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.
File:Order 4 dihedral symmetry subgroup tree.png|thumb|Example symmetry subgroup tree for dihedral symmetry: D4h, ,

Types

;Chiral:
;Achiral:
with C2h symmetry
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.

Examples