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It seems self-evident that there are only two directions an object can rotate in around a linear axis (clockwise and counter-clockwise). But as math has taught me over the years, self-evident is not the same as correct. Can it be proven that there are only two directions of rotation around an axis?

Joe
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  • Yes, it can. Please first define "direction of rotation around an axis". That will be the biggest challenge. – dfeuer Jul 16 '13 at 23:43
  • Not sure how to prove this either, but here are some thoughts that might help this along: I suppose we would need to start with some agreed upon definitions. For rotation, perhaps a "transformation preserving size, orientation and position"? Now what about direction? Arguably, all rotations are counterclockwise rotations by some angle which would indicate only $1$ direction. What do you mean when you say there are two directions? – Ben Grossmann Jul 16 '13 at 23:47
  • Maybe the thing to prove is that all rotations about a given axis can be uniquely characterized by some angle from $-180˚$ to $180˚$. Of course, this requires a workable definition of angle. – Ben Grossmann Jul 16 '13 at 23:48
  • This is more interesting from a dynamic perspective, where an object is rotating and the trick is to show that this motion looks, at a sufficiently small timescale, like (counterclockwise) rotation about some axis. – dfeuer Jul 16 '13 at 23:51
  • Alternatively, you may wish to prove that every transformation with certain properties is indeed a rotation about some axis. – dfeuer Jul 16 '13 at 23:53
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    Perhaps we can state this as follows: For any line $\ell$ in ${\bf R}^3$, the space of nontrivial homomorphisms ${\bf R}\to{\rm Stab}_{{\rm Aff}({\bf R}^3)}(\ell)$ has two connected components. – anon Jul 17 '13 at 01:14

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