# Axes of rotation in 4D

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.

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:

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.

• 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! – safesphere Nov 30 '17 at 4:02
• 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... – safesphere Nov 30 '17 at 20:55
• ... 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. – safesphere Nov 30 '17 at 21:06