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List of planar symmetry groups

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Summary

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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:

  • 2 families of rosette groups – 2D point groups
  • 7 frieze groups – 2D line groups
  • 17 wallpaper groups – 2D space groups.

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.

FamilyIntl
(orbifold)Schön.Geo
CoxeterOrderExamplesCyclic symmetryDihedral symmetry
n
(n•)Cn
[n]+
n[[File:Cyclic symmetry 1.svg80px]]
C1, [ ]+ (•)[[File:Cyclic symmetry 2.svg80px]]
C2, [2]+ (2•)[[File:Cyclic symmetry 3.png80px]]
C3, [3]+ (3•)[[File:Cyclic symmetry 4.png80px]]
C4, [4]+ (4•)
nm
(*n•)Dnn
[n]
2n[[File:Dihedral symmetry domains 1.svg80px]]
D1, [ ] (*•)[[File:Dihedral symmetry domains 2.svg80px]]
D2, [2] (*2•)[[File:Dihedral symmetry domains 3.svg80px]]
D3, [3] (*3•)[[File:Dihedral symmetry domains 4.svg80px]]
D4, [4] (*4•)

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.

IUC
(orbifold)GeoSchönfliesCoxeterFundamental
domainExample
p1m1
(*∞•)p1C∞v[1,∞]
[[File:Frieze group m1.png75px]][[File:Frieze example p1m1.png120px]][[File:Frieze_sidle.png120px]]sidle
p1
(∞•)pC∞[1,∞]+
[[File:Frieze group 11.png75px]][[File:Frieze example p1.png120px]][[File:Frieze_hop.png120px]]hop
IUC
(orbifold)GeoSchönfliesCoxeterFundamental
domainExample
p11m
(∞*)p. 1C∞h[2,∞+]
[[File:Frieze group 1m.png75px]][[File:Frieze example p11m.png120px]][[File:Frieze_jump.png120px]]jump
p11g
(∞×)p.g1S2∞[2+,∞+]
[[File:Frieze group 1g.png75px]][[File:Frieze example p11g.png120px]][[File:Frieze_step.png120px]]step

|

IUC
(orbifold)GeoSchönfliesCoxeterFundamental
domainExample
p2mm
(*22∞)p2D∞h[2,∞]
[[File:Frieze group mm.png75px]][[File:Frieze example p2mm.png120px]][[File:Frieze_spinning_jump.png120px]]spinning jump
p2mg
(2*∞)p2gD∞d[2+,∞]
[[File:Frieze group mg.png75px]][[File:Frieze example p2mg.png120px]][[File:Frieze_spinning_sidle.png120px]]spinning sidle
p2
(22∞)pD∞[2,∞]+
[[File:Frieze group 12.png75px]][[File:Frieze example p2.png120px]][[File:Frieze_spinning_hop.png120px]]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.

IUC
(Orb.)
GeoCoxeterDomainConway name
p1
(°)
p[[Image:Wallpaper group diagram p1 square.svg60px]]Monotropic
p2
(2222)
p[4,1+,4]+
[1+,4,4,1+]+
[[Image:Wallpaper group diagram p2 square.svg60px]]Ditropic
pgg
(22×)
pg2g[4+,4+]
[[Image:Wallpaper group diagram pgg square.svg60px]]Diglide
pmm
(*2222)
p2[4,1+,4]
[1+,4,4,1+]
[[Image:Wallpaper group diagram pmm square.svg60px]]Discopic
cmm
(2*22)
c2[(4,4,2+)]
[[Image:Wallpaper group diagram cmm square.svg60px]]Dirhombic
p4
(442)
p[4,4]+
[[Image:Wallpaper group diagram p4 square.svg60px]]Tetratropic
p4g
(4*2)
pg4[4+,4]
[[Image:Wallpaper group diagram p4g square.svg60px]]Tetragyro
p4m
(*442)
p4[4,4]
[[Image:Wallpaper group diagram p4m square.svg60px]]Tetrascopic

