Cyclic symmetry in three dimensions

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.

Selected point groups in three dimensions

Involutional symmetry Cs, (*) [ ] =	Cyclic symmetry Cnv, (*nn) [n] =	Dihedral symmetry Dnh, (*n22) [n,2] =
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetry Td, (*332) [3,3] =	Octahedral symmetry Oh, (*432) [4,3] =	Icosahedral symmetry Ih, (*532) [5,3] =

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.

Example symmetry subgroup tree for dihedral symmetry: D_4h, [4,2], (*224)

Types

Chiral

• C_n, [n]⁺, (nn) of order n - n-fold rotational symmetry - acro-n-gonal group (abstract group Z_n); for n=1: no symmetry (trivial group)

Achiral

Piece of loose-fill cushioning with C_2h symmetry

• C_nh, [n⁺,2], (n*) of order 2n - prismatic symmetry or ortho-n-gonal group (abstract group Z_n × Dih₁); for n=1 this is denoted by C_s (1*) and called reflection symmetry, also bilateral symmetry. It has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis.
• C_nv, [n], (*nn) of order 2n - pyramidal symmetry or full acro-n-gonal group (abstract group Dih_n); in biology C_2v is called biradial symmetry. For n=1 we have again C_s (1*). It has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.
• S_2n, [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 Z_2n); 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 D_nd, it contains a number of improper rotations without containing the corresponding rotations.
  • for n=1 we have S₂ (1×), also denoted by C_i; this is inversion symmetry.

C_2h, [2,2⁺] (2*) and C_2v, [2], (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C_2v 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.

Frieze groups

Notations				Examples
IUC	Orbifold	Coxeter	Schönflies*	Euclidean plane	Cylindrical (n=6)
p1	∞∞	[∞]+	C∞
p1m1	*∞∞	[∞]	C∞v
p11m	∞*	[∞+,2]	C∞h
p11g	∞×	[∞+,2+]	S∞

Examples

S2/Ci (1x):	C4v (*44):		C5v (*55):
Parallelepiped	Square pyramid	Elongated square pyramid	Pentagonal pyramid

See also

• Dihedral symmetry in three dimensions

References

• Sands, Donald E. (1993). "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0-486-67839-3.
• On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups