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 h2.pngCDel n.pngCDel node h2.png
Cn n Archimedean spiral.svg
[ ]+
Yin and Yang.svg
[2]+ (*)
The armoured triskelion on the flag of the Isle of Man.svg
[3]+
Octagonal star-b2.png
[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]
Octagonal star2.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 h2.pngCDel infin.pngCDel node h2.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 h2.pngCDel 2x.pngCDel node h3.pngCDel infin.pngCDel node h2.png
Frieze group 1g.png Frieze example p11g.png
p11m
(∞*)
p. 1 C∞h [2,∞+]
CDel node.pngCDel 2.pngCDel node h2.pngCDel infin.pngCDel node h2.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 h2.pngCDel 2x.pngCDel node h2.pngCDel infin.pngCDel node h2.png
Frieze group 12.png Frieze example p2.png
p2mg
(2*∞)
p2g D∞d [2+,∞]
CDel node h2.pngCDel 2x.pngCDel node h2.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
[(4,1+,4,2+)]+ 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 h3.pngCDel 4.pngCDel node h2.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
[(4,4,2+)]
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
Parallelogrammatic (oblique)
p1
(°)
p1
Wallpaper group diagram p1.svg
p2
(2222)
p2
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 h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node h.pngCDel infin.pngCDel node h.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 h3.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 h3.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
(Orbifold)
Geometric
Coxeter Fundamental
domain
p1
(°)
p1
[∞+,2+,∞+]
CDel node h2.pngCDel infin.pngCDel node h3.pngCDel 2x.pngCDel node h3.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram p1 rhombic.svg
p2
(2222)
p2
[∞,2+,∞]+
CDel label2.pngCDel branch hh.pngCDel iaib.pngCDel branch h2h2.pngCDel label2.png
Wallpaper group diagram p2 rhombic.svg
cm(h)
(*×)
c1
h: [∞+,2+,∞]
CDel node h2.pngCDel infin.pngCDel node h3.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 h3.pngCDel infin.pngCDel node h2.png
Wallpaper group diagram cm rotated.svg
pgg
(22×)
pg2g
[+(∞,(2),∞)+]
CDel node h2.pngCDel split1-2i.pngCDel nodes h3h3.pngCDel split2-i2.pngCDel node h2.png
Wallpaper group diagram pgg rhombic.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
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 p 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
  • 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. and 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, (2015) Chapter 11: Finite symmetry groups

External links[edit]