Can a rigid body have three independent rotations, about three different, mutually perpendicular axes?

Question

It appears that this is kinematically possible, but it looks quite unprobable. For example, it is quite improbable for a sphere to have rotation about three mutually perpendicular axes (supposedly passing through its centre) simultaneously.

This led me to think if this is even possible kinematically.

As a simple case, consider a particle attached to a three dimensional reference frame. Now, It is possible for the particle to move in all three axes simultanoeusly. Does this necessarily imply that the particle can have rotation about all three axes simultaneously?

Answer 1

Made visualisations so you can see how it looks like.

Rotating X:

Rotating X and Y:

Rotating X,Y and Z:

Answer 2

There is not a simple answer for this since it is both yes and no.

From an internal frame of reference:
Looking at the object, nothing is happening.
Looking at the outside world, it is moving in a predictable pattern.

From an external frame of reference:
The rotation on one axis is easy to see.
The rotation on 2 axes looks like a rotation on a moving axis.
The rotation on 3 axes is difficult to describe.

Answer 3

The answer is that at any moment a rigid body is rotating about an arbitrary axis, with an arbitrary angle. There are two components defining the direction of rotation axis, and one for the magnitude of rotation.

Often we transform these three quantities into three sequential rotations called Euler Angles. If you look up rotation matrix you will find all sorts of ways to go from axis angle angle to three angles. They both describe the same thing, but with different representation.

BTW, The pure motion of a rigid body is a simultaneous rotation about an axis and translation along the same axis. This is called a screw motion. See https://physics.stackexchange.com/a/86020/392 for more details.