1
$\begingroup$

In the Eucledian 3D space, only one axis of rotation is possible at one time for a body, such as a sphere. So we have a 2D plane of rotation perpendicular to the 1D axis of rotation, the total of 3D. Easy to visualize. Now, in 4D, not so easy to visualize anymore. If we still have a 2D plane of rotation, then we have two extra dimensions perpendicular to this plane. Does this mean that a body, such as a 3-sphere:

  1. Can rotate around two orthogonal axes at the same time or:
  2. Can rotate only around one of two axes at a time or:
  3. Rotation in 4D is not in the plane, but in tbe "volume of rotation" around one axis

Sorry for my lack of 4D imagination. I would appreciate your help with pointing out what the correct way of thinking is regarding rotations in 4D.

$\endgroup$
2
$\begingroup$

To clarify, a rotation $P$ in 4-D is defined to be an orthogonal matrix with positive determinant, that is, $P^TP=I$, $\det P=+1$.

For every such rotation, one can find two perpendicular rotation planes meeting only at the origin. So $P$ has a matrix $$\pmatrix{\cos\theta&-\sin\theta&0&0\\\sin\theta&\cos\theta&0&0\\0&0&\cos\phi&-\sin\phi\\0&0&\sin\phi&\cos\phi}$$

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.

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. The projections of a body onto the planes rotate with the planes.

Add: 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.

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). 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 find it hard to imagine stars rotating about a plane, think of the flatman A. Square: all his stars rotate about his circular Earth and it might be inconceivable for him to imagine a fixed polar star.

Add2: Represent the 3-sphere using three angles $(\theta,\phi,\psi)$. Here is a plot of three random stars as they move in this space:

enter image description here enter image description here

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. The two rotations could of course have different rates.

$\endgroup$
3
  • $\begingroup$ Thank you for the answer! While I am trying to comprehend it, I wonder if you could please help me with an example. As mentioned in the question, the rotating object is a 3-sphere. Consider it is the shape of the universe. If I am stationary in it while it rotates in a single rotation, I would see stars approaching me on one side and moving away from me on the other (as if I am flying forward). What woud I see during a double rotation that you've described? Thanks again for your help! $\endgroup$ – safesphere Nov 30 '17 at 4:02
  • $\begingroup$ Thank you for the addition, but it may not be what I meant. My example was a universe with space shaped as a 3-sphere (a 3D surface of a 4D ball) embedded in a flat 4D Eucledian space. There are no stars outside the volume of the 3-sphere, so no 4D vision should be required (at least locally) to observe the stars. Since a 3-dphere is a 4D object, it should rotate around 2 orthogonal planes meeting at the center of the sphere, as you originally described, should it not? I am stationary inside the rotating 3-sphere. I am observing the stars inside the sphere. If the sphere rotates around only... $\endgroup$ – safesphere Nov 30 '17 at 20:55
  • $\begingroup$ ... one plane, I believe, I would see the stars approaching me on one side and moving away from me on the opposite side, as if I were moving straight (along some "meridian") inside the sphere. This seems pretty intuitive, is this not correct? Finally, if the 3-sphere rotates around 2 orthogonal planes, as you described, this is no longer as intuitive. Clearly, I would still see the stars moving somehow. Would I see the same as above plus the universe rotating around the "vector" of my perceived motion forward? Or would it look like something else? To clarify - I do not rotate with the sphere. $\endgroup$ – safesphere Nov 30 '17 at 21:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.