View source for Infinite dihedral group

{{refimprove|date=July 2011}}
{| class=wikitable align=right width=360
!p1m1, ([[Orbifold notation|*∞∞]])
!p2, (22∞)
!p2mg, (2*∞)
|-
|[[File:Frieze group m1.png|120px]]
|[[File:Frieze group 12.png|120px]]
|[[File:Frieze group mg.png|120px]]
|-
|[[File:Frieze example p1m1.png|120px]]<BR>[[File:Frieze sidle.png|120px]]
|[[File:Frieze example p2.png|120px]]<BR>[[File:Frieze spinning hop.png|120px]]
|[[File:Frieze example p2mg.png|120px]]<BR>[[File:Frieze spinning sidle.png|120px]]
|-
|colspan=3|In 2-dimensions three [[frieze group]]s p1m1, p2, and p2mg are isomorphic to the Dih<sub>∞</sub> 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.
|}

[[File:Isogonal apeirogon linear.png|360px|thumb|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<sub>∞</sub> is an [[infinite group]] with properties analogous to those of the finite [[dihedral group]]s.

In [[two-dimensional geometry]], the '''infinite dihedral group''' represents the [[frieze group]] symmetry, ''p''1''m''1, 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 ''r<sup>n</sup>'' is the identity, and we have a finite dihedral group of order 2''n''. If the rotation is ''not'' a rational multiple of a full rotation, then there is no such ''n'' and the resulting group has [[Infinity|infinite]]ly many elements and is called Dih<sub>∞</sub>. It has [[Presentation of a group|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><ref>{{cite journal |last1=Connolly |first1=Francis |last2=Davis |first2=James |title=The surgery obstruction groups of the infinite dihedral group |journal=Geometry & Topology |volume=8 |date=August 2004 |issue=3 |pages=1043–1078 |doi=10.2140/gt.2004.8.1043|arxiv=math/0306054 }}</ref>

and is isomorphic to a [[semidirect product]] of '''Z''' and '''Z'''/2, and to the [[free product]] '''Z'''/2&nbsp;*&nbsp;'''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'':&nbsp;'''Z'''&nbsp;→&nbsp;'''Z''' satisfying |''i''&nbsp;−&nbsp;''j''| = |''α''(''i'') −&nbsp;''α''(''j'')|, for all ''i',&nbsp;''j'' in '''Z'''.<ref>Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups, Issue 1689. Springer, 1998. [{{Google books|plainurl=y|id=lKsEawMzEOQC|page=38|text=Infinite dihedral group}} p. 38]. {{ISBN|978-3-540-64965-6}}</ref>

The infinite dihedral group can also be defined as the [[Holomorph (mathematics)|holomorph]] of the [[infinite cyclic group]].

==Aliasing==
[[File:Aliasing-folding.svg|thumb|300px|When periodically sampling a sinusoidal function at rate {{math|''f''{{sub|s}}}}, 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]] {{math|[0, ''f''{{sub|s}}/2]}}), and they exhibit dihedral symmetry.  The many-to-one phenomenon is known as [[aliasing]].]]
{{details|Aliasing}}
An example of infinite dihedral symmetry is in [[aliasing]] of real-valued signals.  

When sampling a function at frequency {{math|''f''{{sub|s}}}} (intervals {{math|1/''f''{{sub|s}}}}), the following functions yield identical sets of samples: {{math|{sin(2{{pi}}( ''f''&nbsp;+&nbsp;''Nf''{{sub|s}}) ''t''&nbsp;+&nbsp;''φ''), ''N''&nbsp;{{=}}&nbsp;0,&nbsp;±1,&nbsp;±2,&nbsp;±3,&nbsp;.&nbsp;.&nbsp;.&nbsp;}}}.  Thus, the detected value of frequency {{mvar|''f''}} is ''periodic'', which gives the translation element {{math|''r'' {{=}} ''f''{{sub|s}}}}.  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 ({{mvar|''f''}}) element, namely {{mvar|''f''}}&nbsp;↦&nbsp;{{mvar|−''f''}}.&nbsp; For example, with {{math|''f'' {{=}} 0.6''f''{{sub|s}}}}&nbsp; and &nbsp;{{math|''N'' {{=}} −1}},&nbsp; {{math|''f''&nbsp;+&nbsp;''Nf''{{sub|s}} {{=}}&nbsp;−0.4''f''{{sub|s}}}}&nbsp; ''reflects'' to &nbsp;{{math|0.4''f''{{sub|s}}}}, resulting in the two left-most black dots in the figure.<ref group="note">In [[signal processing]], the symmetry about axis {{math|''f''{{sub|s}}/2}} is known as ''[[aliasing#Folding|folding]],'' and the axis is known as the ''folding frequency''.</ref>&nbsp; The other two dots correspond to {{math|''N'' {{=}} −2}}&nbsp; and &nbsp;{{math|''N'' {{=}} 1}}.  As the figure depicts, there are reflection symmetries, at 0.5{{math|''f''{{sub|s}}}},&nbsp; {{math|''f''{{sub|s}}}},&nbsp; 1.5{{math|''f''{{sub|s}}}},&nbsp; etc.&nbsp; Formally, the quotient under aliasing is the ''[[orbifold]]'' [0, 0.5{{math|''f''{{sub|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
* The [[affine symmetric group]], a family of groups including the infinite dihedral group

== Notes ==
{{reflist|group=note}}

== References==
<references/>

[[Category:Infinite group theory]]