Editing Invariant (mathematics) (section)

{{Short description|Property that is not changed by mathematical transformations}}
{{Other uses|Invariant (disambiguation)}}
{{confusing|reason=all this article confuses "invariance" (a property) and "an invariant" (a mathematical object that is left invariant under a [[group action]])|date=January 2024}}{{more inline|date=April 2015}}

[[File:Wallpaper group-p2-3.jpg|thumb|A [[Wallpaper group|wallpaper]] is invariant under some transformations. This one is invariant under horizontal and vertical translation, as well as rotation by 180° (but not under reflection).]]
In [[mathematics]], an '''invariant''' is a property of a [[mathematical object]] (or a [[Class (set theory)|class]] of mathematical objects) which remains unchanged after [[Operation (mathematics)|operations]] or [[Transformation (function)|transformations]] of a certain type are applied to the objects.<ref>{{Cite web|url=https://www.mathsisfun.com/definitions/invariant.html|title=Invariant Definition (Illustrated Mathematics Dictionary)|website=www.mathsisfun.com|access-date=2019-12-05}}</ref><ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Invariant.html|title=Invariant|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-05}}</ref> The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the [[area]] of a [[triangle]] is an invariant with respect to [[isometry|isometries]] of the [[Plane (geometry)|Euclidean plane]]. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an [[equivalence relation]] is a property that is constant on each [[equivalence class]].<ref name=":2">{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Invariant|title=Invariant – Encyclopedia of Mathematics|website=www.encyclopediaofmath.org|access-date=2019-12-05}}</ref>

Invariants are used in diverse areas of mathematics such as [[geometry]], [[topology]], [[algebra]] and [[discrete mathematics]]. Some important classes of transformations are defined by an invariant they leave unchanged. For example, [[conformal map]]s are defined as transformations of the plane that preserve [[angle]]s. The discovery of invariants is an important step in the process of classifying mathematical objects.<ref name=":1" /><ref name=":2" />