Ornament group

Groups of ornaments categorize patterns according to their symmetries. Subtle differences in similar patterns can lead to the distribution of patterns in different groups, while patterns that are significantly different in style, color, scale or orientation may belong to the same group.

Consider the following examples:

• 

  Example A : Fabric, Tahiti
• 

  Example B : Ornament, Nineveh , Assyria
• 

  Example C : Painted Porcelain , China

Examples A and B have the same group of ornaments, in the IUC notation it is called p 4 m , and in the - * 442 . Example C has another group of ornaments called p 4 g , or 4 * 2 . The fact that A and B have the same group means that these ornaments have the same symmetries regardless of the details of the patterns, while C has a different set of symmetries in spite of the external similarity.

A complete list of all seventeen possible groups of patterns can be found below.

Pattern Symmetries

Symmetry of a pattern is, roughly speaking, a way of transforming a pattern in such a way that it looks after the transformation in exactly the same way as it was before the transformation. For example, symmetry of parallel transfer is present if, with a certain shift ( parallel transfer ), the pattern is combined with itself. Imagine the shift of vertical (one width) stripes horizontally by one strip, the pattern will remain the same. Strictly speaking, true symmetry exists only for patterns that repeat exactly and endlessly. A set of, say, only five bands does not have parallel transfer symmetry - when shifting, a strip on one side “disappears” and a new strip is “added” on the other side.

Sometimes there are two ways to categorize a pattern, one based solely on form, and the other using coloring. If you ignore the colors, the pattern may have more symmetries. Among black and white mosaics, there are also 17 groups of ornaments. For example, a painted tile is equivalent to a black and white tile with a color-coded tile in the form of a radially symmetric “bar code” in the center of mass of each tile.

The types of transformations discussed here are called motions . For example:

• If we move 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 . This type of symmetry is called parallel transfer . Examples A and C are similar, but in them the smallest possible shift is located diagonally.
• If we rotate Example B clockwise 90 ° around the center of one of the squares, we again get the same pattern. This is called a turn . Examples A and C also have 90 ° rotations, although it takes a little more ingenuity to find the right center of rotation for C.
• We can reflect Example B with respect to the horizontal axis passing through the middle of the image. This is called (mirror) reflection . Example B has mirror symmetry also with respect to the vertical axis and two diagonal axes. The same can be said of Example A.

However, Example C is different . It has reflections only relative to horizontal and vertical directions, but not relative to diagonal axes. If we flip the pattern around the diagonal axis, we will not get the same pattern. We get the original pattern, offset by a certain distance. This is one of the reasons why the ornamental pattern group A and B is different from the ornamental pattern group C.

Another transformation is moving symmetry , a combination of reflection and parallel transport along the axis of reflection.

Evidence that there are only 17 possible patterns was first carried out by Evgraf Stepanovich Fedorov in 1891 ^[1] , and then, independently, was deduced by Györgye Poia in 1924 ^[2] . The proof that the list of ornament groups is complete only came after it was done for the much more complex case of crystallographic groups.

A group of ornaments, or a flat crystallographic group , is an isometric completely discontinuous cocompact action of a group on the Euclidean plane (cocompactness is equivalent to the fact that the action contains two linearly independent parallel transfers ).

Two such isometry groups are of the same type (the same group of ornaments) if they are transformed into each other during affine transformation of the plane.

So, for example, the shift of the whole pattern (and, therefore, the transfer of the reflection axes and rotation centers) does not affect the group of ornaments. The same applies to a change in the angle between the parallel transport vectors, provided that this does not add or disappear any symmetry (this is only possible if there is no mirror symmetry and moving symmetries , and rotational symmetry is of the order of maximum 2).

Remarks

• In this definition, we can restrict affine transformations to orientation- preserving transformations.
  • Unlike the three-dimensional case , the classification remains the same.

• It follows from Bieberbach's theorem that all groups of ornaments even differ as abstract groups (as opposed to, for example, border groups , of which two groups are isomorphic to Z ).

Isometrics of the Euclidean plane

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

• Parallel transfers are denoted by T _v (from English “translation”), where v is a vector in R ² . The transformation effect is a shift of the plane by the displacement vector v .
• Turns are denoted by R _{c , θ} (from the English “rotation”), where c is the point of the plane (center of rotation), and θ is the angle of rotation.
• Reflections , or mirror isometries , are denoted by F _L (from the English “flip”), where L is a straight line in R ² . The reflection will result in mirror symmetry of the plane with respect to the line L , which is called the axis of reflection or the mirror .
• Sliding symmetries are denoted by G _{L , d} (from the English “glide”), where L is the line in R ² and d is the distance. The transformation is a combination of mirror reflection relative to the straight line L and parallel transfer along L along the distance d .

