Editing Euler characteristic (section)

{{Short description|Topological invariant in mathematics}}
{{About|Euler characteristic number|Euler characteristic class|Euler class|Euler number in 3-manifold topology|Seifert fiber space}}
In [[mathematics]], and more specifically in [[algebraic topology]] and [[polyhedral combinatorics]], the '''Euler characteristic''' (or '''Euler number''', or '''Euler&ndash;Poincaré characteristic''') is a [[topological invariant]], a number that describes a [[topological space]]'s shape or structure regardless of the way it is bent. It is commonly denoted by <math> \chi </math> ([[Greek alphabet|Greek lower-case letter]] [[chi (letter)|chi]]).

The Euler characteristic was originally defined for [[polyhedron|polyhedra]] and used to prove various theorems about them, including the classification of the [[Platonic solid]]s. It was stated for Platonic solids in 1537 in an unpublished manuscript by [[Francesco Maurolico]].<ref>{{cite book|first= Michael|last=Friedman|publisher=Birkhäuser|year=2018|title=A History of Folding in Mathematics: Mathematizing the Margins|title-link=A History of Folding in Mathematics|series=Science Networks. Historical Studies|volume=59|isbn=978-3-319-72486-7|doi=10.1007/978-3-319-72487-4|page=71}}</ref> [[Leonhard Euler]], for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from [[homology (mathematics)|homology]] and, more abstractly, [[homological algebra]].