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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:
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.
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.
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.
Coxeter notation, Geometry, Orbifold, Frieze group, Infinity
Orbifold notation, Coxeter-Dynkin diagram, Translation, Coxeter diagram, Commutator subgroup
Schoenflies notation, Geometry, Crystallography, Diamond, Improper rotation
Architecture, Rotation, Symmetry, Line group, Wallpaper group
Ancient Egypt, China, Spain, Tessellation, Porcelain
Hyperbolic geometry, MathWorld, Uniform tilings in hyperbolic plane, Vertex configuration, Geometry
MathWorld, Vertex configuration, Geometry, Truncated square tiling, Truncated tetrapentagonal tiling
Snub cube, Face configuration, List of spherical symmetry groups, Cube, Catalan solid
Square tiling, Vertex configuration, Cairo pentagonal tiling, Truncated square tiling, Elongated triangular tiling