List of planar symmetry groups

Last updated April 06, 2022

This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane:

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.

Family	Intl (orbifold)	Schön.	Geo ^[1] Coxeter	Order	Examples
Cyclic symmetry	n (n•)	C_n	n [n]⁺	n	C₁, [ ]⁺ (•)	C₂, [2]⁺ (2•)	C₃, [3]⁺ (3•)	C₄, [4]⁺ (4•)	C₅, [5]⁺ (5•)	C₆, [6]⁺ (6•)
Dihedral symmetry	nm (*n•)	D_n	n [n]	2n	D₁, [ ] (*•)	D₂, [2] (*2•)	D₃, [3] (*3•)	D₄, [4] (*4•)	D₅, [5] (*5•)	D₆, [6] (*6•)

Frieze groups

The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each.

[1,∞],
IUC (orbifold)	Geo	Schönflies	Coxeter	Fundamental domain	Example
p1m1 (*∞•)	p1	C_∞v	[1,∞]		sidle
p1 (∞•)	p1	C_∞	[1,∞]⁺		hop

[2,∞⁺],
IUC (orbifold)	Geo	Schönflies	Coxeter	Fundamental domain	Example
p11m (∞*)	p. 1	C_∞h	[2,∞⁺]		jump
p11g (∞×)	p._g1	S_2∞	[2⁺,∞⁺]		step

[2,∞],
IUC (orbifold)	Geo	Schönflies	Coxeter	Example
p2mm (*22∞)	p2	D_∞h	[2,∞]	spinning jump
p2mg (2*∞)	p2_g	D_∞d	[2⁺,∞]	spinning sidle
p2 (22∞)	p2	D_∞	[2,∞]⁺	spinning hop

Wallpaper groups

The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, 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 are given with all classes except oblique.

Square
[4,4],
IUC (Orb.) Geo	Coxeter	Domain Conway name
p1 (°) p1		Monotropic
p2 (2222) p2	[4,1⁺,4]⁺ [1⁺,4,4,1⁺]⁺	Ditropic
pgg (22×) p_g2_g	[4⁺,4⁺]	Diglide
pmm (*2222) p2	[4,1⁺,4] [1⁺,4,4,1⁺]	Discopic
cmm (2*22) c2	[(4,4,2⁺)]	Dirhombic
p4 (442) p4	[4,4]⁺	Tetratropic
p4g (4*2) p_g4	[4⁺,4]	Tetragyro
p4m (*442) p4	[4,4]	Tetrascopic

Rectangular
[∞_h,2,∞_v],
IUC (Orb.) Geo	Coxeter	Domain Conway name
p1 (°) p1	[∞⁺,2,∞⁺]	Monotropic
p2 (2222) p2	[∞,2,∞]⁺	Ditropic
pg(h) (××) p_g1	h: [∞⁺,(2,∞)⁺]	Monoglide
pg(v) (××) p_g1	v: [(∞,2)⁺,∞⁺]	Monoglide
pgm (22*) p_g2	h: [(∞,2)⁺,∞]	Digyro
pmg (22*) p_g2	v: [∞,(2,∞)⁺]	Digyro
pm(h) (**) p1	h: [∞⁺,2,∞]	Monoscopic
pm(v) (**) p1	v: [∞,2,∞⁺]	Monoscopic
pmm (*2222) p2	[∞,2,∞]	Discopic

Rhombic
[∞_h,2⁺,∞_v],
IUC (Orb.) Geo	Coxeter	Domain Conway name
p1 (°) p1	[∞⁺,2⁺,∞⁺]	Monotropic
p2 (2222) p2	[∞,2⁺,∞]⁺	Ditropic
cm(h) (*×) c1	h: [∞⁺,2⁺,∞]	Monorhombic
cm(v) (*×) c1	v: [∞,2⁺,∞⁺]	Monorhombic
pgg (22×) p_g2_g	[((∞,2)⁺)^[2]]	Diglide
cmm (2*22) c2	[∞,2⁺,∞]	Dirhombic

Parallelogrammatic (oblique)
p1 (°) p1		Monotropic
p2 (2222) p2		Ditropic

Hexagonal/Triangular
[6,3], / [3^[3]],
IUC (Orb.) Geo	Coxeter	Domain Conway name
p1 (°) p1		Monotropic
p2 (2222) p2	[6,3]^Δ	Ditropic
cmm (2*22) c2	[6,3]^⅄	Dirhombic
p3 (333) p3	[1⁺,6,3⁺] [3^[3]]⁺	Tritropic
p3m1 (*333) p3	[1⁺,6,3] [3^[3]]	Triscopic
p31m (3*3) h3	[6,3⁺]	Trigyro
p6 (632) p6	[6,3]⁺	Hexatropic
p6m (*632) p6	[6,3]	Hexascopic

Wallpaper subgroup relationships

Subgroup relationships among the 17 wallpaper group^[2]
		o	2222	××	**	*×	22×	22*	*2222	2*22	442	4*2	*442	333	*333	3*3	632	*632
		p1	p2	pg	pm	cm	pgg	pmg	pmm	cmm	p4	p4g	p4m	p3	p3m1	p31m	p6	p6m
o	p1	2
2222	p2	2	2	2
××	pg	2		2
**	pm	2		2	2	2
*×	cm	2		2	2	3
22×	pgg	4	2	2			3
22*	pmg	4	2	2	2	4	2	3
*2222	pmm	4	2	4	2	4	4	2	2	2
2*22	cmm	4	2	4	4	2	2	2	2	4
442	p4	4	2								2
4*2	p4g	8	4	4	8	4	2	4	4	2	2	9
*442	p4m	8	4	8	4	4	4	4	2	2	2	2	2
333	p3	3												3
*333	p3m1	6		6	6	3								2	4	3
3*3	p31m	6		6	6	3								2	3	4
632	p6	6	3											2			4
*632	p6m	12	6	12	12	6	6	6	6	3				4	2	2	2	3

Notes

↑ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF
↑ Coxeter, (1980), The 17 plane groups, Table 4

Related Research Articles

Here a wallpaper is a drawing that covers 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 transformations, with function composition as the group operation. Thus, a wallpaper group is in a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations and tiles as well as wallpaper.

Schläfli symbol Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form $that defines regular polytopes and tessellations.$

In geometry, a point group is a mathematical group of symmetry operations that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.

In geometry, orbifold notation 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 advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

Order-4 pentagonal tiling Regular tiling of the hyperbolic plane

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} and is self-dual.

In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.

In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.

In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,8}.

In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.

In 4-dimensional geometry, there are 9 uniform polytopes with A₄ symmetry. There is one self-dual regular form, the 5-cell with 5 vertices.

In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

References

The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Orbifold notation for polyhedra, Euclidean and hyperbolic tilings)
On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
Kaleidoscopes: Selected Writings of H. S. M. Coxeter , edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.
N. W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 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)
The 17 Wallpaper Groups

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[1] The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF

[2] Coxeter, (1980), The 17 plane groups, Table 4

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