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Rotating 4d sphere7/23/2023 In the rotation in space ( e ) it is rotated around. The two rotations could of course have different rates. Rotation in the fourth dimension ( u ) rotates the u-axis and another spatial axis around any two axes. In general they are seen as moving in a direction (the one you would have moved if you weren't stationary) but rotating about that direction. Here is a plot of three random stars as they move in this space: Square: all his stars rotate about his circular Earth and it might be inconceivable for him to imagine a fixed polar star.Īdd2: Represent the 3-sphere using three angles $(\theta,\phi,\psi)$. If you find it hard to imagine stars rotating about a plane, think of the flatman A. There would be a whole plane (appearing like a great circle) of fixed 'polar' stars, and all other stars rotate about this plane along with the 'equator'. If you are rotating with the sphere and look up at the sky, and you have 4-D vision, then the sky would not look like a dome but a volume (a 3-sphere). The projections of a body onto the planes rotate with the planes.Īdd: A rotating 3-D sphere does have an axis, and there is only one way to extend the rotation to 4-D, namely by fixing the fourth dimension this is the case $\phi=0$ in the above matrix. So none of the suggestions 1-3 hold: there are two orthogonal planes, each rotating independently, and causing the volume of vectors in between them to rotate with them. If by axis you mean non-zero fixed vectors $Px=x$, then there are no axes in general for 4-D rotations, just as there are no axes in 2-D. To clarify, a rotation $P$ in 4-D is defined to be an orthogonal matrix with positive determinant, that is, $P^TP=I$, $\det P= 1$.įor every such rotation, one can find two perpendicular rotation planes meeting only at the origin.
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