**Cyclic Symmetries**

This article deals with the four infinite series of point groups in three dimensions (*n*≥1) with *n*-fold **rotational symmetry** about one axis (rotation by an angle of 360°/*n* does not change the object), and no other rotational symmetry (*n*=1 covers the cases of no rotational symmetry at all):

**Chiral:**

of order*C*(_{n}*nn*)*n*-*n*-fold rotational symmetry (abstract group*C*); for_{n}*n*=1:**no symmetry**(trivial group)

**Achiral:**

of order 2*C*(_{nh}*n**)*n*-**prismatic symmetry**(abstract group*C*×_{n}*C*); for_{2}*n*=1 this is denoted byand called*C*(1*)_{s}**reflection symmetry**, also**bilateral symmetry**.of order 2*C*(*_{nv}*nn*)*n*-**pyramidal symmetry**(abstract group*D*); in biology_{n}*C*is called_{2v}**biradial symmetry**. For*n*=1 we have again*C*(1*)._{s}of order 2*S*(_{2n}*n*×)*n*(not to be confused with symmetric groups, for which the same notation is used; abstract group*C*); for_{2n}*n*=1 we have*S*(_{2}**1×**), also denoted by; this is**C**_{i}**inversion symmetry**

They are the finite symmetry groups on a cone. For *n* = they correspond to four frieze groups. Schönflies notation is used, and, in parentheses, orbifold notation. The terms horizontal (h) and vertical (v) are used with respect to a vertical axis of rotation.

*C _{nh}* (

*n**) has reflection symmetry with respect to a plane perpendicular to the

*n*-fold rotation axis.

*C _{nv}* (*

*nn*) has vertical mirror planes. This is the symmetry group for a regular

*n*-sided pyramid.

*S _{2n}* (

*n*×) has a 2

*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}*C _{2h}* (2*) and

*C*(*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group.

_{2v}*C*applies e.g. for a rectangular tile with its top side different from its bottom side.

_{2v}Read more about Cyclic Symmetries: Examples

### Other articles related to "cyclic symmetries":

**Cyclic Symmetries**- Improper Rotation Groups (

*S*

_{n})

... The S8 table reflects the 2007 discovery of errors in older references ... Specifically, (Rx, Ry) transform not as E1 but rather as E3 ...