List of planar symmetry groups

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's bracket notation.

There are three kinds of symmetry groups of the plane:

• 2 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)	Geo	Schönflies	Coxeter	Order	Example
Cyclic symmetry	n (nn)	n	Cn	[n]+	n	5-fold rotation
Dihedral symmetry	nm (*nn)	n	Dn	[n]	2n	4-fold reflection

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. Simple example images are given as periodic tilings on a cylinder with a periodicity of 6.

[∞,1], IUC (Orbifold) Geo Schönflies Coxeter Fundamental domain Example p1 (∞∞) p1 C∞ [∞,1]+ p1m1 (*∞∞) p1 C∞v [∞,1] [∞+,2], IUC (Orbifold) Geo Schönflies Coxeter Fundamental domain Example p11g (∞x) p.g1 S2∞ [∞+,2+] p11m (∞*) p.1 C∞h [∞+,2]

[∞,2], IUC (Orbifold) Geo Schönflies Coxeter Fundamental domain Example p2 (22∞) p2 D∞ [∞,2]+ p2mg (2*∞) p2g D∞d [∞,2+] p2mm (*22∞) p2 D∞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 (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], IUC (Orbifold) Geometric Coxeter Fundamental domain p1 (o) p1 [∞+,2,∞+] p2 (2222) p2 [1+,4,4]+ p2gg pgg (22x) pg2g [4+,4+] p2mm pmm (*2222) p2 [1+,4,4] c2mm cmm (2*22) c2 [[4+,4+]] p4 (442) p4 [4,4]+ p4gm p4g (4*2) pg4 [4+,4] p4mm p4m (*442) p4 [4,4] Parallelogrammatic (oblique) p1 (o) p1 [∞+,2,∞+] p2 (2222) p2 [∞,2,∞]+

Hexagonal [6,3], IUC (Orbifold) Coxeter Fundamental domain p1 (o) p1 [∞+,2,∞+] p2 (2222) p2 [∞,2,∞]+ p3 (333) p3 [1+,6,3+] p3m1 (*333) p3 [1+,6,3] p31m (3*3) h3 [6,3+] c2mm cmm (2*22) c2 [∞,2+,∞] p6 (632) p6 [6,3]+ p6mm p6m (*632) p6 [6,3] Hexagonal [3[3]], p3 (333) p3 [3[3]]+ p3m1 (*333) p3 [3[3]] p31m (3*3) h3 [3[3[3]]+] p6 (632) p6 [3[3[3]]]+ p6mm p6m (*632) p6 [3[3[3]]]

Rectangular, [∞h,2,∞v], IUC (Orbifold) Coxeter Fundamental domain p1 (o) p1 [∞+,2,∞+] p2 (2222) p2 [∞,2,∞]+ p11g pg(h) (xx) pg1 h: [∞+,(2,∞)+] p1g1 pg(v) (xx) pg1 v: [(∞,2)+,∞+] p2gm pgm (22*) pg2 h: [(∞,2)+,∞] p2mg pmg (22*) pg2 v: [∞,(2,∞)+] p11m pm(h) (**) p1 h: [∞+,2,∞] p1m1 pm(v) (**) p1 v: [∞,2,∞+] p2mm pmm (*2222) p2 [∞,2,∞] Rhombic, [∞h,2+,∞v], p1 (o) p1 [∞+,2,∞+] p2 (2222) p2 [∞,2,∞]+ c11m cm(h) (*x) c1 h: [∞+,2+,∞] c1m1 cm(v) (*x) c1 v: [∞,2+,∞+] p2gg pgg (22x) pg2g [∞+,2+,∞+] c2mm cmm (2*22) c2 [∞,2+,∞]

See also

• List of spherical symmetry groups

References

• 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, editied 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

• "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