From Surf Wiki (app.surf) — the open knowledge base

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.

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:

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:

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.

Examples	IUC	Coxeter	Schönflies*
[[File:Frieze example p1.png	150px]]	[[File:Uniaxial c6.png	150px]]
[[File:Frieze example p1m1.png	150px]]	[[File:Uniaxial c6v.png	150px]]
[[File:Frieze example p11m.png	150px]]	[[File:Uniaxial c6h.png	150px]]
[[File:Frieze example p11g.png	150px]]	[[File:Uniaxial s6.png	150px]]

Examples

S2/Ci (1x):	C4v (*44):	C5v (*55):
Parallelepiped	[[Image:Square pyramid.png	100px]]Square pyramid	[[Image:Elongated square pyramid.png	100px]]Elongated square pyramid

References

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, wiley.com,
N.W. Johnson: Geometries and Transformations, (2018) , Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

symmetry

Want to explore this topic further?

Ask Mako anything about Cyclic symmetry in three dimensions — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report