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'' mea ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

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 Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

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 generally, ...

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 group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...

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 invari ...

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 substances ...

. 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 Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the ...

the symbol ''S_n'' (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 \m ...

s). * In

Hermann–Mauguin notation In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogis ...

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 diagram ...

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 advanta ...

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 ta ...

: ''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 In geometry and crystallography, a glide plane (or transflection) is a symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged. Glide planes are noted by ''a'', ...

. 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 ma ...

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

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s 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 o ...

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 a ...

and

pseudoscalar In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...

s, 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 tenso ...

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 coordinat ...

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