Cyclic symmetries explained

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:

• Cn (nn) of order n - n-fold rotational symmetry (abstract group Cn); for n=1: no symmetry (trivial group)

Achiral:

• Cnh (n*) of order 2n - prismatic symmetry (abstract group Dn × C2); for n=1 this is denoted by Cs (1*) and called reflection symmetry, also bilateral symmetry.
• Cnv (*nn) of order 2n - pyramidal symmetry (abstract group Dn); in biology C2v is called biradial symmetry. For n=1 we have again Cs (1*).
• S2n (n×) of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); for n=1 we have S2 (), also denoted by Ci; this is inversion symmetry

They are the finite symmetry groups on a cone. For n =

infty

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.

Cnh (n*) has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis. Cnv (*nn) has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.

S2n (n×) 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 (2*) and C2v (*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.

Examples

S2/Ci (1x):

C4v (*44):C5v (*55):