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:
“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.)
“By intervening in the Vietnamese struggle the United States was attempting to fit its global strategies into a world of hillocks and hamlets, to reduce its majestic concerns for the containment of communism and the security of the Free World to a dimension where governments rose and fell as a result of arguments between two colonels’ wives.”
—Frances Fitzgerald (b. 1940)