Independence condition for parallel transfers

The condition of linear independence of parallel translations means that there are linearly independent vectors v and w (in R ² ) such that the group contains both T _v and T _w .

The purpose of this condition is to isolate groups of ornaments from border groups that have parallel transfer but not two linearly independent ones, as well as from two-dimensional discrete point groups that do not have parallel transfers at all. In other words, ornament groups represent a pattern that repeats in two different directions, as opposed to border groups that repeat only along one axis.

(This situation can be generalized. For example, we could study discrete isometry groups R ⁿ with m linearly independent parallel translations, where m is any integer in the interval 0 ≤ m ≤ n .)

Complete discontinuity condition

The condition of complete discontinuity (sometimes called discreteness) means that there exists some positive real number ε such that for any parallel transfer T _v in the group, the vector v has a length of at least ε (except, of course, for the case of the zero vector v ).

The purpose of this condition is to ensure that the group has a compact fundamental region , or, in other words, a “cell” of a nonzero finite area that repeats on a plane (in the form of a pattern). Without this condition, we can obtain, for example, a group containing a parallel translation T _x for any rational number x , which does not correspond to any acceptable ornamental pattern.

An important and non-trivial consequence of the discreteness condition in combination with the condition for the independence of parallel transfers is that a group can contain only rotations of the order of 2, 3, 4 or 6. That is, any rotation in the group must be a rotation of 180 °, 120 °, 90 ° or 60 °. This fact is known as , and this theorem can be generalized to cases of higher dimensions.

Crystallographic designation

In crystallography, there are 230 different crystallographic groups , many more than 17 groups of ornaments, but many symmetries in the groups are the same. Thus, you can use similar notations for both types of groups, the notation of and . An example of the full name of an ornament in the Hermann-Mogen style (designations are also called “Designations of the International Union of Crystallographers”, IUC ) - p 31 m with four letters and numbers. A short name is usually used, such as cmm or pg .

For groups of ornaments, the full notation starts with p (from primitive cell - unit cell ) or c (from face-centred cell - face-centered cell). They will be explained below. The letter is followed by the number n , which indicates the highest order of rotational symmetry - 1-fold (no), 2-fold, 3-fold, 4-fold or 6-fold. The following two symbols indicate symmetries with respect to one of the axes of parallel transfer, which is considered to be “main”. If there is mirror symmetry perpendicular to the axis of parallel transfer, select this axis as the main axis (if there are two, select any of them). As symbols, m , g or 1 is chosen, for mirror symmetry, moving symmetry or lack of symmetry. The axis of mirror symmetry or moving symmetry is perpendicular to the main axis for the first letter, and is either parallel or tilted 180 ° / n (if n > 2) for the second letter. Many groups include other symmetries. In a short notation, digits or m are discarded if it is determined logically, if this does not lead to confusion with other groups.

A primitive cell is a minimal area repeated by parallel transfer along the lattice. All but two symmetry groups of ornaments are described by the axes of a primitive cell, a coordinate basis using parallel lattice transfer vectors. In the remaining two cases, symmetry is described by centered cells, which are larger than primitive cells, and therefore have an internal repetition. The directions of their sides are different from the directions of the parallel transport vectors. Herman-Maugen notation for crystals of crystallographic groups uses additional types of cells.

Examples

• p 2 ( p 211 ): Primitive cell, 2-fold rotation symmetry, no specular reflections, nor moving symmetries.
• p 4 gm ( p 4 mm ): Primitive cell, 4-fold rotation symmetry, sliding symmetry perpendicular to the main axis, mirror symmetry axis at an angle of 45 °.
• c 2 mm ( c 2 mm ): Center cell, 2-fold rotation symmetry, the axis of mirror symmetry are perpendicular and parallel to the main axis.
• p 31 m ( p 31 m ): Primitive cell, 3-fold rotation symmetry, mirror symmetry axis at an angle of 60 °.

Names whose short and full appearance are different.

Crystallographic Short and Full Names

A short	p 2	pm	pg	cm	pmm	pmg	pgg	cmm	p 4 m	p 4 g	p 6 m
Full	p 211	p 1 m 1	p 1 g 1	c 1 m 1	p 2 mm	p 2 mg	p 2 gg	c 2 mm	p 4 mm	p 4 gm	p 6 mm

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

Orb Designations

The orb designation for groups of ornaments is popularized by John Conway , based not on crystallography, but on topology. We consider the factor- orbifoldness of the plane by the action of the ornament group and describe it with the help of several symbols.

