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Wallpaper group

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Wallpaper group

Example of an Egyptian design with wallpaper group p4mm

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.

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).

Contents

  • Introduction 1
    • Symmetries of patterns 1.1
  • History 2
  • Formal definition and discussion 3
    • Isometries of the Euclidean plane 3.1
    • The independent translations condition 3.2
    • The discreteness condition 3.3
    • Notations for wallpaper groups 3.4
      • Crystallographic notation 3.4.1
      • Orbifold notation 3.4.2
    • Why there are exactly seventeen groups 3.5
  • Guide to recognizing wallpaper groups 4
  • The seventeen groups 5
    • Group p1 (o) 5.1
    • Group p2 (2222) 5.2
    • Group pm (**) 5.3
    • Group pg (××) 5.4
    • Group cm (*×) 5.5
    • Group p2mm (*2222) 5.6
    • Group p2mg (22*) 5.7
    • Group p2gg (22×) 5.8
    • Group c2mm (2*22) 5.9
    • Group p4 (442) 5.10
    • Group p4mm (*442) 5.11
    • Group p4mg (4*2) 5.12
    • Group p3 (333) 5.13
    • Group p3m1 (*333) 5.14
    • Group p31m (3*3) 5.15
    • Group p6 (632) 5.16
    • Group p6mm (*632) 5.17
  • Lattice types 6
  • Symmetry groups 7
  • Dependence of wallpaper groups on transformations 8
  • Web demo and software 9
  • See also 10
  • Notes 11
  • References 12
  • External links 13

Introduction

Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.

Consider the following examples:

Examples A and B have the same wallpaper group; it is called p4mm in the IUC notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4mg or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

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 that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry — when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.

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 the wallpaper group of A and B is different from the wallpaper group of C.

History

A

  • 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
  • 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.
  • A JavaScript app for drawing wallpaper patterns
  • Beobachtungen zum geometrischen Motiv der Pelta
  • Seventeen Kinds of Wallpaper Patterns the 17 symmetries found in traditional Japanese patterns.

External links

  • 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
  • J. H. Conway; H. Burgiel, C. Goodman-Strauss (2008): "The Symmetries of Things". Worcester MA: A.K. Peters. ISBN 1-56881-220-5.
  • Grünbaum, Branko; Shephard, G. C. (1987): Tilings and Patterns. New York: Freeman. ISBN 0-7167-1193-1.
  • Pattern Design, Lewis F. Day

References

  1. ^ E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, vol. 28, pages 245-291 (in Russian).
  2. ^ George Pólya (1924) "Über die Analogie der Kristallsymmetrie in der Ebene," Zeitschrift für Kristallographie, vol. 60, pages 278–282.
  3. ^ Radaelli, Paulo G. Symmetry in Crystallography. Oxford University Press. 
  4. ^ It helps to consider the squares as the background, then we see a simple patterns of rows of rhombuses.

Notes

See also

  • MadPattern, a free set of Adobe Illustrator templates that support the 17 wallpaper groups
  • Tess, a nagware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.
  • Kali, online graphical symmetry editor applet.
  • Kali, free downloadable Kali for Windows and Mac Classic.
  • 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. (See [2])
  • SymmetryWorks is a commercial plugin for Adobe Illustrator, supports all 17 groups.
  • Arabeske is a free standalone tool, supports a subset of wallpaper groups.

Several software graphic tools will let you create 2D patterns using wallpaper symmetry groups. Usually you can edit the original tile and its copies in the entire pattern are updated automatically.

Web demo and software

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.

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.

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.

  • 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 p6mm, p4mg, and p3m1 into c2mm, p3m1 into cm, and p4mm, depending on direction of expansion/contraction, into p2mm or c2mm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from p6mm to p4mm.

Dependence of wallpaper groups on transformations

  • p1: Z2
  • pm: Z × D
  • p2mm: D × D.

Some symmetry group isomorphisms:

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

  • 6 for p2
  • 5 for p2mm, p2mg, p2gg, and c2mm
  • 4 for the rest.

The numbers of degrees of freedom are:

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.

