A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. The mathematical study of such patterns reveals that exactly 7 different types of patterns can occur. Sphere symmetry group o. ... This article or section does not adequately cite its references or sources. ... The decorative arts are traditionally defined as ornamental and functional works in ceramic, wood, glass, metal, or textile. ...

Frieze groups are related to the more complex wallpaper groups, which classify patterns which are repetitive in two directions. 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. ...

As with wallpaper groups, a frieze group is often visualised by a simple periodic pattern in the category concerned.

General

Formally, a frieze group is a class of infinite discrete symmetry groups for patterns on a strip (infinitely wide rectangle), hence a class of groups of isometries of the plane, or of a strip. There are seven different frieze groups. The actual symmetry groups within a frieze group are characterized by the smallest translation distance, and, for the frieze groups 4-7, by a shifting parameter. In the case of symmetry groups in the plane, additional parameters are the direction of the translation vector, and, for the frieze groups 2, 3, 5, 6, and 7, the positioning perpendicular to the translation vector. Thus there are two degrees of freedom for group 1, three for groups 2, 3, and 4, and four for groups 5, 6, and 7. The symmetry group of an object (e. ... Strip can refer to: as a noun a long narrow piece cut from a sheet material (metal plastic plywood etc) a power strip a landing strip a comic strip other items of a similar shape to that above e. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... 7 (seven) is the natural number following 6 and preceding 8. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

A symmetry group of a frieze group necessarily contains translations and may contain glide reflections. Other possible group elements are reflections along the long axis of the strip, reflections along the narrow axis of the strip and 180° rotations. For two of the seven frieze groups (numbers 1 and 2 below) the symmetry groups are singly-generated, for four (numbers 3–6) they have a pair of generators, and for number 7 the symmetry groups require three generators. Look up translate in Wiktionary, the free dictionary. ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... In mathematics, a reflection (also spelt reflexion) is a map that transforms an object into its mirror image. ... A sphere rotating around its axis. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

A symmetry group in frieze group 1, 3, 4, or 5 is a subgroup of a symmetry group in the last frieze group with the same translational distance. A symmetry group in frieze group 2 or 6 is a subgroup of a symmetry group in the last frieze group with half the translational distance. This last frieze group contains the symmetry groups of the simplest periodic patterns in the strip (or the plane), a row of dots. Any transformation of the plane leaving this pattern invariant can be decomposed into a translation, (x,y) → (n+x,y), optionally followed by a reflection in either the horizontal axis, (x,y) → (x,−y), or the vertical axis, (x,y) → (−x,y), provided that this axis is chosen through or midway between two dots, or a rotation by 180°, (x,y) → (−x,−y) (ditto). Therefore, in a way, this frieze group contains the "largest" symmetry groups, which consist of all such transformations. In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

The inclusion of the discrete condition is to exclude the group containing all translations, and groups containing arbitrarily small translations (e.g. the group of horizontal translations by rational distances). Even apart from scaling and shifting, there are infinitely many cases, e.g. by considering rational numbers of which the denominators are powers of a given prime number.

The inclusion of the infinite condition is to exclude groups that have no translations:

• the group with the identity only (isomorphic to C₁, the trivial group of order 1).
• the group consisting of the identity and reflection in the horizontal axis (isomorphic to C₂, the cyclic group of order 2).
• the groups each consisting of the identity and reflection in a vertical axis (ditto)
• the groups each consisting of the identity and 180° rotation about a point on the horizontal axis (ditto)
• the groups each consisting of the identity, reflection in a vertical axis, reflection in a vertical axis, and 180° rotation about the point of intersection (isomorphic to the Klein four-group)

This picture illustrates how the hours in a clock form a group. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... This article is about the mathematical group. ...

Descriptions of the seven frieze groups

Figure 1. Patterns corresponding to the 7 frieze groups

There are seven distinct subgroups (up to scaling and shifting of patterns) in the discrete frieze group generated by a translation, reflection (along the same axis) and a 180° rotation. Each of these subgroups is the symmetry group of a frieze pattern, and sample patterns are shown in Fig. 1. The seven different groups correspond to the 7 infinite series of point groups in three dimensions, with n = . They are, with Conway's orbifold notation in parentheses: Image File history File links No higher resolution available. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

1. () Translations only. This group is singly-generated, with a generator being a translation by the distance over which the pattern is periodic. Consequently the group is isomorphic to Z, the group of integers.
2. () Glide-reflections and translations. This group is generated by a single glide reflection, with translations being obtained by combining two glide reflections. Consequently, this group is also isomorphic to Z.
3. () Translations, the reflection in the horizontal axis and glide reflections. This group is isomorphic to the direct product Z × C₂, and is generated by a translation and the reflection in the horizontal axis.
4. () Translations and reflections across certain vertical lines. The elements in this group correspond to isometries (or equivalently, bijective affine transformations) of the set of integers, and so it is isomorphic to a semidirect product of the integers with C₂. The group is generated by a translation and a reflection in a vertical axis. It is the same as the non-trivial group in the one-dimensional case
5. () Translations and 180° rotations. Again, the transformations in this group correspond to isometries of the set of integers, and so the group is isomorphic to a semidirect product of Z and C₂. The group is generated by a translation and a 180° rotation.
6. () Reflections across certain vertical lines, glide-reflections, translations and rotations. The translations here arise from the glide reflections, so this group is generated by a glide reflection and a rotation. It is isomorphic to a semi-direct product of Z and C₂.
7. () Translations, glide reflections, reflections in both axes and 180° rotations. This group is the "largest" frieze group and requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis. It is isomorphic to a semidirect product of Z × C₂ with C₂.

An example for each frieze group (in order 1, 4, 7, 2, 3, 6, 5)

Summarized: The integers are commonly denoted by the above symbol. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... A bijective function. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces (strictly speaking, two affine 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... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... A pattern in 1D can be represented as a function f(x) for, say, the color at position x. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

1. T (translation only)
2. TG (translation and glide reflection)
3. THG (translation, horizontal line reflection, and glide reflection)
4. TV (translation and vertical line reflection)
5. TR (translation and 180° rotation)
6. TRVG (translation, 180° rotation, vertical line reflection, and glide reflection)
7. TRHVG (translation, 180° rotation, horizontal line reflection, vertical line reflection, and glide reflection)

As we have seen, up to isomorphism, there are four groups, two abelian, and two non-abelian. In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... In mathematics, a reflection (also spelt reflexion) is a map that transforms an object into its mirror image. ... An object is in a vertical position when it is aligned in an up-down direction, perpendicular to the horizon. ... A sphere rotating around its axis. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

Hierarchy

With the same translation distance, sequences of increasing symmetry are 137, 147, 157, and 126; with halving of the translation distance we also have 23 and 67.

The symmetries of groups 1,3,4,5, and 7 with translation distance t imply those of the same group and translation distance nt, for an integer n. For groups 2 and 6 this is only true if n is odd.

See also

• Symmetry groups in one dimension
• Wallpaper group

A pattern in 1D can be represented as a function f(x) for, say, the color at position x. ... 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. ...

Web demo and software

There exist software graphic tools that will let you create 2D patterns using frieze groups. Usually, you can edit the original strip 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.
• FriezingWorkz, a freeware Hypercard stack for the Classic Mac platform that supports all frieze groups.

Nagware is a term of distinction used to differentiate between types of shareware software. ...

External link

• Frieze Patterns at cut-the-knot