List of symmetry groups on the sphere

Spherical symmetry groups are also called point groups (in 3D). The symmetry group of an object (e. ...

Name	Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order	Fundamental domain
no symmetry (monotropic)	C1	[1]+	11	1
reflection symmetry (monoscopic)	Cs	[1]	*11	2
inversion symmetry (monodromic)	Ci	[2+,2+]	1x	2
discrete rotational symmetry (polytropic)	Cn	[n]+	nn	n
Polyscopic	Cnv	[n]	*nn	2n
Polygyros	Cnh	[2,n+]	n*	2n
Polydromic	S2n	[2+,2n+]	nx	2n
Polyditropic	Dn	[2,n]+	22n	2n
Polydigyros	Dnd	[2+,2n]	2*n	4n
Polydiscopic	Dnh	[2,n]	*22n	4n
Chiral tetrahedral	T	[3,3]+	332	12
Achiral tetrahedral	Td	[3,3]	*332	24
Pyritohedral	Th	[3+,4]	3*2	24
Chiral octahedral	O	[3,4]+	432	24
Achiral octahedral	Oh	[3,4]	*432	48
Chiral icosahedral	I	[3,5]+	532	60
Achiral icosahedral	Ih	[3,5]	*532	120

Arthur Moritz Schönflies (April 17, 1853 Landsberg an der Warthe(Gorzów) – May 27, 1928) was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. ... H(arold). ... In topology, an orbifold is a generalization of manifold. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... Figures with the axes of symmetry drawn in. ... This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. ... This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. ... This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the... The icosahedral rotation group I with fundamental domain Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the...

Relation between orbifold notation and order

The order of each group is 2 divided by the orbifold Euler characteristic; the latter is 2 minus the sum of the feature values, assigned as follows: In topology, an orbifold is a generalization of manifold. ... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...

• n without or before * counts as (n−1)/n
• n after * counts as (n−1)/(2n)
• * and x count as 1

This can also be applied for wallpaper groups: for them, the sum of the feature values is 2, giving an infinite order; see orbifold Euler characteristic for wallpaper groups Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical concept to classify repetitive designs on two-dimensional surfaces, such as walls, based on the symmetries in the pattern. ... (Redirected from 2D crystallographic group) Plane crystallographic groups or wallpaper groups There are seventeen different types of wallpaper patterns. ...

See also

• Point groups in three dimensions
• Overview of point groups by crystal system
• Crystallographic point group
• List of planar symmetry groups

A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ... There are 17 symmetry groups in the plane. ...

References

• [1] Book: "Polyhedra", by Peter R. Cromwell, 1997 - Appendix I
• Finite spherical symmetry groups