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 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. 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 (intervals ), the following functions yield identical sets of samples: }. Thus, the detected value of frequency is periodic, which gives the translation element . 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: . This gives the reflection () element, namely ↦ . For example, with and , reflects to , resulting in the two left-most black dots in the figure. The other two dots correspond to and . As the figure depicts, there are reflection symmetries, at 0.5, , 1.5, etc. Formally, the quotient under aliasing is the orbifold [0, 0.5], 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 Infinite group theory