[1] The unmoved points will constitute a line, a plane, or all of space.
In polar point groups of high symmetry, the polar direction can be a unique axis of rotation, but if the symmetry operations do not allow any rotation at all, such as mirror symmetry, there can be an infinite number of such axes: in that case the only restriction on the polar direction is that it must be parallel to any mirror planes.
Of the 32 crystallographic point groups, 10 are polar:[2] The space groups associated with a polar point group do not have a discrete set of possible origin points that are unambiguously determined by symmetry elements.
[1] When materials having a polar point group crystal structure are heated or cooled, they may temporarily generate a voltage called pyroelectricity.
Molecular crystals which have symmetry described by one of the polar space groups, such as sucrose, may exhibit triboluminescence.