Symmetry groups

  • In the 5 cases of rotational symmetry of order 3 or 6, the unit cell consists of two equilateral triangles (hexagonal lattice, itself p6mm). They form a rhombus with angles 60° and 120°.
  • In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself p4mm).
  • In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself p2mm). Special cases: square.
  • In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself c2mm). It may also be interpreted as a centered rectangular lattice. Special cases: square, hexagonal unit cell.
  • In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell is in general a parallelogram (parallelogrammatic or oblique lattice, itself p2). Special cases: rectangle, square, rhombus, hexagonal unit cell.

There are five lattice types or Bravais lattices, corresponding to the five possible wallpaper groups of the lattice itself. The wallpaper group of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.

Lattice types

Examples of group p6mm

A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with D3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D6 symmetry (with the edges of the tiles not necessarily part of the pattern). Thus the simplest examples are a triangular lattice with or without connecting lines, and a hexagonal tiling with one color for outlining the hexagons and one for the background.

  • Orbifold notation: *632
  • Coxeter notation: [6,3]
  • Lattice: hexagonal
  • Point group: D6
  • The group p6mm 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.
Cell structure for p6mm
Example and diagram for p6mm

Group p6mm (*632)

Examples of group p6

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

  • Orbifold notation: 632
  • Coxeter notation: [6,3]+
  • Lattice: hexagonal
  • Point group: C6
  • The group p6 has one rotation centre of order six, which only differ by a rotation of 60°; it has also two rotation centres of order three, which only differ by a rotation of 120° and three of order two (or, equivalently, 180°). It has no reflections or glide reflections.
Cell structure for p6
Example and diagram for p6

Group p6 (632)

Examples of group p31m

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

  • Orbifold notation: 3*3
  • Coxeter notation: [6,3+]
  • Lattice: hexagonal
  • Point group: D3
  • 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.
Cell structure for p31m
Example and diagram for p31m

Group p31m (3*3)

Examples of group p3m1

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

  • Orbifold notation: *333
  • Coxeter notation: [(3,3,3)] or [3[3]]
  • Lattice: hexagonal
  • Point group: D3
  • 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.
Cell structure for p3m1
Example and diagram for p3m1

Group p3m1 (*333)

Examples of group p3

Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by √3. Then this wallpaper group corresponds to the case that all hexagons are equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror image symmetry (if they have rotational symmetry of order six we have p6, if they are symmetric with respect to the main diagonals we have p31m, if they are symmetric with respect to lines perpendicular to the sides we have p3m1; if two of the three apply then the third also, and we have p6mm). For a given image, three of these tessellations are possible, each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of the hexagons can be the red, the blue or the green triangles.

Imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, but the two are not equal, not each other's mirror image, and not both symmetric (if the two are equal we have p6, if they are each other's mirror image we have p31m, if they are both symmetric we have p3m1; if two of the three apply then the third also, and we have p6mm). 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.

  • Orbifold notation: 333
  • Coxeter notation: [(3,3,3)]+ or [3[3]]+
  • Lattice: hexagonal
  • Point group: C3
  • The group p3 has three different rotation centres of order three (120°), but no reflections or glide reflections.
Cell structure for p3
Example and diagram for p3

Group p3 (333)

Examples of group p4mg

A p4mg pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90° rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4mm (with diagonal translation cells).

  • Orbifold notation: 4*2
  • Coxeter notation: [4+,4]
  • Lattice: square
  • Point group: D4
  • The group p4mg 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.
Cell structure for p4mg
Example and diagram for p4mg

Group p4mg (4*2)

Examples displayed with the smallest translations diagonal:

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

Examples of group p4mm

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

  • Orbifold notation: *442
  • Coxeter notation: [4,4]
  • Lattice: square
  • Point group: D4
  • The group p4mm 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.
Cell structure for p4mm
Example and diagram for p4mm

Group p4mm (*442)

A p4 pattern can be looked upon as a repetition in rows and columns of equal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a checkerboard pattern of two such tiles, a factor \sqrt{2} smaller and rotated 45°.

Examples of group p4
  • Orbifold notation: 442
  • Coxeter notation: [4,4]+
  • Lattice: square
  • Point group: C4
  • 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.
Cell structure for p4
Example and diagram for p4

Group p4 (442)

Examples of group c2mm
  • 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

The pattern corresponds to each of the following:

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.

