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As far as I know right now, a rotation of something in 3D-Euclidean space is always defined by an axis and an angle. However, in higher dimensions, is there any such thing as a rotation that can be defined by a plane and an angle, or is the definition of a rotation restricted to specifically a 1-dimensional axis and a single angle of rotation? Can any sense be made of this notion of rotations about higher-dimensional spaces?

Please excuse my ignorance if the answer to this is obvious.

Arturo don Juan
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  • That can be done: Given a (oriented) plane $L$ and an angle $\theta$ you can rotate all the vectors in $L$ by $\theta$ (you need an orientation to do that even in $\mathbb R^3$) and fixes all vector in $L^\perp$. –  Jul 16 '15 at 14:45

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Yes, you can rotate about a plane in 4 dimensions. To imagine this, first start with rotation about a point in two dimensions. To extend this to rotating about a line in three dimensions, just perform the 2D point-rotation on every cross section perpendicular to the line. Similarly, extend this from 3 dimensions to 4 by taking cross sections perpendicular to the plane of rotation, and rotating about the resulting line on the 3D cross section. I hope that made sense.

msinghal
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  • Although I can't imagine it, I can understand your notion from construction. Do you know if this idea is further developed anywhere else? – Arturo don Juan Jul 17 '15 at 21:20
  • You can rotate in the x-y plane at the same time as you rotate in the z-w plane. If you co-ordinate these rotations to be of equal magnitude, it makes an interesting projection. In general you can do this in n dimensions where n is even. Just rotate dimensions i and i+1 where i is even; for all such at once. For the left-over odd dimension you can rotate about a line you think? – John L Winters Jul 23 '19 at 06:00