Editing 3D rotation group (section)

==Generalizations==
The rotation group generalizes quite naturally to ''n''-dimensional [[Euclidean space]], <math>\R^n</math> with its standard Euclidean structure. 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''), which is a [[Lie group]] of dimension {{nowrap|''n''(''n'' − 1)/2}}.

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 [[metric signature|signature]]. However, one can still define ''generalized rotations'' which preserve this inner product. Such generalized rotations are known as [[Lorentz transformation]]s and the group of all such transformations is called the [[Lorentz group]].

The rotation group SO(3) can be described as a subgroup of [[SE(3)|E<sup>+</sup>(3)]], the [[Euclidean group]] of [[Euclidean group#Direct and indirect isometries|direct isometries]] of Euclidean <math>\R^3.</math> 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, 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 [[chirality (mathematics)|chiral]] objects it is the same as the full symmetry group.