List of planar symmetry groups

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This article summarizes the classes of discrete planar symmetry groups. 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)
Geo
[1]
Coxeter
Schönflies Order Examples
Cyclic symmetry n
(n•)
n
[n]+
CDel node.pngCDel n.pngCDel h.pngCDel node.png
Cn n Archimedean spiral.svg
[ ]+
Yin and Yang.svg
[2]+ (*)
The armoured triskelion on the flag of the Isle of Man.svg
[3]+
Circular-cross-decorative-knot-12crossings.svg
[4]+
Flag of Hong Kong.svg
[5]+
Crop circles Swirl.jpg
[6]+
Dihedral symmetry nm
(*n•)
n
[n]
CDel node.pngCDel n.pngCDel node.png
Dn 2n Eenbruinigherfstblad.jpg
[ ]
Rhombus (polygon).png
[2]
Labeled Triangle Reflections.svg
[3]
Topological Rose with mirrors.png
[4]
4-simplex t0.svg
[5]
Benzene-aromatic-3D-balls.png
[6]
(*) The Yin and Yang symbol has 2-fold cyclic symmetry of geometry but inverted colors.

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.

[∞], CDel node.pngCDel infin.pngCDel node.png
IUC
(Orbifold)
Geo Schönflies Coxeter Fundamental
domain
Example
p1
(∞•)
p1 C [∞]+
CDel node.pngCDel infin.pngCDel h.pngCDel node.png
Frieze group 11.png Frieze example p1.png
p1m1
(*∞•)
p1 C∞v [∞]
CDel node.pngCDel infin.pngCDel node.png
Frieze group m1.png Frieze example p1m1.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.pngCDel 2x.pngCDel h.pngCDel node.pngCDel infin.pngCDel h.pngCDel node.png
Frieze group 1g.png Frieze example p11g.png
p11m
(∞*)
p. 1 C∞h [2,∞+]
CDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel h.pngCDel node.png
Frieze group 1m.png Frieze example p11m.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 g.pngCDel 2hg.pngCDel node g.pngCDel ig.pngCDel node g.png
Frieze group 12.png Frieze example p2.png
p2mg
(2*∞)
p2g D∞d [2+,∞]
CDel node.pngCDel 2x.pngCDel h.pngCDel node.pngCDel infin.pngCDel node.png
Frieze group mg.png Frieze example p2mg.png
p2mm
(*22∞)
p2 D∞h [2,∞]
CDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Frieze group mm.png Frieze example p2mm.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 (60 degree rhombic), rectangular, and centered rectangular (rhombic).

