Cyclic symmetry in three dimensions

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 Polyhedral group, [n,3], (*n32) Involutional symmetry Cs, [1], (*) Cyclic symmetry Cnv, [n], (*nn) Dihedral symmetry Dnh, [n,2], (*n22) Tetrahedral symmetry Td, [3,3], (*332) Octahedral symmetry Oh, [4,3], (*432) Icosahedral symmetry Ih, [5,3], (*532)

This article deals with the 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) 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.

Example symmetry subgroup tree for dihedral symmetry: D4h, [4,2], (*224)

Chiral:

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

Achiral:

• Cnh, [n+,2], (n*) of order 2n - prismatic symmetry (abstract group Cn × C2); 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 (abstract group Dn); 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 (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); for n=1 we have S2 (), also denoted by Ci; this is inversion symmetry. 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*) 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.

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

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-Strass, 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 [1]
• N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 11: Finite symmetry groups