Editing Euler's rotation theorem (section)

==Generalizations==
{{see also|Rotations in 4-dimensional Euclidean space}}

In higher dimensions, any rigid motion that preserves a point in dimension {{math|2''n''}} or {{math|2''n'' + 1}} is a composition of at most {{mvar|n}} rotations in orthogonal [[plane of rotation|planes of rotation]], though these planes need not be uniquely determined, and a rigid motion may fix multiple axes. Also, any rigid motion that preserves {{math|''n''}} linearly independent points, which span an {{math|''n''}}-dimensional body in dimension {{math|2''n''}} or {{math|2''n'' + 1}}, is  a single  [[plane of rotation]]. To put it another way, if two rigid bodies, with identical geometry, share at least {{math|''n''}} points of 'identical' locations within themselves, the convex hull of which is {{math|''n''}}-dimensional, then a single planar rotation can bring one to cover the other accurately in dimension {{math|2''n''}} or  {{math|2''n'' + 1}}.

[[File:Pure screw.svg|thumb|A screw motion.]]
A rigid motion in three dimensions that does not necessarily fix a point is a "screw motion". This is because a composition of a rotation with a translation perpendicular to the axis is a rotation about a parallel axis, while composition with a translation parallel to the axis yields a screw motion; see [[screw axis]]. This gives rise to [[screw theory]].