One-dimensional symmetry group

A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D). A pattern in 1D can be represented as a function f(x) for, say, the color at position x. The only nontrivial point group in 1D is a simple reflection. It can be represented by the simplest Coxeter group, A1, [ ], or Coxeter-Dynkin diagram . Affine symmetry groups represent translation. Isometries which leave the function unchanged are translations x + a with a such that f(x + a) = f(x) and reflections a − x with a such that f(a − x) = f(x). The reflections can be represented by the affine Coxeter group [∞], or Coxeter-Dynkin diagram representing two reflections, and the translational symmetry as [∞]+, or Coxeter-Dynkin diagram as the composite of two reflections. For a pattern without translational symmetry there are the following possibilities (1D point groups): the symmetry group is the trivial group (no symmetry) the symmetry group is one of the groups each consisting of the identity and reflection in a point (isomorphic to Z2) These affine symmetries can be considered limiting cases of the 2D dihedral and cyclic groups: Consider all patterns in 1D which have translational symmetry, i.e., functions f(x) such that for some a > 0, f(x + a) = f(x) for all x. For these patterns, the values of a for which this property holds form a group. We first consider patterns for which the group is discrete, i.e., for which the positive values in the group have a minimum. By rescaling we make this minimum value 1. Such patterns fall in two categories, the two 1D space groups or line groups. In the simpler case the only isometries of R which map the pattern to itself are translations; this applies, e.g., for the pattern − −−− − −−− − −−− − −−− Each isometry can be characterized by an integer, namely plus or minus the translation distance. Therefore the symmetry group is Z. In the other case, among the isometries of R which map the pattern to itself there are also reflections; this applies, e.g.