  • Orbifold notation: 2*22
  • Coxeter notation (rhombic): [∞,2+,∞]
  • Coxeter notation (square): [(4,4,2+)]
  • Lattice: rhombic
  • Point group: D2
  • The group c2mm has reflections in two perpendicular directions, and a rotation of order two (180°) whose centre is not on a reflection axis. It also has two rotations whose centres are on a reflection axis.
  • This group is frequently seen in everyday life, since the most common arrangement of bricks in a brick building (running bond) utilises this group (see example below).
Cell structures for c2mm by lattice type

Rhombic

Square
Example and diagram for c2mm

Group c2mm (2*22)

Examples of group p2gg
  • Orbifold notation: 22×
  • Coxeter notation (rectangular): [((∞,2)+,(∞,2)+)]
  • Coxeter notation (square): [4+,4+]
  • Lattice: rectangular
  • Point group: D2
  • The group p2gg 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.
Cell structures for p2gg by lattice type

Rectangular

Square
Example and diagram for p2gg

Group p2gg (22×)

Examples of group p2mg
  • Orbifold notation: 22*
  • Coxeter notation: [(∞,2)+,∞] or [∞,(2,∞)+]
  • Lattice: rectangular
  • Point group: D2
  • The group p2mg 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.
Cell structures for p2mg

Horizontal mirrors

Vertical mirrors
Example and diagram for p2mg

Group p2mg (22*)

Examples of group p2mm
  • Orbifold notation: *2222
  • Coxeter notation (rectangular): [∞,2,∞] or [∞]×[∞]
  • Coxeter notation (square): [4,1+,4] or [1+,4,4,1+]
  • Lattice: rectangular
  • Point group: D2
  • The group p2mm has reflections in two perpendicular directions, and four rotation centres of order two (180°) located at the intersections of the reflection axes.
Cell structure for p2mm

rectangular

square
Example and diagram for p2mm

Group p2mm (*2222)

Examples of group cm
  • Orbifold notation: *×
  • Coxeter notation: [∞+,2+,∞] or [∞,2+,∞+]
  • Lattice: rhombic
  • Point group: D1
  • The group cm contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis is not a reflection axis; it is halfway between two adjacent parallel reflection axes.
  • This group applies for symmetrically staggered rows (i.e. there is a shift per row of half the translation distance inside the rows) of identical objects, which have a symmetry axis perpendicular to the rows.
Cell structure for cm

Horizontal mirrors

Vertical mirrors
Rhombic
Example and diagram for cm

Group cm (*×)

Ignoring the wavy borders of the tiles, the pavement is p2gg.

Without the details inside the zigzag bands the mat is p2mg; with the details but without the distinction between brown and black it is p2gg.

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

Horizontal glides

Vertical glides
Rectangular
Example and diagram for pg

Group pg (××)

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

Examples of group pm
  • Orbifold notation: **
  • Coxeter notation: [∞,2,∞+] or [∞+,2,∞]
  • Lattice: rectangular
  • Point group: D1
  • The group pm has no rotations. It has reflection axes, they are all parallel.
Cell structure for pm

Horizontal mirrors

Vertical mirrors
Example and diagram for pm

Group pm (**)

Examples of group p2
  • Orbifold notation: 2222
  • Coxeter notation (rectangular): [∞,2,∞]+
  • Lattice: oblique
  • Point group: C2
  • The group p2 contains four rotation centres of order two (180°), but no reflections or glide reflections.
Cell structures for p2 by lattice type

Oblique

Hexagonal

Rectangular

Rhombic

Square
Example and diagram for p2

Group p2 (2222)

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

Examples of group p1
  • Orbifold notation: °
  • Coxeter notation (rectangular): [∞+,2,∞+] or [∞]+×[∞]+
  • Lattice: oblique
  • Point group: C1
  • The group p1 contains only translations; there are no rotations, reflections, or glide reflections.
Cell structures for p1 by lattice type

Oblique

Hexagonal

Rectangular

Rhombic

Square
Example and diagram for p1

Group p1 (o)

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.

The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern that is repeated.

On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated) differently.

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.

Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows (it is the shape that is significant, not the colour):

The seventeen groups

See also .

