Английская Википедия:Infinite dihedral group
| p1m1, (*∞∞) | p2, (22∞) | p2mg, (2*∞) |
|---|---|---|
| Файл:Frieze group m1.png | Файл:Frieze group 12.png | Файл:Frieze group mg.png |
| Файл:Frieze example p1m1.png Файл:Frieze sidle.png |
Файл:Frieze example p2.png Файл:Frieze spinning hop.png |
Файл:Frieze example p2mg.png Файл: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 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
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
- <math>\langle r, s \mid s^2 = 1, srs = r^{-1} \rangle \,\!</math>
- <math>\langle x, y \mid x^2 = y^2 = 1 \rangle \,\!</math>[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
Шаблон:Details An example of infinite dihedral symmetry is in aliasing of real-valued signals.
When sampling a function at frequency Шаблон:Math (intervals Шаблон:Math), the following functions yield identical sets of samples: Шаблон:Math}. Thus, the detected value of frequency Шаблон:Mvar is periodic, which gives the translation element Шаблон:Math. The functions and their frequencies are said to be aliases of each other. Noting the trigonometric identity:
- <math>
\sin(2\pi (f+Nf_s)t + \varphi) = \begin{cases}
+\sin(2\pi (f+Nf_s)t + \varphi), & f+Nf_s \ge 0, \\[4pt] -\sin(2\pi |f+Nf_s|t - \varphi), & f+Nf_s < 0,
\end{cases} </math>
we can write all the alias frequencies as positive values: <math display=inline>|f+Nf_s|</math>. This gives the reflection (Шаблон:Mvar) element, namely Шаблон:Mvar ↦ Шаблон:Mvar. For example, with Шаблон:Math and Шаблон:Math, Шаблон:Math reflects to Шаблон:Math, resulting in the two left-most black dots in the figure.[note 1] The other two dots correspond to Шаблон:Math and Шаблон:Math. As the figure depicts, there are reflection symmetries, at 0.5Шаблон:Math, Шаблон:Math, 1.5Шаблон:Math, etc. Formally, the quotient under aliasing is the orbifold [0, 0.5Шаблон:Math], 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
- The affine symmetric group, a family of groups including the infinite dihedral group
Notes
References
- ↑ Шаблон:Cite journal
- ↑ Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. [[[:Шаблон:Google books]] p. 38]. Шаблон:ISBN
Ошибка цитирования Для существующих тегов <ref> группы «note» не найдено соответствующего тега <references group="note"/>
