# Cyclic symmetry in three dimensions

Selected point groups in three dimensions :
| Involutional symmetry C$_s$, (*) [ ] = | Cyclic symmetry C$_{nv}$, (*nn) [n] = | Dihedral symmetry D$_{nh}$, (*n22) [n,2] = |
| Polyhedral group, [n,3], (*n32) |||
| Tetrahedral symmetry T$_d$, (*332) [3,3] = | Octahedral symmetry O$_h$, (*432) [4,3] = | Icosahedral symmetry I$_h$, (*532) [5,3] = |
In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does not change the object.
They are the finite symmetry groups on a cone. For n = ∞ they correspond to four frieze groups. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation.

## Types
### Chiral

* C$_n$, [n]$^+$, (nn) of order n - n-fold rotational symmetry - acro-n-gonal group (abstract group Z$_n$); for n=1: no symmetry (trivial group)

### Achiral

* C$_{nh}$, [n$^+$,2], (n*) of order 2n - prismatic symmetry or ortho-n-gonal group (abstract group Z$_n$ × Dih$_1$); for n=1 this is denoted by C$_s$ (1*) and called reflection symmetry, also bilateral symmetry. It has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis.
* C$_{nv}$, [n], (*nn) of order 2n - pyramidal symmetry or full acro-n-gonal group (abstract group Dih$_n$); in biology C$_{2v}$ is called biradial symmetry. For n=1 we have again C$_s$ (1*). It has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.
* S$_{2n}$, [2$^+$,2n$^+$], (n×) of order 2n - gyro-n-gonal group (not to be confused with symmetric groups, for which the same notation is used; abstract group Z$_{2n}$); It has a 2n-fold rotoreflection axis, also called 2n-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like D$_{nd}$, it contains a number of improper rotations without containing the corresponding rotations.
** for n=1 we have S$_2$ (1×), also denoted by C$_i$; this is inversion symmetry.
C$_{2h}$, [2,2$^+$] (2*) and C$_{2v}$, [2], (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C$_{2v}$ applies e.g. for a rectangular tile with its top side different from its bottom side.

## Frieze groups

In the limit these four groups represent Euclidean plane frieze groups as C$_∞$, C$_{∞h}$, C$_{∞v}$, and S$_∞$. Rotations become translations in the limit. Portions of the infinite plane can also be cut and connected into an infinite cylinder.
Frieze groups :
| Notations | Notations | Notations | Notations | Examples | Examples |
| IUC | Orbifold | Coxeter | Schönflies$^*$ | Euclidean plane | Cylindrical (n=6) |
| p1 | ∞∞ | [∞]$^+$ | C$_∞$ | | |
| p1m1 | *∞∞ | [∞] | C$_{∞v}$ | | |
| p11m | ∞* | [∞$^+$,2] | C$_{∞h}$ | | |
| p11g | ∞× | [∞$^+$,2$^+$] | S$_∞$ | | |

## Examples

| S$_2$/C$_i$ (1x): | C$_{4v}$ (*44): | C$_{4v}$ (*44): | C$_{5v}$ (*55): |
| Parallelepiped | Square pyramid | Elongated square pyramid | Pentagonal pyramid |