• The figure, n , shows the center of the n- fold rotation corresponding to the vertex of the orbifold cone. By the crystallographic constraint theorem, n must be 2, 3, 4, or 6.
• The asterisk, * , shows the mirror symmetry corresponding to the boundary of the orbifold. It is interconnected with numbers as follows:
  1. The numbers before * mean the centers of simple ( cyclic ) rotation.
  2. The numbers after * indicate the centers of rotation with the mirrors passing through them, which corresponds to the “corners” of the orbifold boundary ( dihedral ).
• A cross, × , appears when moving symmetry is present; he shows a Mobius strip in orbifold. Simple reflections are combined with lattice translation to obtain moving symmetry, but they are already taken into account, so we do not denote them.
• The symbol of "lack of symmetry", o , stands alone and means that there is only symmetry of parallel transfer and no other symmetries. An orbifold with such a symbol is a torus. In the general case, the symbol o corresponds to gluing the handle to the orbifold.

Consider a group with cmm crystallographic notation. In Conway's notation, this will be 2 * 22 . 2 before * says that we have a center of 2-fold rotation without mirrors passing through it. * She herself says that we have a mirror. The first number 2 after * indicates that we have a center of 2-fold rotation on the mirror. Ultimate 2 says that we have an independent second center of 2-fold rotation on the mirror, which does not duplicate the first center at symmetries.

The group with the designation pgg will have the designation Conway 22 × . We have two simple centers of 2-fold rotation and an axis of moving symmetry. The group pmg contrasts with this group, with the symbol of Conway 22 * , where the crystallographic designation mentions moving symmetry, but one that is implied implicitly by other symmetries of orbifolds.

also included. It is based on the Coxeter group and is modified with a plus (in the superscript) for rotations, and parallel translations.

Why there are exactly seventeen groups

An orbifold can be considered as a polygon with a face, edges and vertices, which can be expanded to form, possibly, an infinite set of polygons that til the entire sphere , plane or hyperbolic plane . If a polygon tilts a plane, it gives a group of ornaments, and if a sphere or a hyperbolic plane, then a group of spherical symmetry or a . The type of space that the polygon tilts can be found by calculating the Euler characteristic , χ = V - E + F , where V is the number of angles (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 the Euler characteristic is equal to zero, it has a parabolic structure, i.e. This is a group of ornaments. If the Euler characteristic is negative, the orbifold has a hyperbolic structure. When all possible orbifolds were enumerated, it turned out that only 17 had an Eulerian characteristic of 0.

When an orbifold is copied to fill a plane, its elements create a structure of vertices, edges and faces that must satisfy the Euler characteristic. Inverting the process, we can assign numbers to orbifold elements, but fractional, not integer. Since the orbifold itself is a factor group of the complete surface with respect to the symmetry group, the Euler characteristic of the orbifold is the quotient of dividing the Euler characteristic of the surface by the order of the symmetry group.

The Euler characteristic of the orbifold is 2 minus the sum of the values ​​of the elements assigned as follows:

• The number n before * is considered as ( n - 1) / n .
• The number n after * is considered as ( n - 1) / 2 n .
• * and × count as 1.
• The sign “no symmetry” ° is considered as 2.

For a group of ornaments, the sum for the Euler characteristic should be zero, so the sum of the values ​​of the elements should 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
• 22 ×: 1/2 + 1/2 + 1 = 2

Now the enumeration of all groups of ornaments is reduced to arithmetic, a list of sets of elements giving a total of 2.

Sets of items with a different amount are not meaningless. They comprise non-planar tilings, which we do not discuss here. (If the Euler characteristic of the orbifold is negative, the tiling is , if it is positive, the tiling is spherical or bad ).

To understand which group of ornaments corresponds to a particular mosaic, the following table can be used ^[3] .

See also This View with Charts .

Each of the groups in this section has two diagrams of the cell structure, each of which is interpreted as follows (form is essential here, not color):

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

Brown or yellow areas indicate a fundamental area , i.e. smallest repeating part of the pattern.

The diagrams on the right show the grid cell corresponding to the smallest parallel hyphenation. Left sometimes shows a large area.

Group p 1 (o)

• Orbifold signature: o
• Coxeter designation (rectangle): [∞ ⁺ , 2, ∞ ⁺ ] or [∞] ⁺ × [∞] ⁺
• Lattice: oblique
• Point group: C ₁
• The group p 1 contains only parallel transfers. The group contains neither rotations, nor specular reflections, nor moving symmetries.

Examples of the group p 1

•  

  Created on computer
•  

  Medieval wall drapery

Two parallel transfers (cell sides) can have different lengths and can form any angle.

Group p 2 (2222)

• Orbifold signature: 2222
• Coxeter Designation (rectangle): [∞, 2, ∞] ⁺
• Lattice: oblique
• Point group: C ₂
• The p 2 group contains four rotation centers of the order of two (180 °), but does not contain either reflections or moving symmetries.

