Symmetry Group - One Dimension

One Dimension

The isometry groups in 1D where for all points the set of images under the isometries is topologically closed are:

  • the trivial group C1
  • the groups of two elements generated by a reflection in a point; they are isomorphic with C2
  • the infinite discrete groups generated by a translation; they are isomorphic with Z
  • the infinite discrete groups generated by a translation and a reflection in a point; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D (which is a semidirect product of Z and C2).
  • the group generated by all translations (isomorphic with R); this group cannot be the symmetry group of a "pattern": it would be homogeneous, hence could also be reflected. However, a uniform 1D vector field has this symmetry group.
  • the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group of R, Dih(R).

See also symmetry groups in one dimension.

Read more about this topic:  Symmetry Group

Famous quotes containing the word dimension:

    Authority is the spiritual dimension of power because it depends upon faith in a system of meaning that decrees the necessity of the hierarchical order and so provides for the unity of imperative control.
    Shoshana Zuboff (b. 1951)

    God cannot be seen: he is too bright for sight; nor grasped: he is too pure for touch; nor measured: for he is beyond all sense, infinite, measureless, his dimension known to himself alone.
    Marcus Minucius Felix (2nd or 3rd cen. A.D.)