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?

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    $\begingroup$ Well, this isn't really true. See this wiki article. $\endgroup$ – Ennar Sep 23 '16 at 7:20
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    $\begingroup$ 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. $\endgroup$ – edm Sep 23 '16 at 7:24
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    $\begingroup$ 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. $\endgroup$ – MJD Sep 23 '16 at 15:08

It's not quite clear what you are asking about, but I suppose you mean rigid motions (which preserve the distance between any two points), which also preserve orientation (which rules out mirror reflections).

If so, then as Ennar points out, there is a new kind of motion in three dimensions: the screw translation. This is a rotation about an axis combined with a translation along the same axis.

The simple rotation and the simple translation are both special cases of the screw translation, with the translation part or the rotation part being zero, respectively.


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