Editing Reflection (mathematics) (section)

{{Short description|Mapping from a Euclidean space to itself}}
{{About|reflection in geometry|reflexivity of [[binary relation]]s|reflexive relation}}
[[File:SimmetriainvOK.svg|right|thumb|A reflection through an axis.]] 

In [[mathematics]], a '''reflection''' (also spelled '''reflexion''')<ref>[https://web.archive.org/web/20120829214317/http://oxforddictionaries.com/definition/english/reflexion "Reflexion" is an archaic spelling]</ref> is a [[function (mathematics)|mapping]] from a [[Euclidean space]] to itself that is an [[isometry]] with a [[hyperplane]] as the set of [[Fixed point (mathematics)|fixed point]]s; this set is called the [[Axis of symmetry|axis]] (in dimension 2) or [[plane (mathematics)|plane]] (in dimension 3) of reflection. The image of a figure by a reflection is its [[mirror image]] in the axis or plane of reflection. For example the mirror image of the small Latin letter '''p''' for a reflection with respect to a [[vertical axis]] (a ''vertical reflection'') would look like '''q'''. Its image by reflection in a [[horizontal axis]] (a ''horizontal reflection'') would look like '''b'''. A reflection is an [[involution (mathematics)|involution]]: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.

The term ''reflection'' is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. The set of fixed points (the "mirror") of such an isometry is an [[affine subspace]], but is possibly smaller than a hyperplane. For instance a [[Point reflection|reflection through a point]] is an involutive isometry with just one fixed point; the image of the letter '''p''' under it
would look like a '''d'''. This operation is also known as a [[point reflection|central inversion]] {{harv|Coxeter|1969|loc=§7.2}}, and exhibits Euclidean space as a [[symmetric space]]. In a [[Euclidean vector space]], the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a [[hyperplane]].

Some mathematicians use "'''flip'''" as a synonym for "reflection".<ref>{{Citation |last=Childs |first=Lindsay N. |year=2009 |title=A Concrete Introduction to Higher Algebra |edition=3rd |publisher=Springer Science & Business Media |page=251 |isbn=9780387745275 |url=https://books.google.com/books?id=qyDAKBr_I2YC&q=flip&pg=PA251 }}</ref><ref>
{{Citation |last=Gallian |first=Joseph |author-link=Joseph Gallian |year=2012 |title=Contemporary Abstract Algebra |edition=8th |publisher=Cengage Learning |page=32 |isbn=978-1285402734 |url=https://books.google.com/books?id=Ef4KAAAAQBAJ&q=flip&pg=PA32 }}</ref><ref>
{{Citation |last=Isaacs |first=I. Martin |author-link=Martin Isaacs |year=1994 |title=Algebra: A Graduate Course |publisher=American Mathematical Society |page=6 |isbn=9780821847992 |url=https://books.google.com/books?id=5tKq0kbHuc4C&q=flip&pg=PA6 }}
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