Editing Improper rotation (section)

{{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>.
|}