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.
Point group
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)
Discrete symmetry groups
These affine symmetries can be considered limiting cases of the 2D dihedral and cyclic groups:
Translational symmetry
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., for the pattern
− −−− − − −−− − − −−− −
We choose the origin for x at one of the points of reflection. Now all reflections which map the pattern to itself are of the form a−x where the constant "a" is an integer (the increments of a are 1 again, because we can combine a reflection and a translation to get another reflection, and we can combine two reflections to get a translation). Therefore all isometries can be characterized by an integer and a code, say 0 or 1, for translation or reflection.
Thus:
The latter is a reflection with respect to the point a/2 (an integer or an integer plus 1/2).
Group operations (function composition, the one on the right first) are, for integers a and b:
E.g., in the third case: translation by an amount b changes x into x + b, reflection with respect to 0 gives−x − b, and a translation a gives a − b − x.
This group is called the generalized dihedral group of Z, Dih(Z), and also D∞. It is a semidirect product of Z and C2. It has a normal subgroup of index 2 isomorphic to Z: the translations. Also it contains an element f of order 2 such that, for all n in Z, n f = f n −1: the reflection with respect to the reference point, (0,1).
The two groups are called lattice groups. The lattice is Z. As translation cell we can take the interval 0 ≤ x < 1. In the first case the fundamental domain can be taken the same; topologically it is a circle (1-torus); in the second case we can take 0 ≤ x ≤ 0.5.
The actual discrete symmetry group of a translationally symmetric pattern can be: of group 1 type, for any positive value of the smallest translation distance of group 2 type, for any positive value of the smallest translation distance, and any positioning of the lattice of points of reflection (which is twice as dense as the translation lattice)
The set of translationally symmetric patterns can thus be classified by actual symmetry group, while actual symmetry groups, in turn, can be classified as type 1 or type 2.
These space group types are the symmetry groups “up to conjugacy with respect to affine transformations”: the affine transformation changes the translation distance to the standard one (above: 1), and the position of one of the points of reflections, if applicable, to the origin. Thus the actual symmetry group contains elements of the form gag−1= b, which is a conjugate of a.
Non-discrete symmetry groups
For a homogeneous “pattern” the symmetry group contains all translations, and reflection in all points. The symmetry group is isomorphic to Dih(R).