Doublerotation

Doublerotation is a term used to describe a rotation in four-dimensional Euclidean space that can be decomposed into independent rotations in two orthogonal planes. In this view, a single 4D rotation is characterized by two plane angles, often denoted α and β, governing the rotation in each of the two orthogonal planes.

Mathematically, rotations in four dimensions form the group SO(4). Unlike in 3D, a general 4D rotation can be expressed, up to conjugacy, as a pair of independent planar rotations. Equivalently, any orientation-preserving 4D rotation is conjugate to a block-diagonal form with two 2×2 rotation blocks, one by angle α in one plane and the other by angle β in the orthogonal plane. The structure of SO(4) is intimately linked to the isomorphism SO(4) ≅ (SU(2) × SU(2)) / ±1, which encodes this double-angle decomposition as a pair of SU(2) actions.

Special cases help distinguish different types of 4D rotations. If either α or β is zero, the rotation reduces to a simple rotation in a single plane. If α equals β or α equals −β, the rotation is called isoclinic (left- or right-isoclinic, respectively), rotating all relevant 2D subspaces by the same angle in a coordinated way.

In practice, doublerotations arise in higher-dimensional geometry and physics, where the analysis of 4D rotational symmetries benefits from separating actions on orthogonal planes. They also play a role in quaternion and Clifford-algebra formalisms used to represent rotations in four dimensions. See also isoclinic rotation, quaternion, and SO(4).