List of planar symmetry groups

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•)		 Cn n
[n]+
 n
C1, [ ]+ (•)
C2, [2]+ (2•)
C3, [3]+ (3•)
C4, [4]+ (4•)
C5, [5]+ (5•)
C6, [6]+ (6•)
Dihedral symmetry nm
(*n•)		 Dn 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
(*∞•)		p1C∞v[1,∞]

sidle
p1
(∞•)		p1C∞[1,∞]+

hop
[2,∞+],
IUC
(orbifold)		 Geo Schönflies Coxeter Fundamental
domain		 Example
p11m
(∞*)		p. 1C∞h[2,∞+]

jump
p11g
(∞×)		p.g1S2∞[2+,∞+]

step
[2,∞],
IUC
(orbifold)		 Geo Schönflies Coxeter Fundamental
domain		 Example
p2mm
(*22∞)		p2D∞h[2,∞]

spinning jump
p2mg
(2*∞)		p2gD∞d[2+,∞]

spinning sidle
p2
(22∞)		p2D∞[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×)
pg2g		[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)
pg4		[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)
(××)
pg1
h: [∞+,(2,∞)+]

Monoglide
pg(v)
(××)
pg1		v: [(∞,2)+,∞+]

Monoglide
pgm
(22*)
pg2		h: [(∞,2)+,∞]

Digyro
pmg
(22*)
pg2		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×)
pg2g		[((∞,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]
o2222××***×22×22**22222*224424*2*442333*3333*3632*632
p1p2pgpmcmpggpmgpmmcmmp4p4gp4mp3p3m1p31mp6p6m
op1 2
2222p2 222
××pg 22
**pm 2222
*×cm 2223
22×pgg 4223
22*pmg 4222423
*2222pmm 424244222
2*22cmm 424422224
442p4 422
4*2p4g 84484244229
*442p4m 848444422222
333p3 33
*333p3m1 6663243
3*3p31m 6663234
632p6 6324
*632p6m 12612126666342223

See also

Notes
  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

References
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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
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