Euler Characteristic

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Euler Characteristic


The Euler characteristic of a polyhedron is the number α0 — α1 + α2, where α0 is the number of vertices, α1 is the number of edges, and α2 is the number of faces. According to Euler’s theorem, if the polyhedron is convex or is homeomorphic to a convex polyhedron, then its Euler characteristic is 2. This fact was known to R. Descartes; L. Euler published a proof of the theorem in 1758.

The Euler characteristic of an arbitrary simplicial complex is the number

where n is the dimension of the complex, α0 is the number of vertices, and α1 is the number of edges. In general, k is the number of k-dimensional simplexes belonging to the complex. According to the Euler-Poincaré formula, the Euler characteristic is equal to

where πk is the k-dimensional Betti number of the complex. The topological invariance of the Euler characteristic follows from this fact.

In view of the topological invariance of the Euler characteristic, we speak of the Euler characteristic of a surface and of a polytope, by which we mean the Euler characteristic of any triangulation of the surface or polytope.


Aleksandrov, P. S. Kombinatornaia topologiia. Moscow-Leningrad, 1947.
Pontriagin, L. S. Osnovy kombinatornoi topologii, 2nd ed. Moscow, 1976.
References in periodicals archive ?
In a 3D-Euclidian space, this set corresponds to volume, surface area, mean integral curvature, and the Euler-Poincare characteristic (44).
Different types of microdomains have different values of the local Euler-Poincare characteristic.
Therefore, when dimM = 2, (M, g) admits an orthogonally conformal vector field if, and only if, either M is noncompact or M is compact and its Euler-Poincare characteristic vanishes.
Local contributions to the Euler-Poincare characteristic of a set.
An integralgeometric approach for the Euler-Poincare characteristic of spatial images.
In all what follows we fix a rank n (the same for all components), an Euler-Poincare characteristic [Chi] = d + n(1 - g) and weights [[Lambda].
Let E be a torsion-free sheaf on C of rank n on each component and Euler-Poincare characteristic [Chi].
Minkowski functionals encompass standard geometric parameters such as volume, area, length and the Euler-Poincare characteristic.
Another parameter is the Euler-Poincare characteristic, related to the topology of the structure.
It makes use of the local Euler-Poincare characteristic, which is defined as the expected Euler number of X in a neighborhood of a point x, i.
Sujatha, Euler-Poincare characteristics of abelian varieties, C.
Wintenberger, On the Euler-Poincare characteristics of finite dimensional p-adic Galois representations, Publ.