Euler characteristic

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The Euler characteristic, usually written as , 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 , 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 , 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 for 2 dimensions, for 3 dimensions as seen above, for 4 dimensions (where C is the number of cells in the polychoron), for 5 dimensions (where T is the number of tera in the polyteron), and so on.

Examples: polyhedra
Polyhedron Image V E F
dodecahedron 20 30 12 2
cuboctahedron 12 24 14 2
snub disphenoid 8 18 12 2
Examples: polychora
Polychoron image V E F C
pentachoron 5 10 10 5 0
truncated tesseract 64 128 88 24 0
grand antiprism 100 500 720 320 0

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 , or the great grand 120-cell for which . Toroidal polytopes also have unusual Euler characteristics, since their surfaces are topologically equivalent to tori and not spheres.

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