This article summarizes the classes of

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

s of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

. The symmetry groups are named here by three naming schemes: International notation,

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 ...

, and

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...

. There are three kinds of symmetry groups of the plane: *2 families of rosette groups – 2D

point groups In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...

frieze group In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetri ...

s – 2D line groups *17

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 transformatio ...

s – 2D

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 unchan ...

Rosette groups

There are two families of discrete two-dimensional point groups, and they are specified with parameter ''n'', which is the order of the group of the rotations in the group.

Frieze groups

The 7

s, the 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, with a direction of periodicity are given with five notational names. The

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 ...

is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each.

Wallpaper groups

The 17

s, with finite fundamental domains, are given by International notation,

, and

, classified by the 5 Bravais lattices in the plane:

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular). The ''p1'' and ''p2'' groups, with no reflectional symmetry, are repeated in all classes. The related pure reflectional

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

are given with all classes except oblique.

Wallpaper subgroup relationships

Notes

References

* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (Orbifold notation for polyhedra, Euclidean and hyperbolic tilings) * ''On Quaternions and Octonions'', 2003, John Horton Conway and Derek A. Smith * Kaleidoscopes: Selected Writings of

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380–407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559–591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3–45* * N. W. Johnson: ''Geometries and Transformations'', (2018) Chapter 12: ''Euclidean Symmetry Groups''

External links

"Conway's manuscript" on Orbifold notation
(Notation changed from this original, ''x'' is now used in place of open-dot, and o is used in place of the closed dot)

{{DEFAULTSORT:Planar symmetry groups Euclidean symmetries Mathematics-related lists