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
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