List of planar symmetry groups

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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:

Rosette groups[edit]

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]+
CDel node h2.pngCDel n.pngCDel node h2.png
n Cyclic symmetry 1.svg
C1, [ ]+ (•)
Cyclic symmetry 2.svg
C2, [2]+ (2•)
Cyclic symmetry 3.png
C3, [3]+ (3•)
Cyclic symmetry 4.png
C4, [4]+ (4•)
Cyclic symmetry 5.png
C5, [5]+ (5•)
Cyclic symmetry 6.png
C6, [6]+ (6•)
Dihedral symmetry nm
(*n•)
Dn n
[n]
CDel node.pngCDel n.pngCDel node.png
2n Dihedral symmetry domains 1.png
D1, [ ] (*•)
Dihedral symmetry domains 2.png
D2, [2] (*2•)
Dihedral symmetry domains 3.png
D3, [3] (*3•)
Dihedral symmetry domains 4.png
D4, [4] (*4•)
Dihedral symmetry domains 5.png
D5, [5] (*5•)
Dihedral symmetry domains 6.png
D6, [6] (*6•)

Frieze groups[edit]

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,∞], CDel node h2.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
IUC
(orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p1
(∞•)
p1 C [1,∞]+
CDel node h2.pngCDel 2.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Frieze group 11.png Frieze example p1.png
Frieze hop.png
p1m1
(*∞•)
p1 C∞v [1,∞]
CDel node h2.pngCDel 2.pngCDel node c2.pngCDel infin.pngCDel node c6.png
Frieze group m1.png Frieze example p1m1.png
Frieze sidle.png
[2,∞+], CDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel h.pngCDel node.png
IUC
(orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p11g
(∞×)
p.g1 S2∞ [2+,∞+]
CDel node h2.pngCDel 2x.pngCDel node h4.pngCDel infin.pngCDel node h2.png
Frieze group 1g.png Frieze example p11g.png
Frieze step.png
p11m
(∞*)
p. 1 C∞h [2,∞+]
CDel node c2.pngCDel 2.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Frieze group 1m.png Frieze example p11m.png
Frieze jump.png
[2,∞], CDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
IUC
(orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p2
(22∞)
p2 D [2,∞]+
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Frieze group 12.png Frieze example p2.png
Frieze spinning hop.png
p2mg
(2*∞)
p2g D∞d [2+,∞]
CDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node c2.png
Frieze group mg.png Frieze example p2mg.png
Frieze spinning sidle.png
p2mm
(*22∞)
p2 D∞h [2,∞]
CDel node c5.pngCDel 2.pngCDel node c2.pngCDel infin.pngCDel node c6.png
Frieze group mm.png Frieze example p2mm.png
Frieze spinning jump.png

Wallpaper groups[edit]

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], CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
IUC
(Orb.)
Geo
Coxeter Fundamental
domain
p1
(°)
p1
Wallpaper group diagram p1 square.svg
p2
(2222)
p2
[4,1+,4]+
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[1+,4,4,1+]+
CDel node h0.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node h0.png
Wallpaper group diagram p2 square.svg
pgg
(22×)
pg2g
[4+,4+]
CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png
Wallpaper group diagram pgg square.svg
pmm
(*2222)
p2
[4,1+,4]
CDel node.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node.png
[1+,4,4,1+]
CDel node h0.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node h0.png
Wallpaper group diagram pmm square.svg
cmm
(2*22)
c2
[(4,4,2+)]
CDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
Wallpaper group diagram cmm square.svg
p4
(442)
p4
[4,4]+
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png
Wallpaper group diagram p4 square.svg
p4g
(4*2)
pg4
[4+,4]
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node.png
Wallpaper group diagram p4g square.svg
p4m
(*442)
p4
[4,4]
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Wallpaper group diagram p4m square.svg
Rectangular
[∞h,2,∞v], CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
IUC
(Orb.)
Geo
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2,∞+]
CDel labelinfin.pngCDel branch h2h2.pngCDel 2.pngCDel branch h2h2.pngCDel labelinfin.png
Wallpaper group diagram p1 rect.svg
p2
(2222)
p2
[∞,2,∞]+
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram p2 rect.svg
pg(h)
(××)
pg1
h: [∞+,(2,∞)+]
CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram pg.svg
pg(v)
(××)
pg1
v: [(∞,2)+,∞+]
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram pg rotated.svg
pgm
(22*)
pg2
h: [(∞,2)+,∞]
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node.png
Wallpaper group diagram pmg.svg
pmg
(22*)
pg2
v: [∞,(2,∞)+]
CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram pmg rotated.svg
pm(h)
(**)
p1
h: [∞+,2,∞]
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Wallpaper group diagram pm.svg
pm(v)
(**)
p1
v: [∞,2,∞+]
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram pm rotated.svg
pmm
(*2222)
p2
[∞,2,∞]
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Wallpaper group diagram pmm.svg
Rhombic
[∞h,2+,∞v], CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node.png
IUC
(Orb.)
Geo
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2+,∞+]
CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h4.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram p1 rhombic.svg
p2
(2222)
p2
[∞,2+,∞]+
CDel label2.pngCDel branch h2h2.pngCDel 2.pngCDel iaib.pngCDel 2.pngCDel branch h2h2.pngCDel label2.png
Wallpaper group diagram p2 rhombic.svg
cm(h)
(*×)
c1
h: [∞+,2+,∞]
CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node.png
Wallpaper group diagram cm.svg
cm(v)
(*×)
c1
v: [∞,2+,∞+]
CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h4.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram cm rotated.svg
pgg
(22×)
pg2g
[((∞,2)+)[2]]
CDel node h2.pngCDel split1-2i.pngCDel nodes h4h4.pngCDel split2-i2.pngCDel node h2.png
Wallpaper group diagram pgg.svg
cmm
(2*22)
c2
[∞,2+,∞]
CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node.png
Wallpaper group diagram cmm.svg
Parallelogrammatic (oblique)
p1
(°)
p1
Wallpaper group diagram p1.svg
p2
(2222)
p2
Wallpaper group diagram p2.svg
Hexagonal/Triangular
[6,3], CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png / [3[3]], CDel node.pngCDel split1.pngCDel branch.png
p1
(°)
p1
Wallpaper group diagram p1 half.svg
p2
(2222)
p2
[6,3]Δ Wallpaper group diagram p2 half.svg
cmm
(2*22)
c2
[6,3] Wallpaper group diagram cmm half.svg
p3
(333)
p3
[1+,6,3+]
CDel node h0.pngCDel 6.pngCDel node h2.pngCDel 3.pngCDel node h2.png
[3[3]]+
CDel branch h2h2.pngCDel split2.pngCDel node h2.png
Wallpaper group diagram p3.svg
p3m1
(*333)
p3
[1+,6,3]
CDel node h0.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
[3[3]]
CDel branch.pngCDel split2.pngCDel node.png
Wallpaper group diagram p3m1.svg
p31m
(3*3)
h3
[6,3+]
CDel node.pngCDel 6.pngCDel node h2.pngCDel 3.pngCDel node h2.png
Wallpaper group diagram p31m.svg
p6
(632)
p6
[6,3]+
CDel node h2.pngCDel 6.pngCDel node h2.pngCDel 3.pngCDel node h2.png
Wallpaper group diagram p6.svg
p6m
(*632)
p6
[6,3]
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Wallpaper group diagram p6m.svg

Wallpaper subgroup relationships[edit]

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

See also[edit]

Notes[edit]

  1. ^ The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF [1]
  2. ^ Coxeter, (1980), The 17 plane groups, Table 4

References[edit]

  • 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 [2]
    • (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[edit]