Improper Rotation
   HOME

TheInfoList



OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
are each special case of improper rotation. Any improper rotation is an
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
and, in cases that keep the coordinate origin fixed, a linear transformation.. It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wo ...
, where an object that is unchanged by a combination of rotation and reflection is said to have ''improper rotation symmetry''.


Three dimensions

In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and inversion in a point on the axis. For this reason it is also called a rotoinversion or rotary inversion. The two definitions are equivalent because rotation by an angle θ followed by reflection is the same transformation as rotation by θ + 180° followed by inversion (taking the point of inversion to be in the plane of reflection). In both definitions, the operations commute. A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation. An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis.. This is called an ''n''-fold improper rotation if the angle of rotation, before or after reflexion, is 360°/''n'' (where ''n'' must be even). There are several different systems for naming individual improper rotations: * In the Schoenflies notation the symbol ''Sn'' (German, ', for ''
mirror A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
''), where ''n'' must be even, denotes the symmetry group generated by an ''n''-fold improper rotation. For example, the symmetry operation ''S''6 is the combination of a rotation of (360°/6)=60° and a mirror plane reflection. (This should not to be confused with the same notation for
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
s). * In Hermann–Mauguin notation the symbol is used for an ''n''-fold rotoinversion; i.e., rotation by an angle of rotation of 360°/''n'' with inversion. If ''n'' is even it must be divisible by 4. (Note that would be simply a reflection, and is normally denoted "m", for "mirror".) When ''n'' is odd this corresponds to a 2''n''-fold improper rotation (or rotary reflexion). * The Coxeter notation for ''S''2''n'' is ''n''+,2+and , as an index 4 subgroup of ''n'',2 , generated as the product of 3 reflections. * The Orbifold notation is ''n''×, order 2''n''.


Subgroups

* The direct subgroup of ''S''2''n'' is ''C''''n'', order ''n'',
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
2, being the rotoreflection generator applied twice. * For odd ''n'', ''S''2''n'' contains an
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
, denoted ''C''i or ''S''2. ''S''2''n'' is the direct product: ''S''2''n'' = ''C''''n'' × ''S''2, if ''n'' is odd. * For any ''n'', if odd ''p'' is a divisor of ''n'', then ''S''2''n''/''p'' is a subgroup of ''S''2''n'', index ''p''. For example ''S''4 is a subgroup of ''S''12, index 3.


As an indirect isometry

In a wider sense, an improper rotation may be defined as any
indirect isometry In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...
; i.e., an element of E(3)\E+(3): thus it can also be a pure reflection in a plane, or have a glide plane. An indirect isometry is an
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
with an orthogonal matrix that has a determinant of −1. A proper rotation is an ordinary rotation. In the wider sense, a proper rotation is defined as a direct isometry; i.e., an element of E+(3): it can also be the identity, a rotation with a translation along the axis, or a pure translation. A direct isometry is an affine transformation with an orthogonal matrix that has a determinant of 1. In either the narrower or the wider senses, the composition of two improper rotations is a proper rotation, and the composition of an improper and a proper rotation is an improper rotation.


Physical systems

When studying the symmetry of a physical system under an improper rotation (e.g., if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general between
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
s and pseudotensors), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).


See also

* Isometry *
Orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...


References

{{reflist Euclidean symmetries Lie groups