Euler characteristic

The Euler characteristic, usually written as $$\chi$$, is an integer quantity that can provide information about a polytope. For an n-dimensional polytope whose surface is topologically equivalent to a sphere of the same dimension, its Euler characteristic is 2 if n is odd, and 0 if n is even. This holds true for all convex polytopes, and even some nonconvex ones. A deviation from this value of the Euler characteristic can indicate that the surface of the polytope is self-intersecting or toroidal, or that an error has occurred.

The Euler characteristic is most often used in the context of polyhedra, i.e. 3-dimensional polytopes. There, it is defined as $$\chi=V-E+F$$, where V is the number of vertices of the polyhedron, E is the number of edges, and F is the number of faces.

More generally, the Euler characteristic of an n-dimensional polytope is defined as $$\chi=\sum_{i=0}^{n-1} (-1)^iC_i$$, where Ci is the number of elements of dimension i in the polytope. In other words, the Euler characteristic is the number of even-dimensioned elements minus the number of odd-dimensioned elements, not counting the polytope itself and the null polytope. This gives us $$\chi=V-E$$ for 2 dimensions, $$\chi=V-E+F$$ for 3 dimensions as seen above, $$\chi=V-E+F-C$$ for 4 dimensions (where C is the number of cells in the polychoron), $$\chi=V-E+F-C+T$$ for 5 dimensions (where T is the number of tera in the polyteron), and so on.

Polytopes that deviate from the "usual" value of the Euler characteristic in their dimension do exist. Since their facets intersect one another, the nonconvex uniform polytopes are a common example of this, such as the great dodecahedron or dodecadodecahedron for which $$\chi=-6$$, or the great grand 120-cell for which $$\chi=-480$$. Toroidal polytopes also have unusual Euler characteristics, since their surfaces are topologically equivalent to tori and not spheres.