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 mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...

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 Euclidea ...

that is a combination of a

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

.. It is used as a symmetry operation in the context of geometric symmetry,

molecular symmetry Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain m ...

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 wor ...

, 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 In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is inv ...

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 A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substance ...

. 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 ''S_n'' (German, ', for ''

mirror A mirror or looking glass is an object that 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 direction of the im ...

''), 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 In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagra ...

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 In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The adva ...

is ''n''×, order 2''n''. Rotoreflection_subgroup_tree

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

, denoted ''C''_i or ''S''₂. ''S''_2''n'' is the

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...

: ''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; 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

with an

orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity m ...

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

pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...

s (as well as

scalars Scalar may refer to: * Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...

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 tensor ...

s and

pseudotensor In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordi ...

s), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).

References

{{reflist Euclidean symmetries Lie groups

Three dimensions

Subgroups

As an indirect isometry

Physical systems

See also

References