 Cyclic symmetries

This article deals with the four infinite series of point groups in three dimensions (n≥1) with nfold 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:
 C_{n} (nn) of order n  nfold rotational symmetry (abstract group C_{n}); for n=1: no symmetry (trivial group)
Achiral:
 C_{nh} (n*) of order 2n  prismatic symmetry (abstract group C_{n} × C_{2}); for n=1 this is denoted by C_{s} (1*) and called reflection symmetry, also bilateral symmetry.
 C_{nv} (*nn) of order 2n  pyramidal symmetry (abstract group D_{n}); in biology C_{2v} is called biradial symmetry. For n=1 we have again C_{s} (1*).
 S_{2n} (n×) of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C_{2n}); for n=1 we have S_{2} (1×), also denoted by C_{i}; this is 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 nfold rotation axis.
C_{nv} (*nn) has vertical mirror planes. This is the symmetry group for a regular nsided pyramid.
S_{2n} (n×) has a 2nfold rotoreflection axis, also called 2nfold 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_{nd}, it contains a number of improper rotations without containing the corresponding rotations.
C_{2h} (2*) and C_{2v} (*22) of order 4 are two of the three 3D symmetry group types with the Klein fourgroup as abstract group. C_{2v} applies e.g. for a rectangular tile with its top side different from its bottom side.
Examples
S_{2}/C_{i} (1x): C_{4v} (*44): C_{5v} (*55):
Parallelepiped
Square pyramid
Elongated square pyramid
Pentagonal pyramidCategories:
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