Examples of the group p 2

•  

  Created on computer
•  

  Cloth, Hawaii ( Hawaii )
•  

  The rug on which stood the Egyptian pharaoh
•  

  Egyptian rug (enlarged)
•  

  Ceiling of egyptian tomb
•  

  Wire fence , ( netting ).

Pm group (**)

• Orbifold signature: **
• Coxeter designation: [∞, 2, ∞ ⁺ ] or [∞ ⁺ , 2, ∞]
• Lattice: rectangular
• Point group: D ₁
• The pm group has no rotations. It has axes of reflection, they are all parallel.

Pm group examples

(The first three have vertical axes of symmetry, and the last two have diagonal axes.)

•  

  Computer generated
•  

  Figured clothing in a tomb in the Valley of the Kings , Egypt
•  

  Egyptian Tomb , Thebes
•  

  Ceiling of the tomb in Qurna, Egypt . The axes of specular reflections are diagonal
•  

  Indian metal work at the World Exhibition in 1851. Almost pm (if we ignore the short diagonal segments between the ovals, we get p 1 )

Pg group (×^)

• Orbifold Signature × ×
• Coxeter designation: [(∞, 2) ⁺ , ∞ ⁺ ] or [∞ ⁺ , (2, ∞) ⁺ ]
• Lattice: rectangular
• Point group: D ₁
• The group pg contains only moving symmetries and the axes of these symmetries are all parallel. There are no turns, no mirror reflections.

Pg group examples

•  

  Computer generated
•  

  A rug with a , on which stood an Egyptian pharaoh
•  

  Egyptian rug (partially)
•  

  Bridge with in Salzburg (note that the edges of the tiles are curved and the tiles do not have axial symmetry). Axes of moving symmetry go from the northeast to the southwest
•  

  One of the coloring of a flat-nosed square mosaic . Lines of moving symmetry go from the upper left corner to the lower right. If we ignore the colors, we get a lot more symmetry than just pg , it will be p 4 g (see the same pattern with triangles painted in the same color) ^[4]

Without looking at the details inside the zigzag mat is pmg . If we take into account the details inside the zigzag, but do not distinguish between brown and black stripes, we get pgg .

If you ignore the wavy edges of the tiles, the pavement is pgg .

Group cm (* ×)

• Orbifold signature: * ×
• Coxeter designation: [∞ ⁺ , 2 ⁺ , ∞] or [∞, 2 ⁺ , ∞ ⁺ ]
• Lattice: rhombic
• Point group: D ₁
• The cm group contains no rotations. It has axes of reflection, they are all parallel. There is at least one moving symmetry whose axis is not the reflection axis, and it lies in the middle between two adjacent parallel reflection axes.
• This group refers to the symmetries of step lines (i.e., there is a shift on each line by half the amount of parallel transfer inside the lines) of identical objects that have axis of symmetry perpendicular to the lines.

Cm group examples

•  

  Created on computer
•  

  Amon's clothes from Abu Simbel , Egypt
•  

  from Valley of the Kings , Egypt
•  

  Bronze vessel from Nimrud , Assyria
•  

  Sinus Arches , Alhambra , Spain
•  

  Spotlights arches, Alhambra , Spain
•  

  Persian tapestry
•  

  Indian loop artwork at the 1851 World's Fair
•  

  The clothes of a figure in a tomb in the Valley of the Kings , Egypt

Pmm group (* 2222)

• Orbifold signature: * 2222
• Coxeter designation (rectangle): [∞, 2, ∞] or [∞] × [∞]
• Coxeter designation (square): [4.1 ⁺ , 4] or [1 ⁺ , 4.4.1 ⁺ ]
• Lattice: rectangular
• Point group: D ₂
• The pmm group has reflections in two perpendicular directions and four rotation centers of the order of two (180 °) located at the intersection points of the mirrors.

Pmm group examples

•  

  2D drawing of a fence grid, USA. (in 3D there is additional symmetry)
•  

  Sarcophagus of the Mummy , Louvre Museum
•  

  Sarcophagus of the mummy , Louvre Museum . The pattern would belong to p 4 m , but the coloring does not match in the pattern

Pmg group (22 *)

• Orbifold signature: 22 *
• Coxeter designation: [(∞, 2) ⁺ , ∞] or [∞, (2, ∞) ⁺ ]
• Lattice: rectangular
• Point group: D ₂
• The pmg group has two centers of rotation of the order of two (180 °) and reflections in only one direction. The group has moving symmetry, the axes of which are perpendicular to the axis of reflection. All centers of rotation lie on the axes of moving symmetries.

