Cyclic symmetry in three dimensions explained

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:

• 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:

• 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
	Orbifold	Coxeter	Schönflies	Euclidean plane	Cylindrical (n=6)
p1	∞∞	[∞]^+	C∞
p1m1	∞∞	[∞]	C∞v
p11m	∞*	[∞<sup>+</sup>,2]	C∞h
p11g	∞×	[∞<sup>+</sup>,2<sup>+</sup>]	S∞

See also

• Dihedral symmetry in three dimensions

References

• Book: Sands, Donald E. . Introduction to Crystallography . 1993 . limited . Dover Publications, Inc. . Mineola, New York . 0-486-67839-3 . Crystal Systems and Geometry . 165 .
• On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,
• 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
• N.W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups