Editing Improper rotation

{{Use American English|date=March 2019}}
{{Short description|Rotation composed with a reflection}}

In [[geometry]], an '''improper rotation'''<ref name="Morawiec">{{citation|title=Orientations and Rotations: Computations in Crystallographic Textures|first=Adam|last=Morawiec|publisher=Springer|year=2004|isbn=978-3-540-40734-8|page=7|url=https://books.google.com/books?id=m3RUd7z22M0C&pg=PA7}}.</ref> (also called '''rotation-reflection''',<ref>{{citation|title=Inorganic Chemistry|first1=Gary|last1=Miessler|first2=Paul|last2=Fischer|first3=Donald|last3=Tarr|publisher=Pearson|edition=5|year=2014|page=78}}</ref> '''rotoreflection,'''<ref name="Morawiec"/> '''rotary reflection''',<ref name="sss">{{citation|title=Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry|first1=L. Christine|last1=Kinsey|author1-link=L. Christine Kinsey|first2=Teresa E.|last2=Moore|publisher=Springer|year=2002|isbn=978-1-930190-09-2|page=267|url=https://books.google.com/books?id=0clfF_CFG9EC&pg=PA267}}.</ref> or '''rotoinversion'''<ref>{{Cite book|title=Earth Materials|last=Klein, Philpotts|publisher=Cambridge University Press|year=2013|isbn=978-0-521-14521-3|pages=89–90}}</ref>) is an [[isometry]] in [[Euclidean space]] that is a combination of a [[Rotation (geometry)|rotation]] about an axis and a [[reflection (mathematics)|reflection]] in a plane perpendicular to that axis. Reflection and [[Point reflection|inversion]] are each a special case of improper rotation. Any improper rotation is an [[affine transformation]] and, in cases that keep the coordinate origin fixed, a [[linear transformation]].<ref>{{citation|title=Computer Graphics and Geometric Modeling|first=David|last=Salomon|publisher=Springer|year=1999|isbn=978-0-387-98682-1|page=84|url=https://books.google.com/books?id=9XZgfTmfAwYC&pg=PA84}}.</ref>
It is used as a [[symmetry operation]] in the context of [[Symmetry (geometry)|geometric symmetry]], [[molecular symmetry]] and [[Crystallographic point group|crystallography]], where an object that is unchanged by a combination of rotation and reflection is said to have ''improper rotation symmetry''.

{| class=wikitable align=center width=400
|+ Example polyhedra with rotoreflection symmetry
!Group
! [[v:Symmetric_group_S4|''S''<sub>4</sub>]]
! ''S''<sub>6</sub>
! ''S''<sub>8</sub>
! ''S''<sub>10</sub>
! ''S''<sub>12</sub>
|- align=center
!Subgroups
| ''C''<sub>2</sub>
| ''C''<sub>3</sub>, ''S''<sub>2</sub> = ''C''<sub>i</sub>
| ''C''<sub>4</sub>, ''C''<sub>2</sub>
| ''C''<sub>5</sub>, ''S''<sub>2</sub> = ''C''<sub>i</sub>
| ''C''<sub>6</sub>, ''S''<sub>4</sub>, ''C''<sub>3</sub>, ''C''<sub>2</sub>
|- align=center
!Example
|[[File:2-antiprism rotoreflection.png|80px]]<BR>beveled digonal antiprism
| [[File:3-antiprism_rotoreflection.png|80px]]<BR>[[triangular antiprism]]
| [[File:Rotoreflection_example_square_antiprism.png|80px]]<BR>[[square antiprism]]
| [[File:Rotoreflection_example_antiprism.png|100px]]<BR>[[pentagonal antiprism]]
| [[File:6-antiprism_rotorereflection.png|100px]]<BR>[[hexagonal antiprism]]
|- align=center
| colspan=6 | [[Antiprism]]s with directed edges have rotoreflection symmetry.<BR>''p''-antiprisms for odd ''p'' contain [[inversion symmetry]], ''C''<sub>i</sub>.
|}

== Three dimensions==
{{See|Point groups in 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.<ref name="Morawiec"/> For this reason it is also called a '''rotoinversion''' or '''rotary inversion'''. The two definitions are equivalent because [[angle of rotation|rotation by an angle]] θ followed by reflection is the same transformation as rotation by θ&nbsp;+&nbsp;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 points of isometry groups in Euclidean space|fixed point]] is necessarily an improper rotation.<ref name="sss"/> 

An improper rotation of an object thus produces a rotation of its [[mirror image]]. The axis is called the '''rotation-reflection axis'''.<ref name="Bishop">{{citation|title=Group Theory and Chemistry|first=David M.|last=Bishop|publisher=Courier Dover Publications|year=1993|isbn=978-0-486-67355-4|page=13|url=https://books.google.com/books?id=l4zv4dukBT0C&pg=PA13}}.</ref> 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).<ref name="Bishop"/> There are several different systems for naming individual improper rotations:
* In the [[Schoenflies notation]] the symbol '''''S<sub>n</sub>''''' (German, ''{{lang|de|Spiegel}}'', for ''[[mirror]]''), where ''n'' must be even, denotes the symmetry group generated by an ''n''-fold improper rotation. For example, the symmetry operation ''S''<sub>6</sub> is the combination of a rotation of (360°/6)=60° and a mirror plane reflection. (This should not be confused with the same notation for [[symmetric group]]s).<ref name="Bishop"/> 
* In [[Hermann–Mauguin notation]] the symbol {{overline|''n''}} 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 {{overline|2}} 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''<sub>2''n''</sub> is [2''n''<sup>+</sup>,2<sup>+</sup>] and {{CDD|node_h2|2x|n|node_h4|2x|node_h2}}, as an index 4 subgroup of [2''n'',2], {{CDD|node|2x|n|node|2|node}}, generated as the product of 3 reflections.
* The [[Orbifold notation]] is ''n''×, order 2''n''.[[File:Rotoreflection_subgroup_tree.png|thumb|Subgroups for ''S''<sub>2</sub> to ''S''<sub>20</sub>.<BR>''C''<sub>1</sub> is the [[Trivial group|identity group]].<BR>''S''<sub>2</sub> is the [[central inversion]].<BR>''C''<sub>n</sub> are [[cyclic group]]s.]]
=== Subgroups===
* The [[Coxeter notation#Subgroups|direct subgroup]] of ''S''<sub>2''n''</sub> is ''C''<sub>''n''</sub>, order ''n'', [[index of a subgroup|index]] 2, being the rotoreflection generator applied twice.
* For odd ''n'', ''S''<sub>2''n''</sub> contains an [[Inversion in a point|inversion]], denoted ''C''<sub>i</sub> or ''S''<sub>2</sub>. ''S''<sub>2''n''</sub> is the [[direct product]]: ''S''<sub>2''n''</sub> = ''C''<sub>''n''</sub>&nbsp;×&nbsp;''S''<sub>2</sub>, if ''n'' is odd.
* For any ''n'', if odd ''p'' is a divisor of ''n'', then ''S''<sub>2''n''/''p''</sub> is a subgroup of ''S''<sub>2''n''</sub>, index ''p''. For example ''S''<sub>4</sub> is a subgroup of ''S''<sub>12</sub>, index 3.

==As an indirect isometry ==

In a wider sense, an improper rotation may be defined as any '''[[Euclidean group#Direct and indirect isometries|indirect isometry]]'''; i.e., an element of [[Euclidean group|E]](3)\E<sup>+</sup>(3): thus it can also be a pure reflection in a plane, or have a [[glide reflection|glide plane]]. An indirect isometry is an [[affine transformation]] 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<sup>+</sup>(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 (geometry)|vector]]s and [[pseudovector]]s (as well as [[scalar (mathematics)|scalars]] and [[pseudoscalar]]s, and in general between [[tensor]]s and [[pseudotensor]]s), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).

==See also==
*[[Isometry]]
*[[Orthogonal group]]

==References==
{{reflist}}

[[Category:Euclidean symmetries]]
[[Category:Lie groups]]