Pmg group examples

•  

  Created on computer
•  

  Cloth, Hawaii ( Hawaii )
•  

  Ceiling of egyptian tomb
•  

  Mosaic floor in Prague , Czech Republic
•  

  Bowl of kerma
•  

  Laying pentagons

Pgg group (22 ×)

• Orbifold signature: 22 ×
• Coxeter designation (rectangle): [((∞, 2) ⁺ , (∞, 2) ⁺ )]
• Coxeter Designation (square): [4 ⁺ , 4 ⁺ ]
• Lattice: rectangular
• Point group: D ₂
• The pgg group contains two rotation centers of the order of two (180 °) and sliding symmetries in two perpendicular directions. Turn centers are not located on axes of moving symmetry. The group does not contain mirror reflections.

Pgg group examples

•  

  Created on computer
•  

  Bronze vessel from Nimrud , Assyria
•  

  Pavement in Budapest , Hungary . The axes of sliding symmetries are diagonal

Group cmm (2 * 22)

• Orbifold signature: 2 * 22
• Coxeter designation (rhombus): [∞, 2 ⁺ , ∞]
• Coxeter Designation (square): [(4.4.2 ⁺ )]
• Lattice: rhombic
• Point group: D ₂
• The cmm group has reflections in two perpendicular directions and a rotation of the order of two (180 °), the center of which does not lie on the axes of symmetry. The group also has two turns, the centers of which lie on the axis of reflection.
• This group is often observed in everyday life, since most bricklayings in brick buildings use this pattern (brickwork in half a brick) (see example below).

Rotation symmetries of order 2 with centers of rotation at the centers of the sides of the rhombus are a consequence of other properties.

The pattern corresponds to:

• symmetrically stepped lines of identical twice symmetric objects
• a pattern in the form of a checkerboard arrangement of two rectangular tiles, each of which, by itself, is twice symmetrical
• a pattern in the form of a checkerboard arrangement of two rectangular tiles with double rotational symmetry and their mirror reflections

Cmm group examples

•  

  Created on computer
•  

  One of 8 semi-correct tilings
•  

  Bonded Brick Country Wall, USA
•  

  The ceiling of the Egyptian tomb . If you ignore the colors, it would be a group p 4 g
•  

  Egyptian pattern
•  

  Persian tapestry
•  

  egyptian tomb
•  

  Turkic plate
•  

  Compact circles in two sizes
•  

  Another compact package of circles of two sizes
•  

  Another compact package of circles of two sizes

Group p 4 (442)

• Orbifold signature: 442
• Coxeter Designation: [4.4] ⁺
• Lattice: square
• Point group: C ₄
• The p 4 group has two rotation centers of the order of four (90 °) and one rotation center of the order of two (180 °). The group has neither reflections nor moving symmetries.

Examples of p 4

The pattern p 4 can be considered as a repetition in the rows and columns of a square tile with 4-fold symmetry of rotation. It can also be considered as a chess cell of two such tiles smaller in${\displaystyle \sqrt{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}}$ {\ displaystyle {\ sqrt {2}}}   times and rotated 45 °.

•  

  Created on computer
•  

  The ceiling of the Egyptian tomb . If you ignore the colors, it is p 4 , otherwise - p 2
•  

  Ceiling of egyptian tomb
•  

  Pattern overlay
•  

  Border, Alhambra , Spain . Careful consideration is required to understand why there are no reflections.
•  

  Venetian Reed Weaving
•  

  Renaissance pottery
•  

  Pythagorean mosaic
•  

  Derived from a photograph

Group p 4 m (* 442)

• Orbifold signature: * 442
• Coxeter Designation: [4.4]
• Lattice: square
• Point group: D ₄
• The p 4 m group has two centers of rotation of the order of four (90 °) and reflections in four different directions (horizontal, vertical and diagonal). The group has additional sliding symmetries whose axes are not reflection axes. Turns of the order of two (180 °) have centers at the intersections of the axes of moving symmetry. All centers of rotation lie on the axes of reflection.

This corresponds to a rectangular grid of rows and columns of identical squares with four axes of symmetry. This also matches the checkerboard pattern of two such squares.

