Cyclic symmetry in three dimensions

Selected point groups in three dimensions
Involutional symmetry
Cs, (*)
[ ] =  |
Cyclic symmetry
Cnv, (*nn)
[n] =  |
Dihedral symmetry
Dnh, (*n22)
[n,2] =

|
|

Polyhedral group, [n,3], (*n32)

Tetrahedral symmetry
Td, (*332)
[3,3] =  |
Octahedral symmetry
Oh, (*432)
[4,3] =  |
Icosahedral symmetry
Ih, (*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.

Example symmetry subgroup tree for dihedral symmetry: D4h, [4,2], (*224)

## Types

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Chiral

* Cn, [n]+, (nn) of order n \- n-fold rotational symmetry - acro-n-gonal group (abstract group Zn); for n=1: no symmetry (trivial group)

Achiral

Piece of loose-fill cushioning with C2h symmetry

* Cnh, [n+,2], (n*) of order 2n \- prismatic symmetry or ortho-n-gonal group (abstract group Zn × Dih1); for n=1 this is denoted by Cs (1*) and called reflection symmetry, also bilateral symmetry. It has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis.
* Cnv, [n], (*nn) of order 2n \- pyramidal symmetry or full acro-n-gonal group (abstract group Dihn); in biology C2v is called biradial symmetry. For n=1 we have again Cs (1*). It has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.
* S2n, [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 Z2n); 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 Dnd, it contains a number of improper rotations without containing the corresponding rotations.
* for n=1 we have S2 (1×), also denoted by Ci; this is inversion symmetry.

C2h, [2,2+] (2*) and C2v, [2], (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C2v applies e.g. for a rectangular tile with its top side different from its bottom side.

## Frieze groups

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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  | Examples

|

IUC | Orbifold | Coxeter | Schönflies* | Euclidean plane  | Cylindrical (n=6)
p1 | ∞∞ | [∞]+ | C∞ |  |
p1m1 | *∞∞ | [∞] | C∞v |  |
p11m | ∞* | [∞+,2] | C∞h |  |
p11g | ∞× | [∞+,2+] | S∞ |  |

## Examples

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S2/Ci (1x):  | C4v (*44):  | C5v (*55):

|
|

Parallelepiped |
Square pyramid |
Elongated square pyramid |
Pentagonal pyramid

## See also

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* Dihedral symmetry in three dimensions

## References

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* 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-Strauss, 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 [1]
* N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups