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In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
and decorative art. Frieze patterns can be classified into seven types according to their symmetries. The set of
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of a frieze pattern is called a frieze group. Frieze groups are two-dimensional
line group A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, ...
s, having repetition in only one direction. They are related to the more complex
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
s, which classify patterns that are repetitive in two directions, and
crystallographic group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unc ...
s, which classify patterns that are repetitive in three directions.


General

Formally, a frieze group is a class of infinite discrete symmetry groups of patterns on a strip (infinitely wide rectangle), hence a class of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of the plane, or of a strip. A symmetry group of a frieze group necessarily contains
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
s and may contain
glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection ...
s,
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
s along the long axis of the strip, reflections along the narrow axis of the strip, and 180°
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s. There are seven frieze groups, listed in the summary table. Many authors present the frieze groups in a different order. The actual symmetry groups within a frieze group are characterized by the smallest translation distance, and, for the frieze groups with vertical line reflection or 180° rotation (groups 2, 5, 6, and 7), by a shift parameter locating the reflection axis or point of rotation. In the case of symmetry groups in the plane, additional parameters are the direction of the translation vector, and, for the frieze groups with horizontal line reflection, glide reflection, or 180° rotation (groups 3–7), the position of the reflection axis or rotation point in the direction 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. For two of the seven frieze groups (groups 1 and 4) the symmetry groups are singly generated, for four (groups 2, 3, 5, and 6) they have a pair of generators, and for group 7 the symmetry groups require three generators. A symmetry group in frieze group 1, 2, 3, or 5 is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of a symmetry group in the last frieze group with the same translational distance. A symmetry group in frieze group 4 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, , optionally followed by a reflection in either the horizontal axis, , or the vertical axis, , provided that this axis is chosen through or midway between two dots, or a rotation by 180°, (ditto). Therefore, in a way, this frieze group contains the "largest" symmetry groups, which consist of all such transformations. 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 In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
of order 1). *the group consisting of the identity and reflection in the horizontal axis (isomorphic to C2, the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
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 the horizontal axis, and 180° rotation about the point of intersection (isomorphic to the
Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
)


Descriptions of the seven 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 axial point groups in three dimensions, with ''n'' = ∞. They are identified in the table below using
Hermann–Mauguin notation In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogis ...
(or
IUC notation The International Union of Crystallography (IUCr) is an organisation devoted to the international promotion and coordination of the science of crystallography. The IUCr is a member of the International Council for Science (ICSU). Objectives ...
), Coxeter notation,
Schönflies notation The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...
,
orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...
, nicknames created by mathematician
John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
, and finally a description in terms of translation, reflections and rotations. Of the seven frieze groups, there are only four up to
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. Two are singly generated and isomorphic to \mathbb; four of them are doubly generated, among which one is abelian and three are nonabelian and isomorphic to D_\infty, the
infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1'', ...
; and one of them has three generators.


Lattice types: Oblique and rectangular

The groups can be classified by their type of two-dimensional grid or lattice. The lattice being oblique means that the second direction ''need not be orthogonal'' to the direction of repeat.


See also

*
Symmetry groups in one dimension A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D). A pattern in 1D can be represented as a function ''f''(''x'') for, say, the color at position ''x''. The only nontrivial point group in 1 ...
*
Line group A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, ...
*
Rod group In mathematics, a rod group is a three-dimensional line group whose point group is one of the axial crystallographic point groups. This constraint means that the point group must be the symmetry of some three-dimensional lattice. Table of the 75 r ...
*
Wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
*
Space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it uncha ...


Web demo and software

There exist software graphic tools that create 2D patterns using frieze groups. Usually, the entire pattern is updated automatically in response to edits of the original strip.
EscherSketch
A free online program for drawing, saving, and exporting tessellations. Supports all wallpaper groups.

a
free and open source software Free and open-source software (FOSS) is a term used to refer to groups of software consisting of both free software and open-source software where anyone is freely licensed to use, copy, study, and change the software in any way, and the source ...
application for wallpaper, frieze and other patterns.
Kali
, free downloadable Kali for Windows and Mac Classic.
Tess
a
nagware Shareware is a type of proprietary software that is initially shared by the owner for trial use at little or no cost. Often the software has limited functionality or incomplete documentation until the user sends payment to the software developer ...
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.


References


External links


Frieze Patterns
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Illuminations: Frieze Patterns
{{DEFAULTSORT:Frieze Group Euclidean symmetries Discrete groups