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).
For a pattern without translational symmetry there are the following possibilities (1D point groups): 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.
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 all isometries can be characterized by an integer and a code, say 0 or 1, for translation or reflection.
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 actual discrete symmetry group of a translationally symmetric pattern can be: 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.
For a homogeneous “pattern” the symmetry group contains all translations, and reflection in all points.
However, 2-fold rotational symmetry of the graph does not imply any symmetry (in the sense of this article) of the function: function values (in a pattern representing colors, grey shades, etc.)
are nominal data, i.e. grey is not between black and white, the three colors are simply all different.
Even with nominal colors there can be a special kind of symmetry, as in: (reflection gives the negative image).
The action of G on X is called Consider a group G acting on a set X.
The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by Gx: Case that the group action is on R: Case that the group action is on patterns: The set of all orbits of X under the action of G is written as X/G.
The image of this map is the orbit of x and the coimage is the set of all left cosets of Gx.
The standard quotient theorem of set theory then gives a natural bijection between