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
(Orbifold)
Geometric
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2,∞+] Wallpaper group diagram p1 square.svg
p2
(2222)
p2
[(4,1+,4,2+)]
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch.pngCDel label2.pngCDel labelh.png
[1+,4,1+,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.pngCDel 4.pngCDel h.pngCDel node.pngCDel 4.pngCDel h.pngCDel node.png
Wallpaper group diagram pgg square2.svg
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+]]
[(4,4,2+)]
CDel node.pngCDel split1-44.pngCDel branch.pngCDel label2.pngCDel labelh.png
Wallpaper group diagram cmm square.svg
p4
(442)
p4
[4,4]+
CDel node g.pngCDel 4hg.pngCDel node g.pngCDel 4g.pngCDel node g.png
Wallpaper group diagram p4 square.svg
p4g
(4*2)
pg4
[4+,4]
CDel node.pngCDel 4.pngCDel h.pngCDel node.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
Parallelogrammatic (oblique)
p1
(°)
p1
[∞+,2,∞+] Wallpaper group diagram p1.svg
p2
(2222)
p2
[∞,2,∞]+ Wallpaper group diagram p2.svg
Rectangular
[∞h,2,∞v], CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
IUC
(Orbifold)
Geometric
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2,∞+]
CDel node.pngCDel infin.pngCDel h.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel h.pngCDel node.png
Wallpaper group diagram p1 rect.svg
p2
(2222)
p2
[∞,2,∞]+
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
Wallpaper group diagram p2 rect.svg
pgg
(22×)
pg2g
[[∞,2,∞]+] Wallpaper group diagram pgg.svg
pg(h)
(××)
pg1
h: [∞+,(2,∞)+]
CDel node.pngCDel infin.pngCDel h.pngCDel node g.pngCDel 2g.pngCDel node g.pngCDel ihg.pngCDel node g.png
Wallpaper group diagram pg.svg
pg(v)
(××)
pg1
v: [(∞,2)+,∞+]
CDel node g.pngCDel ihg.pngCDel node g.pngCDel 2g.pngCDel node g.pngCDel infin.pngCDel h.pngCDel node.png
Wallpaper group diagram pg rotated.svg
pgm
(22*)
pg2
h: [(∞,2)+,∞]
CDel node g.pngCDel ihg.pngCDel node g.pngCDel 2g.pngCDel node g.pngCDel infin.pngCDel node.png
Wallpaper group diagram pmg.svg
pmg
(22*)
pg2
v: [∞,(2,∞)+]
CDel node.pngCDel infin.pngCDel node g.pngCDel 2g.pngCDel node g.pngCDel ihg.pngCDel node g.png
Wallpaper group diagram pmg rotated.svg
pm(h)
(**)
p1
h: [∞+,2,∞]
CDel node.pngCDel infin.pngCDel h.pngCDel node.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.pngCDel infin.pngCDel h.pngCDel node.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.pngCDel 2x.pngCDel h.pngCDel node.pngCDel infin.pngCDel node.png
p1
(°)
p1
[∞+,2,∞+]
CDel node.pngCDel infin.pngCDel h.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel h.pngCDel node.png
Wallpaper group diagram p1 rhombic.svg
p2
(2222)
p2
[∞+,2+,∞+]+
[(∞+,2+,∞+,2+)]
Wallpaper group diagram p2 rhombic.svg
cm(h)
(*×)
c1
h: [∞+,2+,∞]
CDel node.pngCDel infin.pngCDel h.pngCDel node.pngCDel 2x.pngCDel h.pngCDel node.pngCDel infin.pngCDel node.png
Wallpaper group diagram cm.svg
cm(v)
(*×)
c1
v: [∞,2+,∞+]
CDel node.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel h.pngCDel node.pngCDel infin.pngCDel h.pngCDel node.png
Wallpaper group diagram cm rotated.svg
pgg
(22×)
pg2g
[∞+,2+,∞+]
CDel node.pngCDel infin.pngCDel h.pngCDel node.pngCDel 2x.pngCDel h.pngCDel node.pngCDel infin.pngCDel h.pngCDel node.png
Wallpaper group diagram pgg rhombic.svg
cmm
(2*22)
c2
[∞,2+,∞]
CDel node.pngCDel infin.pngCDel node.pngCDel 2x.pngCDel h.pngCDel node.pngCDel infin.pngCDel node.png
Wallpaper group diagram cmm.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
IUC
(Orbifold)
Geometric
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2,∞+] Wallpaper group diagram p1 half.svg
p2
(2222)
p2
[∞,2,∞]+ Wallpaper group diagram p2 half.svg
cmm
(2*22)
c2
[∞,2+,∞] Wallpaper group diagram cmm half.svg
p3
(333)
p3
[1+,6,3+]
CDel node h0.pngCDel 6.pngCDel node.pngCDel 3x.pngCDel h.pngCDel node.png
[3[3]]+
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 node.pngCDel split1.pngCDel branch.png
Wallpaper group diagram p3m1.svg
p31m
(3*3)
h3
[6,3+]
CDel node.pngCDel 6.pngCDel node.pngCDel 3x.pngCDel h.pngCDel node.png
[3+[3[3]]]
Wallpaper group diagram p31m.svg
p6
(632)
p6
[6,3]+
CDel node g.pngCDel 6hg.pngCDel node g.pngCDel 3g.pngCDel node g.png
[3[3[3]]]+
Wallpaper group diagram p6.svg
p6m
(*632)
p6
[6,3]
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
[3[3[3]]]
Wallpaper group diagram p6m.svg

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]

References[edit]

  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5
  • 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]
  • N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 11: Finite symmetry groups

External links[edit]