Screw Axis of A Spatial Displacement
A screw displacement (also screw operation) is the composition of a rotation by an angle φ about an axis (called the screw axis) with a translation by a distance d along this axis. A positive rotation direction usually means one that corresponds to the translation direction by the right-hand rule. Except for φ = 180°, we have to distinguish a screw displacement from its mirror image. Unlike for rotations, a righthand and lefthand screw operation generate different groups.
The combination of a rotation about an axis and a translation in a perpendicular direction is a rotation about a parallel axis. However, a screw operation with a nonzero translation vector along the axis cannot be reduced like that. Thus the effect of a rotation combined with any translation is a screw operation in the general sense, with as special cases a pure translation. a pure rotation, and the identity. Together these are all the direct isometries in 3D.
Screw axis symmetry is invariance under a screw displacement. If φ = 360°/n for some positive integer n, then screw axis symmetry implies translational symmetry with a translation vector which is n times that of the screw displacement.
Applicable for space groups is a rotation by 360°/n about an axis, combined with a translation along the axis by a multiple of the distance of the translational symmetry, divided by n. This multiple is indicated by a subscript. So, 63 is a rotation of 60° combined with a translation of 1/2 of the lattice vector, implying that there is also 3-fold rotational symmetry about this axis. The possibilities are 21, 31, 41, 42, 61, 62, and 63, and the enantiomorphous 32, 43, 64, and 65.
A non-discrete screw axis isometry group contains all combinations of a rotation about some axis and a proportional translation along the axis (in rifling, the constant of proportionality is called the twist rate); in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold helix; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.
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