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Cyclic symmetry in three dimensions#Types

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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.

File:Order 4 dihedral symmetry subgroup tree.png

Types

;Chiral:

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

;Achiral:

File:S shaped packing.jpeg with C2h symmetry]]

  • Cnh, [n+,2], (n*) of order 2n - prismatic symmetry or ortho-n-gonal group (abstract group Zn × Dih1); 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 or full acro-n-gonal group (abstract group Dihn); 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 - gyro-n-gonal group (not to be confused with symmetric groups, for which the same notation is used; abstract group Z2n); 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.
  • for n=1 we have S2 (1×), also denoted by Ci; this is inversion symmetry.

C2h, [2,2+] (2*) and C2v, [2], (*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.

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|+ Frieze groups

colspan=4|Notations

!colspan=2|Examples

IUC

!Orbifold

!Coxeter

!Schönflies*

!Euclidean plane

!Cylindrical (n=6)

p1||∞∞||[∞]+||C∞

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p1m1||*∞∞||[∞]||C∞v

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p11m||∞*||[∞+,2]||C∞h

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p11g||∞×||[∞+,2+]||S∞

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Examples

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!S2/Ci (1x):

!colspan=2|C4v (*44):

!C5v (*55):

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Parallelepiped

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Square pyramid

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Elongated square pyramid

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Pentagonal pyramid

See also

References

  • {{cite book|last=Sands |first=Donald E. |title=Introduction to Crystallography |year=1993 |url=https://archive.org/details/introductiontocr00desa|url-access=limited |publisher=Dover Publications, Inc. |location=Mineola, New York |isbn=0-486-67839-3 |chapter=Crystal Systems and Geometry |page=[https://archive.org/details/introductiontocr00desa/page/n173 165] }}
  • 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • N.W. Johnson: Geometries and Transformations, (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups

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Category:Symmetry