Infinite dihedral group

Summary

p1m1, (*∞∞) p2, (22∞) p2mg, (2*∞)
Frieze group m1.png Frieze group 12.png Frieze group mg.png
Frieze example p1m1.png
Frieze sidle.png
Frieze example p2.png
Frieze spinning hop.png
Frieze example p2mg.png
Frieze spinning sidle.png
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 one dimension, the infinite dihedral group is seen in the symmetry of an apeirogon alternating two edge lengths, containing reflection points at the center of each edge.

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.

DefinitionEdit

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 rn 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] [2]

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: ZZ satisfying |i - j| = |α(i) - α(j)|, for all i, j in Z.[3]

The infinite dihedral group can also be defined as the holomorph of the infinite cyclic group.

AliasingEdit

 
When periodically sampling a sinusoidal function at rate fs, the abscissa above represents its frequency, and the ordinate represents another sinusoid that could produce the same set of samples. An infinite number of abscissas have the same ordinate (an equivalence class with the fundamental domain [0, fs/2]),and they exhibit dihedral symmetry. The many-to-one phenomenon is known as aliasing.

An example of infinite dihedral symmetry is in aliasing of real-valued signals.

When sampling a function at frequency fs (intervals 1/fs), the following functions yield identical sets of samples: {sin(2π( f+Nfs) t + φ), N = 0, ±1, ±2, ±3,...}. Thus, the detected value of frequency f is periodic, which gives the translation element r = fs. 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 fs|.  This gives the reflection (f) element, namely ff.  For example, with f = 0.6fs  and  N = −1f+Nfs = −0.4fs  reflects to  0.4fs, 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 0.5fsfs,  1.5fs,  etc.  Formally, the quotient under aliasing is the orbifold [0, 0.5fs], with a Z/2 action at the endpoints (the orbifold points), corresponding to reflection.

See alsoEdit

NotesEdit

  1. ^ In signal processing, the symmetry about axis fs/2 is known as folding, and the axis is known as the folding frequency.

ReferencesEdit

  1. ^ Connolly, Francis; Davis, James (August 2004). "The surgery obstruction groups of the infinite dihedral group". Geometry & Topology. 8 (3): 1043–1078. arXiv:math/0306054. doi:10.2140/gt.2004.8.1043.
  2. ^ tern, arctic; user384354 (31 Oct 2016). "Infinite dihedral group isomorphic to semidirect product". Mathematics Stack Exchange. Retrieved 2021-07-21. {{cite web}}: |last2= has generic name (help)CS1 maint: url-status (link)
  3. ^ 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