Islands was the name given to Hawaii by Captain James Cook on his discovery of the islands on January 18, 1778. ... This article or section does not cite its references or sources. ... Download high resolution version (1066x1050, 489 KB)Example of wallpaper group type pmg. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (1050x968, 429 KB)Example of wallpaper group type pmg. ... This article or section does not cite its references or sources. ... Image File history File links Download high resolution version (1894x1650, 313 KB) Summary Computer-enhanced photograph of a bowl on display in the Museum of Fine Arts, Boston (item number 13. ... The Kingdom of Kerma was a state in Nubia from around 2000 BCE to about 1520 BCE. It was based in the city of Kerma in Upper Nubia and emerged as a major centre during the Middle Kingdom period in Egypt The town is marked by large and expensive tombs. ...

Group pgg

Cell structure for pgg

Cell structure for 22x

• 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 Cell diagram for wallpaper group type pgg. ... Image File history File links Wallpaper_group_diagram_pgg. ... Image File history File links Wallpaper_group_diagram_pgg. ...

Computer generated

Bronze vessel in Nimroud, Assyria

Pavement in Budapest, Hungary. Glide reflection axes are diagonal.

Example of wallpaper plane class pgg File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1041x1069, 589 KB)Example of wallpaper group type pgg. ... ... Nimrud is a ancient Assyrian city located south of Nineveh on the river Tigris. ... Relief from Assyrian capital of Dur Sharrukin, showing transport of Lebanese cedar (8th c. ... Download high resolution version (1150x910, 393 KB)Example of wallpaper group type pgg. ... This article is about the American English usage of pavement as the durable surfacing of roads and walkways. ... Nickname: Paris of the East, Pearl of the Danubeor Queen of the Danube Location of Budapest in Hungary Country Hungary County Pest Mayor Gábor Demszky (SZDSZ) Area    - City 525,16 km²  - Land n/a km²  - Water n/a km² Population    - City (2006) 1,695,000  - Density 3570/km...

Group cmm

Cell structure for cmm

Cell structure for 2*22

• 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. Cell diagram for wallpaper group type cmm. ... Image File history File links Wallpaper_group_diagram_cmm. ... Image File history File links Wallpaper_group_diagram_cmm. ... An old brick wall in English bond laid with alternating courses of headers and Brick is an artificial stone made by forming clay into rectangular blocks which are hardened, either by burning in a kiln or sometimes, in warm and sunny countries, by sun-drying. ...

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

Computer generated	one of the 8 semi-regular tessellations; ignoring color this is this group cmm, otherwise group p1	Suburban brick wall, U.S.	Ceiling of Egyptian tomb. It is group cmm if the colors are taken into account (they reduce the symmetry), otherwise it is p4g.
Egyptian	Persian tapestry	Egyptian tomb	Turkish dish

Example of wallpaper class cmm File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Tile_33344. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Download high resolution version (1528x1528, 928 KB)Illustration of cmm type Wallpaper group (bricks on side of garage) File links The following pages link to this file: Wallpaper group Talk:Wallpaper group Categories: Public domain images ... An old brick wall in English bond laid with alternating courses of headers and Brick is an artificial stone made by forming clay into rectangular blocks which are hardened, either by burning in a kiln or sometimes, in warm and sunny countries, by sun-drying. ... For other uses, see United States (disambiguation) and US (disambiguation). ... Download high resolution version (1058x1050, 591 KB)Example of wallpaper group type cmm. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (1095x1056, 417 KB)Example of wallpaper group type cmm. ... Download high resolution version (1086x1752, 469 KB)Example of wallpaper group type cmm. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... This article is about tapestry the textile. ... Download high resolution version (1084x1054, 527 KB)Example of wallpaper group type cmm. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Image File history File links Download high resolution version (1656x1701, 841 KB) Summary Computer-enhanced photograph of a dish on display in the Metropolitan Museum of Art in New York City, item number 14. ...

Group p4

Cell structure for p4

Cell structure for 442

• 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 Cell diagram for wallpaper group type p4. ... Image File history File links Wallpaper_group_diagram_p4. ... Image File history File links Wallpaper_group_diagram_p4. ...

Computer generated	one of the 8 semi-regular tessellations	Ceiling of Egyptian tomb; ignoring colors this is p4, otherwise p2	Ceiling of Egyptian tomb
Frieze, the Alhambra, Spain. Requires close inspection to see why there are no reflections.	Viennese cane	Renaissance earthernware

Example of wallpaper plane class p4 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Tile_33434. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Download high resolution version (1064x1070, 572 KB)Example of wallpaper group type p4. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (1063x1073, 515 KB)Example of wallpaper group type p4. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (1498x1205, 1103 KB)Example of wallpaper group type p4. ... View of the Alhambra from the Mirador St Nicolas in the Albaycin of Granada. ... Download high resolution version (1050x958, 325 KB)Example of wallpaper group type p4. ... Download high resolution version (1098x1076, 551 KB)Example of wallpaper group type p4. ...

