7/24/2023 0 Comments Rotating 4d sphere![]() It's not too tough to write down proofs of these claims, at least if you know about eigenvalues and eigenvectors if you want, I can expand my answer to do so. Such rotations are relatively easy to write down, but they do not describe all possible rotations in 4-space: there are also ones that rotate by some amount in a plane spanned by vectors $v_1, v_2$, and by some (possibly different) amount in a plane spanned by vectors $u_1, u_2$, with the $u$s orthogonal to the $v$s. If we did the latter, then generalizing would work fine: we could talk (in 4-space) about rotating in the $xy$-plane, which leaves two perpendicular dimensions ( $z$ and $w$) fixed. The trick is to understand that our standard way of describing rotations in 3-space is slightly flawed: instead of saying that "we rotated 20 degrees about the $z$ axis", for instance, we could say "we rotated in the $xy$-plane by $20$ degrees" (i.e., naming the plane of rotation rather than the single axis that happens to be orthogonal to the plane). Perhaps I should say "the reason is that any rotation that leaves fixed a single axis also leaves fixed another axis orthogonal to it, and hence leaves fixed an entire plane." You were probably hoping to find something with the property thatįor any $v$ in the "axis" $L$, and the property thatįor any vector $w$ orthogonal to $L$, i.e., all vectors $w$ in the plane orthogonal to $L$ are rotated by an angle of $\theta$. The reason you haven't read about "rotation about an axis" in 4D is that there is no such thing.
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