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A frieze group is a mathematical concept used to classify designs on two-dimensional
surfaces that are repetitive in one direction, according to the symmetries of the pattern.
Such patterns occur frequently in architecture and decorative art. The mathematical
study of such patterns reveals that exactly seven types of symmetry can occur.
Frieze groups are two-dimensional line groups, having repetition in only one direction.
They are related to the more complex wallpaper groups, which classify patterns that
are repetitive in two directions, and crystallographic groups, which classify patterns
that are repetitive in three directions.
Examples of frieze group patterns
1 General
2 Descriptions of the seven frieze groups
2.1 Lattice types: Oblique and rectangular
4 Web demo and software
5 References
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 of isometries of the plane, or of a strip. A symmetry group
of a frieze group necessarily contains translations and may contain
glide reflections, reflections along the long axis of the strip,
reflections along the narrow axis of the strip, and 180&deg; rotations.
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&deg; 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&deg; 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.
The seven frieze groups
1. p1: T (translation only, in the horizontal
direction)
2. p1m1: TV (translation and vertical line
reflection)
3. p11m: THG (translation, horizontal line
reflection, and glide reflection)
4. p11g: TG (translation and glide reflection)
5. p2: TR (translation and 180&deg; rotation)
6. p2mg: TRVG (translation, 180&deg; rotation,
vertical line reflection, and glide reflection)
7. p2mm: TRHVG (translation, 180&deg; rotation,
horizontal line reflection, vertical line reflection,
and glide reflection)
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 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.
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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&deg;, (x,y) → (−x,−y) (ditto). Therefore, in a
way, this frieze group contains the &quot;largest&quot; 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 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&deg; 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&deg; rotation
about the point of intersection (isomorphic to the Klein four-group)
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&deg; 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 using Hermann–Mauguin notation or IUC notation, orbifold notation, Coxeter notation, and Sch&ouml;nflies
notation:
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Frieze groups
IUC
Cox
p2mm
[∞,2]
Sch&ouml;n*
Struct.
Diagram&sect;
Orbifold
[∞,2+]
[∞,2]+
p2
(TR) Translations and 180&deg; Rotations:
The group is generated by a translation and a 180&deg;
rotation.
[∞]
[∞+,2]
[∞+,2+]
[∞]+
jump
S∞
Z∞
∞&times;
p1
sidle
C∞h
Z∞&times;Dih1
∞*
p11g
spinning hop
C∞v
Dih∞
*∞∞
p11m
spinning sidle
(TRVG) Vertical reflection lines, Glide reflections,
Translations and 180&deg; Rotations:
The translations here arise from the glide reflections,
so this group is generated by a glide reflection and
either a rotation or a vertical reflection.
D∞
Dih∞
22∞
p1m1
spinning jump
D∞d
Dih∞
2*∞
Description
(TRHVG) Horizontal and Vertical reflection lines,
Translations and 180&deg; Rotations:
This group requires three generators, with one
generating set consisting of a translation, the
reflection in the horizontal axis and a reflection
across a vertical axis.
D∞h
Dih∞&times;Dih1
*22∞
p2mg
Examples
nickname
step
C∞
Z∞
∞∞
hop
(TV) Vertical reflection lines and Translations:
The group is the same as the non-trivial group in the
one-dimensional case; it is generated by a translation
and a reflection in the vertical axis.
(THG) Translations, Horizontal reflections, Glide
reflections:
This group is generated by a translation and the
reflection in the horizontal axis. The glide reflection
here arises as the composition of translation and
horizontal reflection
(TG) Glide-reflections and Translations:
This group is singly generated, by a glide reflection,
with translations being obtained by combining two
glide reflections.
(T) Translations only:
This group is singly generated, by a translation by the
smallest distance over which the pattern is periodic.
*Sch&ouml;nflies's
point group notation is extended here as infinite cases of the equivalent dihedral points symmetries
diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed
green, translation normals in red, and 2-fold gyration points as small green squares.
&sect;The
As we have seen, up to isomorphism, there are four groups, two abelian, and two non-abelian.
Lattice types: Oblique and rectangular
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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.
Lattice type
Oblique
Groups
p1, p2
Rectangular p1m1, p11m, p11g, p2mm, p2mg
Symmetry groups in one dimension
Line group
Rod group
Wallpaper group
Space group
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.
Kali (http://www.geom.uiuc.edu/java/Kali/welcome.html), a free and open source software application for wallpaper,
frieze and other patterns.
Tess (http://www.peda.com/tess/), a nagware tessellation program for multiple platforms, supports all wallpaper, frieze,
and rosette groups, as well as Heesch tilings.
FriezingWorkz (http://apronyms.com/software/friezingworkz.html), a freeware Hypercard stack for the Classic Mac
platform that supports all frieze groups.
1. Coxeter, H. S. M. (1969). Introduction to Geometry. New York: John Wiley &amp; Sons. pp. 47–49. ISBN 0-471-50458-0.
2. Cederberg, Judith N. (2001). A Course in Modern Geometries, 2nd ed. New York: Springer-Verlag. pp. 117–118, 165–171.
ISBN 0-387-98972-2.
3. Fisher, G.L.; Mellor, B. (2007), &quot;Three-dimensional finite point groups and the symmetry of beaded beads&quot; (PDF), Journal for
Mathematics and the Arts
4. Radaelli, Paolo G., Fundamentals of Crystallographic Symmetry (PDF)
5. Hitzer, E.S.M.; Ichikawa, D. (2008), &quot;Representation of crystallographic subperiodic groups by geometric algebra&quot; (PDF), Electronic
Proc. of AGACSE (Leipzig, Germany) (3, 17–19 Aug. 2008)
Frieze Patterns (http://www.cut-the-knot.org/triangle/Frieze.shtml) at cut-the-knot
Illuminations: Frieze Patterns (http://illuminations.nctm.org/ActivityDetail.aspx?id=168)