Euler characteristic

related topics
{math, number, function}
{math, energy, light}
{@card@, make, design}
{specie, animal, plant}

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by χ (Greek letter chi).

The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. Leonhard Euler, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic arises from homology and connects to many other invariants.

Contents

Polyhedra

The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula

where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. Any convex polyhedron's surface has Euler characteristic

This result is known as Euler's formula. This corresponds to the Euler characteristic of the sphere (which is 2), and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below.

Full article ▸

related documents
Wiener process
Dihedral group
Fourier analysis
Binary relation
E (mathematical constant)
Abstract interpretation
Pushdown automaton
Random variable
Algebraic structure
Linear combination
Axiom schema of replacement
Heine–Borel theorem
Euclidean algorithm
Glossary of topology
Presentation of a group
Elliptic curve
Factorization
Ideal class group
Sequence
IEEE 754-1985
Linear independence
Gaussian quadrature
Type theory
Fuzzy logic
Natural transformation
Absolute convergence
Galois theory
Partition (number theory)
Ultrafilter
Complex analysis