Facts about rotations in high dimensions: See this as an introduction to Clifford algebra. Notice that in 4D space, it's possible to have two planes which meet

Facts about rotations in high dimensions: See this as an introduction to Clifford algebra. Notice that in 4D space, it's possible to have two planes which meet at only one point, and for which every vector on one of the planes is perpendicular to every vector on the other. This implies that for each rotation in n dimensions, it is possible to pick floor(n/2) mutually perpendicular planes which are each invariant under the rotation. This can be proved using eigendecomposition. These sets of floor(n/2) invariant planes, weighed by their angles of rotation, form the "bivectors" in exterior and Clifford algebra. ([EDIT] It's slightly more accurate to say that bivectors are the angular velocities in n dimensions, which means that the angular speeds attached to each plane are not necessarily between [0,2pi] but can be any real.) Also, notice that in even dimensions there is a rotation which sends every vector to a vector perpendicular to it. But in odd dimensions, there isn't even a continuous function which sends every vector to a vector perpendicular to it; this follows from the hairy-ball theorem. However, notice that there is still an algorithm for finding perpendicular vectors in any number of dimensions; one such algorithm is an application of the Gram-Schmidt process (also called QR decomposition).