Euler characteristic

Euler characteristic

In <a href="/en/Mathematics" title="Mathematics" class="wl">mathematics</a>, and more specifically in <a href="/en/Algebraic_topology" title="Algebraic topology" class="wl">algebraic topology</a> and <a href="/en/Polyhedral_combinatorics" title="Polyhedral combinatorics" class="wl">polyhedral combinatorics</a>, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a <a href="/en/Topological_invariant" title="Topological invariant" class="mw-redirect wl">topological invariant</a>, a number that describes a <a href="/en/Topological_space" title="Topological space" class="wl">topological space</a>'s shape or structure regardless of the way it is bent. It is commonly denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
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 <mi>χ</mi>
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 <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="\chi "> (<a href="/en/Greek_alphabet" title="Greek alphabet" class="wl">Greek lower-case letter</a> <a href="/en/Chi_(letter)" title="Chi (letter)" class="wl">chi</a>).

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Topological invariant in mathematics</div>
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The Euler characteristic was originally defined for <a href="/en/Polyhedron" title="Polyhedron" class="wl">polyhedra</a> and used to prove various theorems about them, including the classification of the <a href="/en/Platonic_solid" title="Platonic solid" class="wl">Platonic solids</a>. It was stated for Platonic solids in 1537 in an unpublished manuscript by <a href="/en/Francesco_Maurolico" title="Francesco Maurolico" class="wl">Francesco Maurolico</a>.<a href="#cite_note-1" style="counter-reset: mw-Ref 1;">[1]</a> <a href="/en/Leonhard_Euler" title="Leonhard Euler" class="wl">Leonhard Euler</a>, 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 <a href="/en/Homology_(mathematics)" title="Homology (mathematics)" class="wl">homology</a> and, more abstractly, <a href="/en/Homological_algebra" title="Homological algebra" class="wl">homological algebra</a>.

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler 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 
  
    
      
        χ
      
    
    {\displaystyle \chi } . The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. 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 and, more abstractly, homological algebra. 