This article needs additional citations for verification. (July 2011) (Learn how and when to remove this template message) |
p1m1, (*∞∞) | p2, (22∞) | p2mg, (2*∞) |
---|---|---|
In 2-dimensions three frieze groups p1m1, p2, and p2mg are isomorphic to the Dih_{∞} group. They all have 2 generators. The first has two parallel reflection lines, the second two 2-fold gyrations, and the last has one mirror and one 2-fold gyration. |
In mathematics, the infinite dihedral group Dih_{∞} is an infinite group with properties analogous to those of the finite dihedral groups.
In two dimensional geometry, the infinite dihedral group represents the frieze group symmetry, p1m1, seen as an infinite set of parallel reflections along an axis.
Contents
Definition
Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that r^{n} is the identity, and we have a finite dihedral group of order 2n. If the rotation is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called Dih_{∞}. It has presentations
- ^{[1]}
and is isomorphic to a semidirect product of Z and Z/2, and to the free product Z/2 * Z/2. It is the automorphism group of the graph consisting of a path infinite to both sides. Correspondingly, it is the isometry group of Z (see also symmetry groups in one dimension), the group of permutations α: Z → Z satisfying |i - j| = |α(i) - α(j)|, for all i, j in Z.^{[2]}
The infinite dihedral group can also be defined as the holomorph of the infinite cyclic group.
Aliasing
An example of infinite dihedral symmetry is in aliasing of real-valued signals.
When sampling a function at frequency f_{s} (intervals 1/f_{s}), the following functions yield identical sets of samples: {sin(2π( f+Nf_{s}) t + φ), N = 0, ±1, ±2, ±3,...}. Thus, the detected value of frequency f is periodic, which gives the translation element r = f_{s}. The functions and their frequencies are said to be aliases of each other. Noting the trigonometric identity:
we can write all the alias frequencies as positive values: | f+N f_{s}|. This gives the reflection (f) element, namely f ↦ −f. For example, with f = 0.6f_{s} and N = −1, f+Nf_{s} = −0.4f_{s} reflects to 0.4f_{s}, resulting in the two left-most black dots in the figure.^{[note 1]} The other two dots correspond to N = −2 and N = 1. As the figure depicts, there are reflection symmetries, at 0.5f_{s}, f_{s}, 1.5f_{s}, etc. Formally, the quotient under aliasing is the orbifold [0, 0.5f_{s}], with a Z/2 action at the endpoints (the orbifold points), corresponding to reflection.
See also
- The orthogonal group O(2), another infinite generalization of the finite dihedral groups
Notes
References
- ^ Connolly, Francis; Davis, James (August 2004). "The surgery obstruction groups of the infinite dihedral group" (PDF). Geometry & Topology. 8: 1043–1078. arXiv:math/0306054. doi:10.2140/gt.2004.8.1043. Retrieved 2 May 2013.
- ^ Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. p. 38. ISBN 978-3-540-64965-6