Infinite dihedral group

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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 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 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.

Definition[edit]

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

\langle r, s \mid s^2 = 1, srs = r^{-1} \rangle \,\!
\langle x, y \mid x^2 = y^2 = 1 \rangle \,\![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: ZZ 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[edit]

When periodically sampling a signal, the detected frequency exhibits infinite dihedral symmetry, as exhibited in the graph of the detected frequency (equivalence class using the fundamental domain [0, 0.5fs]). Here a horizontal line (demonstrating the many-to-one phenomenon of aliasing) is shown, passing through 0.4fs, 0.6fs, 1.4fs and 1.6fs all of which thus lie in the same orbit (they are "aliases" of one another).
For more details on this topic, see Aliasing.

A concrete example of infinite dihedral symmetry is in aliasing of real-valued signals; this is realized as follows. If sampling a signal (signal processing term for a function) at frequency fs, then the functions sin(ft) and sin((f + fs)t) cannot be distinguished (and likewise for cos), which gives the translation (r) element – translation by fs (the detected frequency is periodic). Further, for a real signal, cos(−ft) = cos(ft) and sin(−ft) = −sin(ft), so every negative frequency has a corresponding positive frequency (this is not true for complex signals), and gives the reflection (f) element, namely f ↦ −f. Together these give further reflection symmetries, at 0.5fs, fs, 1.5fs, etc.; this phenomenon is called folding, as the graph of the detected signal "folds back" on itself, as depicted in the diagram at right.

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.

Notes[edit]

  1. ^ Connolly, Francis; Davis, James (August 2004). "The surgery obstruction groups of the infinite dihedral group" (PDF). Geometry & Topology 8: 1043–1078. doi:10.2140/gt.2004.8.1043. Retrieved 2 May 2013. 
  2. ^ 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