Examples of the group p 4 m

Examples are shown with the smallest horizontal and vertical parallel hyphenation (as in the diagram):

•  

  Created on computer
•  

  One of 3 correct tilings
•  

  Semi-controlled mosaic of triangles . If you ignore the colors, it is p 4 m , otherwise - c 2 m
•  

  One of 8 semi-correct tilings (if you ignore the color, it is also p 4 m , but with lower values ​​of parallel transfer)
•  

  Ornamental drawing, Nineveh , Assyria
•  

  Rain drain , USA
•  

  Egyptian mummy sarcophagus
•  

  Persian glazed mosaic
•  

  Compact circles in two sizes

Examples with the smallest parallel diagonal translation:

•  

  Chess cell
•  

  Fabric, Tahiti )
•  

  Egyptian tomb
•  

  Bourges Cathedral
•  

  Plate from Ottoman Turkey

Group p 4 g (4 * 2)

• Orbifold signature: 4 * 2
• Coxeter Designation: [4 ⁺ , 4]
• Lattice: square
• Point group: D ₄
• The p 4 g group has two rotation centers of the order of four (90 °), which are mirror images of each other, but it has reflections in only two perpendicular directions. There are rotations of the order of two (180 °), the centers of which are located at the intersection of the reflection axes. The group has axes of moving symmetries parallel to the axes of reflection (between them), and also at an angle of 45 ° to them.

The p 4 g pattern can be considered as a checkerboard pattern of copies of square tiles with 4-fold rotational symmetry and their mirror images. Alternatively, the pattern can be considered (when shifted by half the tile) as a checkerboard pattern of copies of horizontally or vertically symmetrical tiles and their versions rotated 90 °. Note that both methods of consideration are not applicable to a simple checkerboard pattern of black and white tiles, in this case it is a group p 4 m (with diagonal parallel cell transfer).

Examples of the group p 4 g

•  

  Linoleum in the bathroom, USA
•  

  Painted Porcelain , China
•  

  Mosquito net, USA.
•  

  Figure, China
•  

  one of the coloring of a flat-nosed square mosaic (see also pg )

Group p 3 (333)

• Orbifold signature: 333
• Coxeter designation: [(3,3,3)] ⁺ or [3 ^[3] ] ⁺
• Lattice: hexagonal
• Point group: C ₃
• The p 3 group has three different centers of order three (120 °), but does not have mirror or moving symmetries.

Imagine a mosaic of a plane with equilateral triangles of the same size with the side corresponding to the smallest parallel translation. Then half of the triangles have one orientation, and the other half are symmetrical. A group of ornaments corresponds to the case when all triangles of the same orientation are equal, while both types have rotational symmetry of the order of three, but these two are not equal, are not mirror images of each other and both are not symmetrical (if both types are equal, we have p 6 if they are mirror images of each other, we have p 31 m , if both types are symmetrical, we have p 3 m 1 , if two of these three properties hold, then the third one takes place, and we get p 6 m ). For a given pattern, three of these tilings are possible, each with centers of rotation at the vertices, that is, two shifts are possible for any tiling. In terms of the figure: the vertices can be red, blue or green triangles.

Equivalently, imagine tiling a plane with regular hexagons with a side equal to the smallest parallel translation divided by √3. Then this group of wallpapers corresponds to the case when all the hexagons are equal (and have the same orientation) and have rotation symmetry of the order of three, but there is no mirror reflection (if they have rotational symmetry of the order of six, we obtain p 6 if there is symmetry with respect to of the main diagonal, we have p 31 m , if there is symmetry with respect to straight lines perpendicular to the sides, we have p 3 m 1 ; if two of these three properties are satisfied, then the third is also true and we have p 6 m ). For a given image, there are three tilings, each obtained when the centers of the hexagons are located at the centers of rotation of the pattern. In terms of the figure, the centers of the hexagon can be red, blue, and green triangles.

Examples of p 3

•  

  Received by computer
•  

  One of 8 semi-regular mosaics (if you ignore the colors: p 6 ). The parallel transfer vectors are slightly shifted relative to the directions of the underlying hexagonal lattice of the pattern
•  

  Street pavement in Zakopane , Poland
•  

  Mosaic on the wall in the city of Alhambra , Spain (here the wall is full ). If we ignore all the colors, we get p 3 (if we ignore only the colors of the stars, we get p 1 )

Group p 3 m 1 (* 333)

• Orbifold signature: * 333
• Coxeter designation: [(3,3,3)] or [3 ^[3] ]
• Lattice: hexagonal
• Point group: D ₃
• The p 3 m 1 group has three different centers of rotation of the order of three (120 °). The group has reflections relative to the three sides of an equilateral triangle. The centers of any rotation lie on the axes of reflection. There are additional sliding symmetries in three different directions, whose axes are located halfway between adjacent parallel reflection axes.

Like the group p 3 , imagine a plane with equilateral triangles of the same size, with a side equal to the smallest amount of parallel transfer. Then half of the triangles has one orientation, and the other half has a reverse orientation. This group of wallpapers corresponds to the case when all triangles of the same orientation are equal. Both types have rotational symmetry of the order of three, both types are symmetrical, but they are not equal and are not mirror images of each other. Three tilings are possible for a given image; each has vertices at the centers of rotation. In terms of the figure, the vertices can be red, dark blue or green triangles.

