# Cyclic symmetry in three dimensions

 Polyhedral group, [n,3], (*n32) Involutional symmetryCs, (*)[ ] =  Cyclic symmetryCnv, (*nn)[n] =    Dihedral symmetryDnh, (*n22)[n,2] =      Tetrahedral symmetryTd, (*332)[3,3] =      Octahedral symmetryOh, (*432)[4,3] =      Icosahedral symmetryIh, (*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.

## Types

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
• 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.

C2h, [2,2+] (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.

## 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.