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Concept
One-dimensional symmetry group
Summary
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.
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