# Euler characteristic

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 C_{i} 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.

Polyhedron | Image | V | E | F | |
---|---|---|---|---|---|

dodecahedron | 20 | 30 | 12 | 2 | |

cuboctahedron | 12 | 24 | 14 | 2 | |

snub disphenoid | 8 | 18 | 12 | 2 |

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.

## External links[edit | edit source]

- Wikipedia Contributors. "Euler characteristic".

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