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
- 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} × Dih_{1}); 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 S_{2} (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.
Notations | Examples | ||||
---|---|---|---|---|---|
IUC | Orbifold | Coxeter | Schönflies^{*} | Euclidean plane | Cylindrical (n=6) |
p1 | ∞∞ | [∞]^{+} | C_{∞} | ||
p1m1 | *∞∞ | [∞] | C_{∞v} | ||
p11m | ∞* | [∞^{+},2] | C_{∞h} | ||
p11g | ∞× | [∞^{+},2^{+}] | S_{∞} |
Examples
S_{2}/C_{i} (1x): | C_{4v} (*44): | C_{5v} (*55): | |
---|---|---|---|
Parallelepiped |
Square pyramid |
Elongated square pyramid |
Pentagonal pyramid |
See also
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. https://archive.org/details/introductiontocr00desa.
- 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, (2018) ISBN:978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups
Original source: https://en.wikipedia.org/wiki/Cyclic symmetry in three dimensions.
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