Dan is correct. In R^3 every rotation is in some plane, and the unique direction left over is the axis of rotation. -- Gene On Sunday, November 1, 2020, 5:04:30 PM PST, Allan Wechsler <acwacw@gmail.com> wrote: Maybe ceil(n/2)? The statement appears to fail for R^3. On Sun, Nov 1, 2020 at 7:35 PM Dan Asimov <dasimov@earthlink.net> wrote:
Henry Baker wrote ----- It is also well known that every 4D rotation can be expressed as the *independent* rotations of 2 planes orthogonal to one another, each with their own separate rotation. -----
In fact any rotation of R^n (orientation-preserving isometry taking the origin to itself) is in fact the result of floor(n/2) rotations on mutually orthogonal 2-dimensional planes. If the angles are all distinct and not 0 or π, then this decomposition is unique.
Coxeter wrote a great paper on quaternions and rotations and reflections of 4-dimensional space: "Quaternions and Reflections", the Monthly, Vol. 53, No. 3 (Mar., 1946), pp. 136-146.
—Dan
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