Group p4m

Cell structure for p4m

Cell structure for *442

• 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 Cell diagram for wallpaper group type p4m. ... Image File history File links Wallpaper_group_diagram_p4m. ... Image File history File links Wallpaper_group_diagram_p4m. ... 5 by 5 checkerboard pattern A checkerboard (or chequerboard) is a board on which American checkers is played. ...

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

Computer generated	one of the 3 regular tessellations (in this checkerboard coloring, smallest translations are diagonal)	Demiregular tiling with triangles (in this coloring, the smallest translations are orthogonal)	one of the 8 semi-regular tessellations (ignoring color also, with smaller translations)
Ornamental painting, Nineveh, Assyria	Storm drain, U.S.	Egyptian mummy case	Persian glazed tile

Examples displayed with the smallest translations diagonal (like on a checkerboard): Example of wallpaper plane class p4m File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Tile_4,4. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Image File history File links Tile_V488. ... In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. ... Image File history File links Tile_488. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Download high resolution version (1065x1057, 424 KB)Example of wallpaper group type p4m. ... , For other uses, see Nineveh (disambiguation). ... Relief from Assyrian capital of Dur Sharrukin, showing transport of Lebanese cedar (8th c. ... Download high resolution version (1389x1242, 490 KB)Example of wallpaper group type p4m. ... A storm drain, storm sewer, or stormwater drain (in Australia) system is designed to drain excess rain and ground water from an area. ... For other uses, see United States (disambiguation) and US (disambiguation). ... Download high resolution version (1074x1063, 563 KB)Example of wallpaper group type p4m. ... A mummy is a corpse whose skin and dried flesh have been preserved by either intentional or accidental exposure to chemicals, extreme cold or dryness, or airlessness. ... Download high resolution version (1070x1074, 398 KB)Example of wallpaper group type p4m. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... Glaze is a term for painting with a transparent medium. ...

Cloth, Otaheite (Tahiti)

Egyptian tomb

Cathedral of Bourges

Dish from Turkey, Ottoman period

Download high resolution version (997x986, 432 KB)Example of wallpaper group type p4m. ... Map of French Polynesia Map of Tahiti and Moorea Tahiti is the largest island of French Polynesia, located in the archipelago of Society Islands in the southern Pacific Ocean at . ... Download high resolution version (1090x1078, 432 KB)Example of wallpaper group type p4m. ... A tomb is a small building (or vault) for the remains of the dead, with walls, a roof, and (if it is to be used for more than one corpse) a door. ... Download high resolution version (1390x935, 551 KB)Example of wallpaper group type p4m. ... Bourges is a town and commune in central France. ... Image File history File links Download high resolution version (1479x1485, 283 KB) Summary Computer-enhanced photograph of a dish on display in the Museum of Fine Arts, Boston (item number 1885. ... Imperial motto (Ottoman Turkish) دولت ابد مدت Devlet-i Ebed-müddet (The Eternal State) The Ottoman Empire at the height of its power (1683) Official language Ottoman Turkish Capital Söğüt (1299-1326), Bursa (1326-1365), Edirne (1365-1453), Constantinople (modern-day Istanbul) (1453-1922) Imperial anthem Ottoman imperial anthem Sovereigns Padishah...

Group p4g

Cell structure for p4g

Cell structure for 4*2

• 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 4-fold 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). Cell diagram for wallpaper group type p4g. ... Image File history File links Wallpaper_group_diagram_p4g. ... Image File history File links Wallpaper_group_diagram_p4g. ... 5 by 5 checkerboard pattern A checkerboard (or chequerboard) is a board on which American checkers is played. ...

Note that the diagram on the left represents in area twice the smallest square that is repeated by translation.
Examples of group p4g

Computer generated	Bathroom linoleum, U.S.	Painted porcelain, China	Fly screen, U.S.
Painting, China

Example of wallpaper plane class p4G File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1587x1587, 940 KB)Illustration of p4g type Wallpaper group (bathroom floor tiling) File links The following pages link to this file: Wallpaper group Talk:Wallpaper group Categories: Public domain images ... A linoleum kitchen floor Linoleum is a floor covering made from solidified linseed oil (linoxyn) in combination with wood flour or cork dust over a burlap or canvas backing. ... For other uses, see United States (disambiguation) and US (disambiguation). ... Download high resolution version (1054x1069, 435 KB)Example of wallpaper group type p4g. ... It has been suggested that Porcelain tile be merged into this article or section. ... Example of wallpaper group type p4g. ... For other uses, see United States (disambiguation) and US (disambiguation). ... Download high resolution version (1068x1080, 471 KB)Example of wallpaper group type p4g. ...

