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.

Family Intl(orbifold) Schön. Geo The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF Coxeter OrderExamplesCyclic symmetry n(n•) Cn [n]+ nC1, [ ]+ (•)C2, + (2•)C3, + (3•)C4, + (4•)C5, + (5•)C6, + (6•)Dihedral symmetrynm(*n•) Dn n[n]2nD1, [ ] (*•)D2, (*2•)D3, (*3•)D4, (*4•)D5, (*5•)D6, (*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)GeoSchönfliesCoxeterFundamentaldomainExamplep1m1(*∞•)p1C∞v[1,∞]sidlep1(∞•)pC∞[1,∞]+hop

+ [2,∞+], IUC(orbifold)GeoSchönfliesCoxeterFundamentaldomainExamplep11m(∞*)p. 1C∞h[2,∞+]jumpp11g(∞×)p.g1S2∞[2+,∞+]step+ [2,∞], IUC(orbifold)GeoSchönfliesCoxeterFundamentaldomainExamplep2mm(*22∞)p2D∞h[2,∞]spinning jumpp2mg(2*∞)p2gD∞d[2+,∞]spinning sidlep2(22∞)pD∞[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.)GeoCoxeterDomainConway name p1(°)p Monotropic p2(2222)p[4,1+,4]+[1+,4,4,1+]+Ditropicpgg(22×)pg2g[4+,4+]Diglidepmm(*2222)p2[4,1+,4][1+,4,4,1+]Discopiccmm(2*22)c2[(4,4,2+)]Dirhombicp4(442)p[4,4]+Tetratropicp4g(4*2)pg4[4+,4]Tetragyrop4m(*442)p4[4,4]Tetrascopic+ Rectangular[∞h,2,∞v], IUC(Orb.)GeoCoxeterDomainConway name p1(°)p[∞+,2,∞+]Monotropic p2(2222)p[∞,2,∞]+Ditropicpg(h)(××)pg1h: [∞+,(2,∞)+]Monoglidepg(v)(××)pg1v: [(∞,2)+,∞+]Monoglidepgm(22*)pg2h: [(∞,2)+,∞]Digyropmg(22*)pg2v: [∞,(2,∞)+]Digyropm(h)(**)p1h: [∞+,2,∞]Monoscopicpm(v)(**)p1v: [∞,2,∞+]Monoscopicpmm(*2222)p2[∞,2,∞]Discopic+ Rhombic[∞h,2+,∞v], IUC(Orb.)GeoCoxeterDomainConway name p1(°)p[∞+,2+,∞+]Monotropic p2(2222)p[∞,2+,∞]+Ditropiccm(h)(*×)c1h: [∞+,2+,∞]Monorhombiccm(v)(*×)c1v: [∞,2+,∞+]Monorhombicpgg(22×)pg2g[((∞,2)+)]Diglidecmm(2*22)c2[∞,2+,∞]Dirhombic + Parallelogrammatic (oblique) p1(°)p Monotropic p2(2222)p Ditropic+ Hexagonal/Triangular[6,3], / [3], IUC(Orb.)GeoCoxeterDomainConway name p1(°)p Monotropic p2(2222)p[6,3]ΔDitropiccmm(2*22)c2[6,3]⅄Dirhombicp3(333)p[1+,6,3+][3]+Tritropic p3m1(*333)p3[1+,6,3][3]Triscopic p31m(3*3)h3[6,3+]Trigyro p6(632)p[6,3]+Hexatropicp6m(*632)p6[6,3]Hexascopic

Wallpaper subgroup relationships + Subgroup relationships among the 17 wallpaper groupCoxeter, (1980), The 17 plane groups, Table 4o2222××***×22×22**22222*224424*2*442333*3333*3632*632p1p2pgpmcmpggpmgpmmcmmp4p4gp4mp3p3m1p31mp6p6mop122222p222××pg22**pm2222*×cm222322×pgg422322*pmg4222423*2222pmm4242442222*22cmm424422223442p44224*2p4g84484244229*442p4m848444422222333p333*333p3m166632433*3p31m6663234632p66324*632p6m12612126666342223

See also List of spherical symmetry groups Orbifold notation#Hyperbolic plane - Hyperbolic symmetry groups

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, (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

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