LIST OF PLANAR SYMMETRY GROUPS

This article summarizes the classes of discrete planar symmetry groups:
#2 infinite set of point groups
#7 Frieze groups
#17 wallpaper groups

Contents

Point groups

Frieze Groups

Wallpaper groups

See also

External references

Point groups

There are two classes of point groups, rotational and reflectional.
Point groups:

Example	Symbols
Example C5	Cn (''n'') Cyclic group
Example D6	Dn ('' ★ n'') Dihedral group

Frieze Groups

There are also 7 Frieze groups in the plane which have a fundamental line of symmetry and infinite fundamental domains.

Example pattern	Conway symbols
patterns for the 7 frieze groups	# (∞∞)# (∞x)# (∞ ★ )# ( ★ ∞∞)# (22∞)# (2 ★ ∞)# ( ★ 22∞)

Wallpaper groups

There are 17 wallpaper groups in the plane with finite fundamental domains.

'Rotation' ''p2'' (2222) parallelogrammetic ''p4'' (442) ''p3'' (333) ''p6'' (632)	'Maximum symmetry per lattice type' ''pmm'' ( ★ 2222) rectangular ''cmm'' (2 ★ 22) rhombic ''p4m'' ( ★ 442) square ''p6m'' ( ★ 632) hexagonal
'Mixed' ''p3m1'' ( ★ 333) ''p31m'' (3 ★ 3) ''p4g'' (4 ★ 2)	'Other' ''pm'' ( ★ ★ ) ''p1'' (o) ''pg'' (xx) ''pmg'' (22 ★ ) ''cm'' ( ★ x) ''pgg'' (22x)

See also

★ relation between orbifold notation and order

External references

★ "Conway's manuscript" on Orbifold notation

★ http://www.xahlee.org/Wallpaper_dir/c5_17WallpaperGroups.html