Screwing

A screw connection turns an object around an axis of rotation by a fixed angle and shifts the result parallel to the axis of rotation.

In the geometry of the three-dimensional space V, a screwing is understood to be a mapping that consists of a sequential execution of a parallel displacement with a displacement vector and a rotation around a straight line in which is parallel . ${\ displaystyle v}$ ${\ displaystyle g}$ ${\ displaystyle v}$ ${\ displaystyle g}$

In crystallography, screw axes are possible symmetry elements of a space group .

A screw connection represents an isometric drawing on V because it is a link between two isometric drawings. Screws play a role especially in discrete geometry, for example when classifying isometrics in dimension 3. Isometrics in three-dimensional vector spaces can be divided into 7 types according to geometrical aspects. Next to the screw you can find:

Screw axes as an element of a space group

Spiral chain of tellurium atoms along the 3 ₁ screw axis (highlighted in blue). Every third atom is congruent (highlighted in blue). The distance between the dark, medium and light blue atoms is one lattice constant.

A space group can only contain screw axes that are compatible with the translation grid of the group. Therefore there can only be n-fold axes of rotation in a space group, with n = 2, 3, 4 or 6. Since these give the identity again after n-times repetition, they can only be linked with a translation vector that is n-fold Repetition corresponds to a vector of the grid. This is only the case if its length is m times the nth fraction of the lattice translation in the direction of the axis of rotation. The Hermann Mauguin symbol for these screw axes is a subscript m after the symbol for the axis of rotation n.

4 ₁ means a four-fold screw axis, with each rotation of 90 ° a translation in the direction of the axis of rotation of ¼ lattice constant is added. All screw axes occurring in the 230 room groups are listed below.

2 ₁
(3 ₁ 3 ₂ )
(4 ₁ 4 ₃ ) 4 ₂
(6 ₁ 6 ₅ ) (6 ₂ 6 ₄ ) 6 ₃

Pairs of enantiomorphic screw axes are summarized in brackets. These screw axes only differ in the direction of rotation. The first screw is a right screw, the second the corresponding left screw. These two symmetry elements are particularly difficult to distinguish from one another.

Screwing of rigid bodies

The Florentine mathematician Giulio Mozzi (1730–1813) was the first to recognize that every movement of a rigid body can be represented as a screw. H. as translation of a reference point and rotation about the reference point with an axis of rotation that is given by the speed of the reference point.

The reference point is determined as follows from the movement of the rigid body, which can always be represented as the translation of a point and the angular velocity of the rigid body around this point: ${\ displaystyle {\ vec {r}}}$ ${\ displaystyle {\ vec {b}}}$ ${\ displaystyle {\ vec {\ omega}}}$

{\ displaystyle {\ vec {v}} ({\ vec {p}}, t) = {\ dot {\ vec {b}}} + {\ vec {\ omega}} \ times ({\ vec {p }} - {\ vec {b}})}

Here, at time t, is the velocity of the particle at the location , the superpoint is a time derivative and “×” is the cross product . Then is too ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {p}}}$

{\ displaystyle {\ vec {v}} ({\ vec {p}}, t) = {\ dot {\ vec {r}}} + {\ vec {\ omega}} \ times ({\ vec {p }} - {\ vec {r}})}

With

{\ displaystyle {\ begin {aligned} {\ vec {r}} = & {\ vec {b}} + {\ frac {{\ vec {\ omega}} \ times {\ dot {\ vec {b}} }} {{\ vec {\ omega}} \ cdot {\ vec {\ omega}}}} + \ rho {\ vec {\ omega}} \\ {\ dot {\ vec {r}}} = & { \ dot {\ vec {b}}} + {\ vec {\ omega}} \ times ({\ vec {r}} - {\ vec {b}}) = {\ frac {{\ vec {\ omega} } \ cdot {\ dot {\ vec {b}}}} {{\ vec {\ omega}} \ cdot {\ vec {\ omega}}}} {\ vec {\ omega}} \ end {aligned}} }

and anything . The arithmetic symbol "·" forms the scalar product . ${\ displaystyle \ rho \ in \ mathbb {R}}$

Individual evidence

↑ Giulio Mozzi Giuseppe. Wikipedia , January 25, 2020, accessed April 15, 2020 (Italian).
↑ Giulio Mozzi Giuseppe: Mathematical discourse on the current rotation of bodies . Print shop of Donato Campo, Naples 1763 (Italian, archive.org [accessed on April 15, 2020] Original title: Discorso matematico sopra il rotamento mementaneo dei corpi . Quoted from Marcolongo (1911), p. 122.).

literature

D. Schwarzenbach: Crystallography . Springer Verlag, Berlin 2001, ISBN 3-540-67114-5 .
Roberto Marcolongo : Theoretical Mechanics . Kinematics and statics. 1st volume. BG Teubner , Leipzig and Berlin 1911, p. 122 ( archive.org [accessed April 15, 2020]).

Web links

Geometry script by Eschenburg, p. 56. (PDF file; 676 kB) Crystallographic screw axis

[1] Giulio Mozzi Giuseppe. Wikipedia , January 25, 2020, accessed April 15, 2020 (Italian).

[2] Giulio Mozzi Giuseppe: Mathematical discourse on the current rotation of bodies . Print shop of Donato Campo, Naples 1763 (Italian, archive.org [accessed on April 15, 2020] Original title: Discorso matematico sopra il rotamento mementaneo dei corpi . Quoted from Marcolongo (1911), p. 122.).