Size of smallest
rotation
Has reflection?
Yes No
360° / 6 p6mm (*632) p6 (632)
360° / 4 Has mirrors at 45°? p4 (442)
Yes: p4mm (*442) No: p4mg (4*2)
360° / 3 Has rot. centre off mirrors? p3 (333)
Yes: p31m (3*3) No: p3m1 (*333)
360° / 2 Has perpendicular reflections? Has glide reflection?
Yes No
Has rot. centre off mirrors? p2mg (22*) Yes: p2gg (22×) No: p2 (2222)
Yes: c2mm (2*22) No: p2mm (*2222)
none Has glide axis off mirrors? Has glide reflection?
Yes: cm (*×) No: pm (**) Yes: pg (××) No: p1 (O)

To work out which wallpaper group corresponds to a given design, one may use the following table.[3]

Guide to recognizing wallpaper groups

Feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).

Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2.

  • 632: 5/6 + 2/3 + 1/2 = 2
  • 3*3: 2/3 + 1 + 1/3 = 2
  • 4*2: 3/4 + 1 + 1/4 = 2
  • 22×: 1/2 + 1/2 + 1 = 2
Examples

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

  • A digit n before a * counts as (n − 1)/n.
  • A digit n after a * counts as (n − 1)/2n.
  • Both * and × count as 1.
  • The "no symmetry" ° counts as 2.

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

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.

An orbifold can be viewed as a polygon with face, edges, and vertices, which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a spherical symmetry group or Hyperbolic symmetry group. The type of space the polygons tile can be found by calculating the Euler characteristic, χ = V − E + F, where V is the number of corners (vertices), E is the number of edges and F is the number of faces. If the Euler characteristic is positive then the orbifold has an elliptic (spherical) structure; if it is zero then it has a parabolic structure, i.e. a wallpaper group; and if it is negative it will have a hyperbolic structure. When the full set of possible orbifolds is enumerated it is found that only 17 have Euler characteristic 0.

Why there are exactly seventeen groups

Conway, Coxeter and crystallographic correspondence
Conway o ×× ** 632 *632
Coxeter [∞+,2,∞+] [(∞,2)+,∞+] [∞,2+,∞+] [∞,2,∞+] [6,3]+ [6,3]
Crystal. p1 pg cm pm p6 p6mm
Conway 333 *333 3*3 442 *442 4*2
Coxeter [3[3]]+ [3[3]] [3+,6] [4,4]+ [4,4] [4+,4]
Crystal. p3 p3m1 p31m p4 p4mm p4mg
Conway 2222 22× 22* *2222 2*22
Coxeter [∞,2,∞]+ [((∞,2)+,(∞,2)+)] [(∞,2)+,∞] [∞,2,∞] [∞,2+,∞]
Crystal. p2 p2gg p2mg p2mm c2mm

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

The group denoted by p2gg will be 22×. We have two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with p2mg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.

Consider the group denoted in crystallographic notation by c2mm; 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 that is not a duplicate of the first one under symmetries.

  • The "no symmetry" symbol, o, stands alone, and indicates we have only lattice translations with no other symmetry. The orbifold with this symbol is a torus; in general the symbol o denotes a handle on the orbifold.
  • A cross, ×, occurs when a glide reflection is present and indicates a crosscap on the orbifold. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so we do not notate them.
  • An asterisk, *, indicates a mirror symmetry corresponding to a boundary of the orbifold. It interacts with the digits as follows:
  • Digits before * denote centres of pure rotation (cyclic).
  • Digits after * denote centres of rotation with mirrors through them, corresponding to "corners" on the boundary of the orbifold (dihedral).
  • A digit, n, indicates a centre of n-fold rotation corresponding to a cone point on the orbifold. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.

Orbifold notation for wallpaper groups, advocated by John Horton Conway (Conway, 1992) (Conway 2008), 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.

Orbifold notation

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

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 p4mg p6mm

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

  • p2 (p211): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
  • p4mg (p4mm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.
  • c2mm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.
  • p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.
Examples

A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells that are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types.

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.

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 (also called IUC notation) is p31m, with four letters or digits; more usual is a shortened name like c2mm or pg.

Crystallographic notation

Notations for wallpaper groups

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 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 Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern.

The discreteness condition means that there is some positive real number ε, such that for every translation Tv in the group, the vector v has length at least ε (except of course in the case that v is the zero vector).

The discreteness condition

(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 purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups, which only repeat along a single axis.

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

The independent translations condition

  • Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane applying displacement vector v.
  • Rotations, denoted by Rc,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 FL, 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 GL,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.

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

Isometries of the Euclidean plane

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

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).

Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation.

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).

Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations.

Formal definition and discussion

The proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. [2]

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