List of Planar Symmetry Groups - Wallpaper Groups

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
,
IUC
(Orbifold)		 Geometric
Coxeter		 Fundamental
domain
p1
(o)		 p1
p2
(2222)		 p2
+
+
+
p2gg
pgg
(22×)		 pg2g
p2mm
pmm
(*2222)		 p2
c2mm
cmm
(2*22)		 c2
]
p4
(442)		 p4
+
p4gm
p4g
(4*2)		 pg4
p4mm
p4m
(*442)		 p4
Parallelogrammatic (oblique)
p1
(o)		 p1
p2
(2222)		 p2
+
Rectangular
,
IUC
(Orbifold)		 Geometric
Coxeter		 Fundamental
domain
p1
(o)		 p1
p2
(2222)		 p2
+
p11g
pg(h)
(××)		 pg1
h:
p1g1
pg(v)
(××)		 pg1
v:
p2gm
pgm
(22*)		 pg2
h:
p2mg
pmg
(22*)		 pg2
v:
p11m
pm(h)
(**)		 p1
h:
p1m1
pm(v)
(**)		 p1
v:
p2mm
pmm
(*2222)		 p2
Rhombic
,
p1
(o)		 p1
p2
(2222)		 p2
+
c11m
cm(h)
(*×)		 c1
h:
c1m1
cm(v)
(*×)		 c1
v:
p2gg
pgg
(22×)		 pg2g
c2mm
cmm
(2*22)		 c2
Hexagonal//Triangular
,
],
IUC
(Orbifold)		 Geometric
Coxeter		 Fundamental
domain
p1
(o)		 p1
p2
(2222)		 p2
+
c2mm
cmm
(2*22)		 c2
p3
(333)		 p3

]+
p3m1
(*333)		 p3

]
p31m
(3*3)		 h3

]+]
p6
(632)		 p6
+
]]+
p6mm
p6m
(*632)		 p6

]]

Read more about this topic:  List Of Planar Symmetry Groups

Famous quotes containing the words wallpaper and/or groups:

    “Either that wallpaper goes, or I do.”
    —Oscar Wilde (1854–1900)

    “Trees appeared in groups and singly, revolving coolly and blandly, displaying the latest fashions. The blue dampness of a ravine. A memory of love, disguised as a meadow. Wispy clouds—the greyhounds of heaven.”
    —Vladimir Nabokov (1899–1977)