List of planar symmetry groups

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

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]+
 n
C1, [ ]+ (•)
C2, [2]+ (2•)
C3, [3]+ (3•)
C4, [4]+ (4•)
C5, [5]+ (5•)
C6, [6]+ (6•)
Dihedral symmetry nm
(*n•)		 Dn n
[n]
 2n
D1, [ ] (*•)
D2, [2] (*2•)
D3, [3] (*3•)
D4, [4] (*4•)
D5, [5] (*5•)
D6, [6] (*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)		 Geo Schönflies Coxeter Fundamental
domain		 Example
p1
(∞•)		p1C∞[1,∞]+

p1m1
(*∞•)		p1C∞v[1,∞]

[2,∞+],
IUC
(Orbifold)		 Geo Schönflies Coxeter Fundamental
domain		 Example
p11g
(∞×)		p.g1S2∞[2+,∞+]

p11m
(∞*)		p. 1C∞h[2,∞+]
[2,∞],
IUC
(Orbifold)		 Geo Schönflies Coxeter Fundamental
domain		 Example
p2
(22∞)		p2D∞[2,∞]+

p2mg
(2*∞)		p2gD∞d[2+,∞]

p2mm
(*22∞)		p2D∞h[2,∞]

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.)
Geo		 Coxeter Fundamental
domain
p1
(°)
p1
p2
(2222)
p2		[4,1+,4]+

[1+,4,4,1+]+
pgg
(22×)
pg2g		[4+,4+]
pmm
(*2222)
p2		[4,1+,4]

[1+,4,4,1+]
cmm
(2*22)
c2		[(4,4,2+)]
p4
(442)
p4		[4,4]+
p4g
(4*2)
pg4		[4+,4]
p4m
(*442)
p4		[4,4]
Rectangular
[∞h,2,∞v],
IUC
(Orb.)
Geo		 Coxeter Fundamental
domain
p1
(°)
p1		[∞+,2,∞+]
p2
(2222)
p2		[∞,2,∞]+
pg(h)
(××)
pg1
h: [∞+,(2,∞)+]
pg(v)
(××)
pg1		v: [(∞,2)+,∞+]
pgm
(22*)
pg2		h: [(∞,2)+,∞]
pmg
(22*)
pg2		v: [∞,(2,∞)+]
pm(h)
(**)
p1		h: [∞+,2,∞]
pm(v)
(**)
p1		v: [∞,2,∞+]
pmm
(*2222)
p2		[∞,2,∞]
Rhombic
[∞h,2+,∞v],
IUC
(Orb.)
Geo		 Coxeter Fundamental
domain
p1
(°)
p1		[∞+,2+,∞+]
p2
(2222)
p2		[∞,2+,∞]+
cm(h)
(*×)
c1		h: [∞+,2+,∞]
cm(v)
(*×)
c1		v: [∞,2+,∞+]
pgg
(22×)
pg2g		[((∞,2)+)[2]]
cmm
(2*22)
c2		[∞,2+,∞]
Parallelogrammatic (oblique)
p1
(°)
p1
p2
(2222)
p2
Hexagonal/Triangular
[6,3], / [3[3]],
p1
(°)
p1
p2
(2222)
p2		[6,3]Δ
cmm
(2*22)
c2		[6,3]⅄
p3
(333)
p3		[1+,6,3+]

[3[3]]+
p3m1
(*333)
p3		[1+,6,3]

[3[3]]
p31m
(3*3)
h3		[6,3+]
p6
(632)
p6		[6,3]+
p6m
(*632)
p6		[6,3]

Wallpaper subgroup relationships

Subgroup relationships among the 17 wallpaper group[2]
o2222××***×22×22**22222*224424*2*442333*3333*3632*632
p1p2pgpmcmpggpmgpmmcmmp4p4gp4mp3p3m1p31mp6p6m
op1 2
2222p 222
××pg 22
**pm 2222
*×cm 2223
22×pgg 4223
22*pmg 4222423
*2222pmm 424244222
2*22cmm 424422224
442p4 422
4*2p4g 84484244229
*442p4m 848444422222
333p3 33
*333p3m1 6663243
3*3p31m 6663234
632p6 6324
*632p6m 12612126666342223

See also

Notes

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

References

External links

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