|

IUC
(Orb.)
GeoCoxeterDomainConway name
p1
(°)
p[∞+,2,∞+]
[[Image:Wallpaper group diagram p1 rect.svg100px]]Monotropic
p2
(2222)
p[∞,2,∞]+
[[Image:Wallpaper group diagram p2 rect.svg100px]]Ditropic
pg(h)
(××)
pg1
h: [∞+,(2,∞)+]
[[Image:Wallpaper group diagram pg.svg100px]]Monoglide
pg(v)
(××)
pg1v: [(∞,2)+,∞+]
[[Image:Wallpaper group diagram pg rotated.svg90px]]Monoglide
pgm
(22*)
pg2h: [(∞,2)+,∞]
[[Image:Wallpaper group diagram pmg.svg100px]]Digyro
pmg
(22*)
pg2v: [∞,(2,∞)+]
[[Image:Wallpaper group diagram pmg rotated.svg90px]]Digyro
pm(h)
(**)
p1h: [∞+,2,∞]
[[Image:Wallpaper group diagram pm.svg100px]]Monoscopic
pm(v)
(**)
p1v: [∞,2,∞+]
[[Image:Wallpaper group diagram pm rotated.svg90px]]Monoscopic
pmm
(*2222)
p2[∞,2,∞]
[[Image:Wallpaper group diagram pmm.svg100px]]Discopic

|

IUC
(Orb.)
GeoCoxeterDomainConway name
p1
(°)
p[∞+,2+,∞+]
[[Image:Wallpaper group diagram p1 rhombic.svg90px]]Monotropic
p2
(2222)
p[∞,2+,∞]+[[Image:Wallpaper group diagram p2 rhombic.svg90px]]Ditropic
cm(h)
(*×)
c1h: [∞+,2+,∞]
[[Image:Wallpaper group diagram cm.svg100px]]Monorhombic
cm(v)
(*×)
c1v: [∞,2+,∞+]
[[Image:Wallpaper group diagram cm rotated.svg90px]]Monorhombic
pgg
(22×)
pg2g[((∞,2)+)[2]][[Image:Wallpaper group diagram pgg.svg90px]]Diglide
cmm
(2*22)
c2[∞,2+,∞]
[[Image:Wallpaper group diagram cmm.svg100px]]Dirhombic
p2
(2222)
p[[Image:Wallpaper group diagram p2.svg110px]]Ditropic

|

IUC
(Orb.)
GeoCoxeterDomainConway name
p1
(°)
p[[Image:Wallpaper group diagram p1 half.svg100px]]Monotropic
p2
(2222)
p[6,3]Δ[[Image:Wallpaper group diagram p2 half.svg100px]]Ditropic
cmm
(2*22)
c2[6,3]⅄[[Image:Wallpaper group diagram cmm half.svg100px]]Dirhombic
p3
(333)
p[1+,6,3+]
[3[3]]+[[Image:Wallpaper group diagram p3.svg100px]]Tritropic
p3m1
(*333)
p3[1+,6,3]
[3[3]]
[[Image:Wallpaper group diagram p3m1.svg100px]]Triscopic
p31m
(3*3)
h3[6,3+]
[[Image:Wallpaper group diagram p31m.svg100px]]Trigyro
p6
(632)
p[6,3]+
[[Image:Wallpaper group diagram p6.svg100px]]Hexatropic
p6m
(*632)
p6[6,3]
[[Image:Wallpaper group diagram p6m.svg100px]]Hexascopic

|}

Wallpaper subgroup relationships

o2222××***×22×22**22222*224424*2*442333*3333*3632*632p1p2pgpmcmpggpmgpmmcmmp4p4gp4mp3p3m1p31mp6p6mop12222p2××pg**pm*×cm22×pgg22*pmg*2222pmm2*22cmm442p44*2p4g*442p4m333p3*333p3m13*3p31m632p6*632p6m
2
22
22
2222
2223
4223
4222423
424244222
424422224
422
84484244229
848444422222
33
6663243
6663234
6324
12612126666342223

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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (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]
  • N. W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean Symmetry Groups

References

  1. ''The Crystallographic Space groups in Geometric algebra'', [[David Hestenes. D. Hestenes]] and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) [[PDF]] [https://davidhestenes.net/geocalc/pdf/CrystalGA.pdf]
  2. Coxeter, (1980), The 17 plane groups, Table 4
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