# Cyclic symmetry in three dimensions

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.

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] = |

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

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

- Achiral

of order 2*C*, [n_{nh}^{+},2], (*n**)*n*-**prismatic symmetry**or**ortho-n-gonal group**(abstract group*Z*×_{n}*Dih*); for_{1}*n*=1 this is denoted byand called*C*(1*)_{s}**reflection symmetry**, also**bilateral symmetry**. It has reflection symmetry with respect to a plane perpendicular to the*n*-fold rotation axis.of order 2*C*, [n], (*_{nv}*nn*)*n*-**pyramidal symmetry**or**full acro-n-gonal group**(abstract group*Dih*); in biology_{n}*C*is called_{2v}**biradial symmetry**. For*n*=1 we have again*C*(1*). It has vertical mirror planes. This is the symmetry group for a regular_{s}*n*-sided pyramid.of order 2*S*, [2_{2n}^{+},2n^{+}], (*n*×)*n*-**gyro-n-gonal group**(not to be confused with symmetric groups, for which the same notation is used; abstract group*Z*); It has a 2_{2n}*n*-fold rotoreflection axis, also called 2*n*-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*, it contains a number of improper rotations without containing the corresponding rotations._{nd}- for
*n*=1 we have*S*(_{2}**1×**), also denoted by; this is inversion symmetry.**C**_{i}

- for

** C_{2h}, [2,2^{+}] (2*)** and

**of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group.**

*C*, [2], (*22)_{2v}*C*applies e.g. for a rectangular tile with its top side different from its bottom side.

_{2v}## 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.

Notations | 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 (1x):_{i} |
C (*44):_{4v} |
C (*55):_{5v} | |
---|---|---|---|

Parallelepiped |
Square pyramid |
Elongated square pyramid |
Pentagonal pyramid |

## References

- Sands, Donald E. (1993). "Crystal Systems and Geometry".
*Introduction to Crystallography*. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0-486-67839-3. *On Quaternions and Octonions*, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5*The Symmetries of Things*2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5**Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6- N.W. Johnson:
*Geometries and Transformations*, (2018) ISBN 978-1-107-10340-5 Chapter 11:*Finite symmetry groups*, 11.5 Spherical Coxeter groups