Hi guys,


I have a simple question


Linear movement can be found in 1D, 2D and 3D world objects


Rotation can be found in 2D and 3D world objects.


Now, are there any kind of motion can *only* be found in 3D world objects? This kind of motion should be as simple as linear movement or rotation, and very basic that other 3D motions can be described based upon.


If there is not, why -


1. There are only 2 kinds of basic movements?
2. perhaps there are more advanced movements in 4D or higher dimensions?



Which motion is exclusive in 3D or higher dimensions?
I interpret your question to be asking for a description of 1-parameter groups of isometries.


The generic kind of motion in 3 dimensions is a screw-motion, which is translation along an axis combined with rotation about the same axis. Up to changing the speed of a parametrization, translations and rotations in 2 and 3 dimensions are all alike, but screw motions have an invariant, the pitch of the screw: how many complete turns are made with one unit of translation along the axis? Rotation is the limiting case pitch $\rightarrow 0$, while translation is the limiting case pitch $\rightarrow \infty$.


In 4 dimensions, in addition to rotations, translations and screw motions, there are compound rotations, which involve rotating about one 2-plane at one speed while rotating about the perpendicular plane at another speed. In 5 dimensions, there is also a compound screw-motion, which combines translation along a 1-dimensional axis with compound rotation in the perpendicular 4-plane.


The general pattern is similar. In dimension n, you can take any collection of mutually orthogonal 2-planes intersecting at a point and independently rotate each of them, and you can independently combine these rotations with translation in any direction that is a common perpendicular.


