page_title: Screw axis
section_title: Screw axis symmetry
hierarchical_section_title: Screw axis / Screw axis symmetry
caption_attribution_description: The chains in the crystal structure of tellurium along the 31-screw axis. The chain is highlighted in blue colors where the dark blue atom is situated on c = 1/3, middle blue on c = 2/3 and light blue on c = 0. Thick red bonds represent covalent bonds between atoms in the chain (d = 284 pm), dashed green bonds secondary contacts between chains (d = 349 pm) and dashed purple bonds represent the hexagonal surrounding within a "layer" of tellurium (d = 446 pm).
context_page_description: A screw axis is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis.
  Plücker coordinates are used to locate a screw axis in space, and consist of a pair of three-dimensional vectors. The first vector identifies the direction of the axis, and the second locates its position. The special case when the first vector is zero is interpreted as a pure translation in the direction of the second vector. A screw axis is associated with each pair of vectors in the algebra of screws, also known as screw theory.
  The spatial movement of a body can be represented by a continuous set of displacements. Because each of these displacements has a screw axis, the movement has an associated ruled surface known as a screw surface. This surface is not the same as the axode, which is traced by the instantaneous screw axes of the movement of a body.
context_section_description: A screw displacement (also screw operation or rotary translation) 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.
  In crystallography, a screw axis symmetry is a combination of rotation about an axis and a translation parallel to that axis which leaves a crystal unchanged. 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 disubscript. 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.
caption: 31 screw axis in crystal structure of tellurium
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