Rotation Group SO(3) - Generalizations

Generalizations

The rotation group generalizes quite naturally to n-dimensional Euclidean space, Rn. The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).

In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.

The rotation group SO(3) can be described as a subgroup of E+(3), the Euclidean group of direct isometries of R3. This larger group is the group of all motions of a rigid body: each of these is a combination of a rotation about an arbitrary axis and a translation along the axis, or put differently, a combination of an element of SO(3) and an arbitrary translation.

In general, the rotation group of an object is the symmetry group within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.