# List of spherical symmetry groups

- List of spherical symmetry groups
= List of symmetry groups on the sphere =

Spherical symmetry groups are also called point groups in three dimensions. This article is about the finite ones.

There are four fundamental symmetry classes which have triangular fundamental domains: dihedral, tetrahedral, octahedral, icosahedral. There are infinitely many dihedral symmetry groups.

The final classes, under "other" have digonal or monogonal fundamental domains.

= Dihedral symmetry [2,n] =

There are an infinite set of dihedral symmetries. **n** can be any positive integer **2** or greater (n = 1 is also possible, but these three symmetries are equal to C_{2}, C_{2v}, and C_{2h}).

** Icosahedral symmetry [3,5] **

** Other **

These final forms have digonal or monogonal fundamental regions with Cyclic symmetries and reflection symmetry. There are four infinite sets with index **n** being any positive integer; for n=1 two cases are equal, so there are three; they are separately named.

**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:

*"n" without or before * counts as ("n"−1)/"n"

*"n" after * counts as ("n"−1)/(2"n")

* * and x count as 1This 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

**ee also**

*Overview of point groups by crystal system

*Crystallographic point group

*List of planar symmetry groups

*Triangle group

** References **

* Peter R. Cromwell, "Polyhedra", (1997) "(See Appendix I.)"

* [*http://www.geom.uiuc.edu/~math5337/Orbifolds/costs.html Finite spherical symmetry groups*]

*

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**List of mathematics articles (L)** — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… … Wikipedia

**List of mathematics articles (S)** — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia

**List of geometry topics** — This is list of geometry topics, by Wikipedia page.*Geometric shape covers standard terms for plane shapes *List of mathematical shapes covers all dimensions *List of differential geometry topics *List of geometers *See also list of curves, list… … Wikipedia

**List of mathematics articles (T)** — NOTOC T T duality T group T group (mathematics) T integration T norm T norm fuzzy logics T schema T square (fractal) T symmetry T table T theory T.C. Mits T1 space Table of bases Table of Clebsch Gordan coefficients Table of divisors Table of Lie … Wikipedia

**List of mathematics articles (D)** — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… … Wikipedia

**List of mathematics articles (P)** — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia

**List of mathematics articles (F)** — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… … Wikipedia

**List of mathematical functions** — In mathematics, several functions or groups of functions are important enough to deserve their own names. This is a listing of pointers to those articles which explain these functions in more detail. There is a large theory of special functions… … Wikipedia

**Point groups in three dimensions** — In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries… … Wikipedia

**Dihedral symmetry in three dimensions** — This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn ( n ≥ 2 ). See also point groups in two dimensions. Chiral: Dn (22n) of… … Wikipedia