Group p3

Cell structure for p3 (the rotation centres at the centres of the triangles are not shown)

Cell structure for 333

• 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. Cell diagram for wallpaper group type p3. ... Image File history File links Wallpaper_group_diagram_p3. ... Image File history File links Wallpaper_group_diagram_p3. ...

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

Computer generated

one of the 8 semi-regular tessellations (ignoring the colors: p6); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image

Street pavement in Zakopane, Poland

Example of wallpaper plane class p3 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Tile_33336. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Download high resolution version (1000x960, 431 KB)Example of wallpaper group type p3. ... Zakopane - view from Gubałówka Hill over Zakopane (Tatra mountains in background) Zakopane - Gubałówka Hill Funicular entrance Zakopane - Gubałówka Hill Funicular Zakopane - Gubałówka Hill ski run Zakopane - Gubałówka Hill a nursery ski run Zakopane (pronounce: [zakopanε]) is a town in southern Poland with approximately 28...

Group p3m1

Cell structure for p3m1

Cell structure for *333

• Orbifold notation: *333.
• The group p3m1 has three different rotation centres of order three (120°). It has reflections in three distinct directions. 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. Cell diagram for wallpaper group type p3m1. ... Image File history File links Wallpaper_group_diagram_p3m1. ... Image File history File links Wallpaper_group_diagram_p3m1. ...

Examples of group p3m1

one of the 3 regular tessellations (ignoring colors: p6m)	another regular tessellation (ignoring colors: p6m)	one of the 8 semi-regular tessellations (ignoring colors: p6m)	Persian glazed tile (ignoring colors: p6m)
Persian ornament	Floor tiling in Budapest, Hungary (ignoring colors: p6m)	Painting, China (see detailed image)

Image File history File links Tile_3,6. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Image File history File links Tile_6,3. ... Image File history File links Tile_3bb. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Download high resolution version (1077x1317, 478 KB)Example of wallpaper group type p3m1. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... Glaze is a term for painting with a transparent medium. ... Download high resolution version (1058x1028, 398 KB)Example of wallpaper group type p3m1. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... Download high resolution version (1000x1010, 377 KB)Example of wallpaper group type p3m1. ... Nickname: Paris of the East, Pearl of the Danubeor Queen of the Danube Location of Budapest in Hungary Country Hungary County Pest Mayor Gábor Demszky (SZDSZ) Area    - City 525,16 km²  - Land n/a km²  - Water n/a km² Population    - City (2006) 1,695,000  - Density 3570/km... Download high resolution version (1075x1075, 575 KB)Example of wallpaper group type p3m1. ...

Group p31m

Cell structure for p31m

Cell structure for 3*3

• 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. Cell diagram for wallpaper group type p31m. ... Image File history File links Wallpaper_group_diagram_p31m. ... Image File history File links Wallpaper_group_diagram_p31m. ...

Examples of group p31m

Persian glazed tile

Painted porcelain, China

Painting, China

Download high resolution version (1090x1246, 550 KB)Example of wallpaper group type p31m. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... Glaze is a term for painting with a transparent medium. ... Download high resolution version (1075x1086, 435 KB)Example of wallpaper group type p31m. ... It has been suggested that Porcelain tile be merged into this article or section. ... Download high resolution version (1074x1059, 463 KB)Example of wallpaper group type p31m. ...

Group p6

Cell structure for p6

Cell structure for 632

• 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 Cell diagram for wallpaper group type p6. ... Image File history File links Wallpaper_group_diagram_p6. ... Image File history File links Wallpaper_group_diagram_p6. ...

Computer generated

Wall panelling, the Alhambra, Spain

Persian ornament

Example of wallpaper plane class p6 File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Download high resolution version (1612x1152, 1068 KB)Example of wallpaper group type p6. ... View of the Alhambra from the Mirador St Nicolas in the Albaycin of Granada. ... Download high resolution version (1022x1036, 393 KB)Example of wallpaper group type p6. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ...

Group p6m

Cell structure for p6m

Cell structure for *632

• Orbifold notation: *632.
• The group p6m has one rotation 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 Cell diagram for wallpaper group type p6m. ... Image File history File links Wallpaper_group_diagram_p6m. ... Image File history File links Wallpaper_group_diagram_p6m. ...

