Cyclic symmetry in three dimensions


In three dimensional geometry, there are four infinite series of point groups in three dimensions with n-fold rotational or reflectional symmetry about one axis 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 and vertical 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:Cn, +, of order n'' - n-fold rotational symmetry - acro-n-gonal group ; for n = 1: no symmetry
;Achiral:Cnh,, of order 2n'' - prismatic symmetry or ortho-n-gonal group ; for n = 1 this is denoted by Cs and called reflection symmetry, also bilateral symmetry. It has reflection symmetry with respect to a plane perpendicular to the n''-fold rotation axis.Cnv,, of order 2n'' - pyramidal symmetry or full acro-n-gonal group ; in biology C2v is called biradial symmetry. For n = 1 we have again Cs. It has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.S2n,, of order 2n'' - gyro-n-gonal group ; 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.
C2h, and C2v,, 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

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.