# Why are there only translations and rotations?

In 1D, we get translations. In 2D, we get rotations. Why doesn't a 3rd spatial dimension introduce anything.

Followup: Do higher dimensions introduce anything?

Are there rigid motions of interest which cannot be described as a combination of translations and rotations?

• Well, this isn't really true. See this wiki article. – Ennar Sep 23 '16 at 7:20
• Actually, we have many types of functions. I suggest that you tell us which kind of functions you are interested. I think you may be interested in linear transformations, isometries and rigid motions. – edm Sep 23 '16 at 7:24
• In four dimensions there is a double rotation. In three dimensions two orthogonal planes must intersect in a line, but in four dimensions they may intersect only in a point. (Consider the planes $(x,y,0,0)$ and $(0,0,z,w)$ for example.) Take two such planes and compose one rotation in each plane. – MJD Sep 23 '16 at 15:08