Examples of the group p 3 m 1

•  

  One of 3 correct mosaics (ignoring colors: p 6 m )
•  

  Another correct mosaic (ignoring colors: p 6 m )
•  

  One of 8 semi-regular mosaics (ignoring colors: p 6 m )
•  

  Persian glazed mosaic (ignoring colors: p 6 m )
•  

  Persian ornament
•  

  Figure, China (see detailed image)

Group p 31 m (3 * 3)

• Orbifold signature: 3 * 3
• Coxeter Designation: [6.3 ⁺ ]
• Lattice: hexagonal
• Point group: D ₃
• The p 31 m group has three different centers of rotation of the order of three (120 °), of which two are mirror images of each other. The group has three reflections in three different directions. It has at least one rotation, the center of which does not lie on the axis of mirror symmetry. There are additional sliding symmetries in three directions, the axes of which are located in the middle between adjacent parallel reflection axes.

As for p 3 and p 3 m 1 , imagine the tiling of a plane by equilateral triangles of the same size, with a side equal to the smallest parallel translation. Then half of the triangles has one orientation, and the other half has the opposite. A group of wallpapers corresponds to the case when all triangles of the same orientation are equal, while both types have rotational symmetry of the order of three and each is a mirror image of the other, but the triangles are not symmetrical and are not equal to themselves. For this image, only one tiling is possible. In terms of the figure, the vertices cannot be dark blue triangles.

Examples of the group p 31 m

•  

  Persian glazed mosaic
•  

  Painted Porcelain , China
•  

  Figure, China
•  

  Compact circles in two sizes

Group p 6 (632)

• Orbifold signature: 632
• Coxeter Designation: [6.3] ⁺
• Lattice: hexagonal
• Point group: C ₆
• Group p 6 has one center of rotation of the order of six, which differ only in rotation by 60 °. It also has two centers of rotation of the order of three, which differ only in rotation by 120 ° and three orders of two (180 °). The group has no reflections or moving symmetries.

A pattern with such symmetry can be considered a mosaic of a plane with equal triangular tiles with C ₃ symmetry, or equivalently, tiling of a plane with equal hexagonal tiles with C ₆ symmetry (while the edges of the tiles will not necessarily be part of the pattern).

Examples of group p 6

•  

  Computer generated
•  

  Regular Polygons
•  

  Wall Cladding, Alhambra , Spain
•  

  Persian ornament

Group p 6 m (* 632)

• Orbifold signature: * 632
• Coxeter Designation: [6.3]
• Lattice: hexagonal
• Point group: D ₆
• The group p 6 m has one center of rotation of the order of six (60 °). It also has two centers of rotation of the order of three, which differ only in rotation by 60 °, and three orders of two, which differ only in rotation by 60 °. The group also has reflections in six different directions. There are additional sliding symmetries in six different directions, whose axes are located in the middle between two adjacent parallel reflection axes.

A pattern with this symmetry can be considered as a mosaic on a plane with equal triangular tiles with D ₃ symmetry, or equivalently, tiling a plane with equal hexagonal tiles with D ₆ symmetry (the edges of the tiles are not necessarily part of the pattern). The simplest examples are a hexagonal grid with connecting straight lines or without them and a hexagonal mosaic with one color for the contours of hexagons and another for the background.

Examples of the group p 6 m

•  

  Computer generated
•  

  One of 8 semi-regular mosaics
•  

  Another semi-regular mosaic
•  

  Another semi-regular mosaic
•  

  Persian glazed mosaic
•  

  Clothing of the King, Dur-Sharrukin , Assyria . This is almost p 6 m (if we ignore the internal parts of the flowers, we get cmm )
•  

  Bronze vessel from Nimrud , Assyria
•  

  Byzantine marble pavement, Rome
•  

  Painted Porcelain , China
•  

  Painted Porcelain , China
•  

  Compact circles in two sizes
•  

  Another compact package of circles of two sizes

There are five types of lattices ( Bravais lattices ), corresponding to the five groups of ornaments of the lattices themselves. A group of pattern ornaments with this parallel transfer symmetry lattice cannot have more, but can have less symmetries than the lattice itself.

