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.
Family | Intl (orbifold) | Schön. | Geo [1] Coxeter | Order | Examples | |||||
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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•) |
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.
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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.
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o | 2222 | ×× | ** | *× | 22× | 22* | *2222 | 2*22 | 442 | 4*2 | *442 | 333 | *333 | 3*3 | 632 | *632 | ||
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p1 | p2 | pg | pm | cm | pgg | pmg | pmm | cmm | p4 | p4g | p4m | p3 | p3m1 | p31m | p6 | p6m | ||
o | p1 | 2 | ||||||||||||||||
2222 | p2 | 2 | 2 | 2 | ||||||||||||||
×× | pg | 2 | 2 | |||||||||||||||
** | pm | 2 | 2 | 2 | 2 | |||||||||||||
*× | cm | 2 | 2 | 2 | 3 | |||||||||||||
22× | pgg | 4 | 2 | 2 | 3 | |||||||||||||
22* | pmg | 4 | 2 | 2 | 2 | 4 | 2 | 3 | ||||||||||
*2222 | pmm | 4 | 2 | 4 | 2 | 4 | 4 | 2 | 2 | 2 | ||||||||
2*22 | cmm | 4 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ||||||||
442 | p4 | 4 | 2 | 2 | ||||||||||||||
4*2 | p4g | 8 | 4 | 4 | 8 | 4 | 2 | 4 | 4 | 2 | 2 | 9 | ||||||
*442 | p4m | 8 | 4 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | |||||
333 | p3 | 3 | 3 | |||||||||||||||
*333 | p3m1 | 6 | 6 | 6 | 3 | 2 | 4 | 3 | ||||||||||
3*3 | p31m | 6 | 6 | 6 | 3 | 2 | 3 | 4 | ||||||||||
632 | p6 | 6 | 3 | 2 | 4 | |||||||||||||
*632 | p6m | 12 | 6 | 12 | 12 | 6 | 6 | 6 | 6 | 3 | 4 | 2 | 2 | 2 | 3 |
Here a wallpaper is a drawing that covers a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformations, with function composition as the group operation. Thus, a wallpaper group is in a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations and tiles as well as wallpaper.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, a point group is a mathematical group of symmetry operations that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.
In geometry, orbifold notation is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal tiling in a bicolored quasiregular form.
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.
In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.
In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.
In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.
In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} and is self-dual.
In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.
In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.
In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,8}.
In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.
In 4-dimensional geometry, there are 9 uniform polytopes with A4 symmetry. There is one self-dual regular form, the 5-cell with 5 vertices.
In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.