Euler Characteristic


Also found in: Wikipedia.

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.

REFERENCES

Aleksandrov, P. S. Kombinatornaia topologiia. Moscow-Leningrad, 1947.
Pontriagin, L. S. Osnovy kombinatornoi topologii, 2nd ed. Moscow, 1976.
References in periodicals archive ?
Along the Euler characteristic approach, we choose as the nonlinear threshold in (3) the observed normalized Draupner crest height [xi] = 6.
In this article we give the implementation of two algorithms to compute the Euler characteristic of Milnor fibre of reduced plane curve singularities in computer algebra system SINGULAR.
Equivalently, there are no normal surfaces with non-negative Euler characteristic, except for peripheral tori and Klein bottles.
They study the simplicial homology groups of certain minimal simple closed surfaces, extended an earlier definition of the Euler characteristics of a digital image, and computed the Euler characteristic of several digital surfaces.
The idea of enumeration using the Euler characteristic was suggested throughout Rota's work and influenced by Schanuel's categorical viewpoint [21, 23, 24, 25].
R] [intersection] K with K [member of] K fulfils the requirements listed above, the density of the Euler characteristic can be interpreted as [(-l).
applies the Euler characteristic to map coloring, proves the spectral theorem for hermitian and normal linear transformations on a finite-dimensional Hilbert space, and develops the structure of a finitely generated torsion module.
The Euler characteristic is easy to compute: Just add up the number of polygons, subtract the number of lines, and add the number of corner points.
Since the Euler characteristic of M, [chi](M), is twice the Euler characteristic of S = M/{-1, 1} and [chi](M) is zero, S = M/{-1, 1} is a Klein bottle.
Our main result is an explicit closed formula for the Euler characteristic of each line of the second page of that spectral sequence.
The reduced Euler characteristic [chi]([DELTA]) is given by [chi]([DELTA]) := [[summation].
Estimate the local Euler characteristic in the reduced observation window W [?