Computer generated	one of the 8 semi-regular tessellations	another semi-regular tessellation	another semi-regular tessellation
Persian glazed tile	King's dress, Khorsabad, Assyria	Bronze vessel in Nimroud, Assyria	Byzantine marble pavement, Rome
Painted porcelain, China	Painted porcelain, China

Example of wallpaper plane class p6m File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Tile_3636. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Image File history File links Tile_3464. ... Image File history File links Tile_46b. ... Download high resolution version (1590x1073, 692 KB)Example of wallpaper group type p6m. ... The Persian Empire was a series of historical empires that ruled over the Iranian plateau (Irān - Land of the Aryans) and beyond. ... Glaze is a term for painting with a transparent medium. ... Download high resolution version (1042x1056, 431 KB)Example of wallpaper group type p6m. ... Khorsabad (Khursabad), village in Iraq, 15 km northeast of Mosul, with well-preserved ruins of the large, rectangular Dur-Sharrukin. ... Relief from Assyrian capital of Dur Sharrukin, showing transport of Lebanese cedar (8th c. ... Download high resolution version (1050x1057, 543 KB)Example of wallpaper group type p6m. ... ... Nimrud is a ancient Assyrian city located south of Nineveh on the river Tigris. ... Relief from Assyrian capital of Dur Sharrukin, showing transport of Lebanese cedar (8th c. ... Download high resolution version (1134x1134, 435 KB)Example of wallpaper group type p6m. ... The most famous of the surviving Byzantine mosaics of the Hagia Sophia in Constantinople - the image of Christ Pantocrator on the walls of the upper southern gallery. ... Venus de Milo, front. ... Nickname: The Eternal City Location of the city of Rome (yellow) within the Province of Rome (red) and region of Lazio (grey) Coordinates: Region Lazio Province Province of Rome Founded 8th century BC Mayor Walter Veltroni Area    - City 1,285 km²  (496. ... Download high resolution version (1055x1078, 443 KB)Example of wallpaper group type p6m. ... It has been suggested that Porcelain tile be merged into this article or section. ... Download high resolution version (1064x1044, 498 KB)Example of wallpaper group type p6m. ... It has been suggested that Porcelain tile be merged into this article or section. ...

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. See lattice for other meanings of this term, both within and without mathematics. ...

• 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.

Symmetry groups

The actual symmetry group should be distinguished from the wallpaper group. The latter is a category of symmetry groups. There are 17 of these categories, but for each 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 and the orientation and position of the reflection axes and rotation centres. The symmetry group of an object (e. ...

The numbers of degrees of freedom are: Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

• 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: Z²
• pm: Z × D_∞
• pmm: D_∞ × D_∞

This article may be confusing for some readers, and should be edited to enhance clarity. ...

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 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 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 4-fold symmetry) is not counted as a form of extra symmetry. In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ... Several equivalence relations in mathematics are called similarity. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, which can be written as...

Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change.

If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change.

Web demo and software

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.

Nagware is a term of distinction used to differentiate between types of shareware software. ... Inkscape is a Free Software vector graphics editor. ... This article is about free software as defined by the sociopolitical free software movement; for information on software distributed without charge, see freeware. ... A screenshot of the xfig vector graphics editor A screenshot of the modern vector graphics editor Xara Xtreme A vector graphics editor is a computer program that allows users to compose and edit vector graphics images interactively on the computer screen (compare with MetaPost) and save them in one of... Adobe Illustrator is a vector-based drawing program developed and marketed by Adobe Systems. ...

See also

Wikimedia Commons has media related to:

Wallpaper group diagrams

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

Image File history File links Commons-logo. ... Wikimedia Commons logo by Reid Beels The Wikimedia Commons (also called Commons or Wikicommons) is a repository of free content images, sound and other multimedia files. ... There are 17 symmetry groups in the plane. ... A tessellated plane seen in street pavement. ... In mathematics, point group is a group of geometric symmetries (isometries) leaving a point fixed. ... Crystallography (from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and graphein = write) is the experimental science of determining the arrangement of atoms in solids. ... A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ... A pattern in 1D can be represented as a function f(x) for, say, the color at position x. ... Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), 1935. ... An aperiodic tiling is a tiling of the plane by a set of prototiles that can only be tiled in a non-repeating (non-periodic) pattern. ...

References

• 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 0-7167-1193-1.

Owen Jones (15 February 1809 - 19 April 1874) was a British architect and decorative artist. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ...

External links

• Article "The Discontinuous Groups of Rotation and Translation in the Plane" by Xah Lee
• Article "The 17 plane symmetry groups" 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