• In 5 cases of rotational symmetry of the order of 3 or 6, the unit cell consists of two equilateral triangles (a hexagonal lattice, p6m itself). They form diamonds with angles of 60 ° and 120 °.
• In 3 cases of rotational symmetry of order 4, the cell is a square (a square lattice, in itself p4m ).
• In 5 cases of reflection or moving symmetry, but not simultaneously, the cell is a rectangle (a rectangular lattice, in itself pmm ). Special occasions: square.
• In 2 cases of reflection, together with moving symmetry, the cell is a rhombus (rhombic lattice, cmm by itself). The lattice can be interpreted as a centered rectangular lattice. Special cases: square, hexagonal cell.
• In the case of only rotational symmetry of order 2 and the absence of other symmetries other than parallel transfer, the cell, in the general case, is a parallelogram (parallelogram or oblique lattice, p 2 itself ). Special cases: a cell in the form of a rectangle, square, rhombus, hexagon.

The actual symmetry group must be distinguished from the ornamental group. Ornament groups are a set of symmetry groups. There are 17 such sets, but for each set there are infinitely many symmetry groups in the sense of the actual isometry groups. They depend, separately from the group of ornaments, the number of parameters of the parallel transport vectors, the orientation and position of the axes of mirror symmetry and rotation centers.

The number of degrees of freedom is:

• 6 for p 2
• 5 for pmm , pmg , pgg , and cmm
• 4 for the rest.

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

Some isomorphisms of symmetry groups:

• p 1 : Z ²
• pm : Z × D _∞
• pmm : D _∞ × D _∞ .

• The group of pattern ornaments is invariant in isometry and uniform ( similarity transformation ).
• Parallel transfer is preserved during an arbitrary bijective affine transformation .
• Rotational symmetry of order two is the same. This means that the centers of 4- and 6-fold rotations retain at least 2-fold rotations.
• The reflection with respect to the direct line and the moving symmetry are preserved during tension / compression along the axis of symmetry or perpendicular to it. This changes p 6 m , p 4 g and p 3 m 1 in cmm , p 3 m 1 in cm and p 4 m depending on the direction of tension / compression, in pmm or cmm .

Note that if a transformation reduces symmetry, a transformation of the same kind (inverse), obviously, increases symmetry for the same pattern. Such a pattern property (for example, expanding in one direction gives a pattern with four-fold symmetry) is not considered a form of additional symmetry.

Replacing colors does not affect the group of ornaments if any two dots having the same color before changing also have the same color after replacing, and if any two dots having different colors before replacing have different colors after replacing.

If the first is executed, but the second is not, as in the case of reducing the image to black / white, the symmetries will remain, but may increase, so the group of wallpapers can change.

Some software products allow you to create two-dimensional patterns using symmetry groups of ornaments. You can usually edit the original tile and all copies of the tiles in the pattern are updated automatically.

• MadPattern , a free set of Adobe Illustrator templates that support 17 groups of patterns
• Tess , the nagware tiling program for a number of platforms, supports all ornaments, borders, and outlet groups, as well as Hiisha mosaics.
• Kali , an online symmetry editing applet.
• Kali , a free download program for Windows and Mac Classic.
• Inkscape , a free vector graphics editor , supports all 17 groups, plus arbitrary scaling, shifts, rotations, and color changes by row or column. (See [1] )
• SymmetryWorks is a commercial plug-in for Adobe Illustrator , supports all 17 groups.
• Arabeske is a free standalone product that supports a subset of ornament groups.

• List of planar symmetry groups (summary of this page)
• Aperiodic mosaic
• Crystallography
• Maurits Cornelis Escher
• Point symmetry group
• Mosaics

• E. Fedorov. Symmetry on the plane // Notes of the Imperial St. Petersburg Mineralogical Society. - 1891. - T. 28 .
• George Pólya. Über die Analogie der Kristallsymmetrie in der Ebene // Zeitschrift für Kristallographie. - 1924.- T. 60 .
• Paulo G. Radaelli. Symmetry in Crystallography. - Oxford University Press, 2011. - (Crystallography). - ISBN 0-19-955065-4 .
• Owen Jones. The Grammar of Ornament . - 1856. Many of the images in this article are taken from this book. The book contains many more examples.
• John H. Conway . The Orbifold Notation for Surface Groups // Groups, Combinatorics and Geometry / MW Liebeck, J. Saxl (eds.). Proceedings of the LMS Durham Symposium, July 5–15, Durham, UK, 1990; London Math. Soc .. - Cambridge: Cambridge University Press, 1992. - T. 165. - S. 438–447. - (Lecture Notes Series).
• John H. Conway , Heidi Burgiel, Chaim Goodman-Strauss. The Symmetries of Things. - Worcester MA: AK Peters, 2008 .-- ISBN 1-56881-220-5 .
• Branko Grünbaum , GC Shephard. Tilings and Patterns. - New York: Freeman, 1987 .-- ISBN 0-7167-1193-1 .
• Lewis F. Day. Pattern Design. - Mineola, New York: Dover Publications, Inc., 1933. - ISBN 0-486-40709-8 .

• 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.