In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and more specifically in algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

 and polyhedral combinatorics

Polyhedral combinatorics

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher dimensional convex polytopes....

, the Euler characteristic (or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

's shape or structure regardless of the way it is bent. It is commonly denoted by (Greek letter

Greek alphabet

The Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...

 chi

Chi (letter)

Chi is the 22nd letter of the Greek alphabet, pronounced as in English.-Greek:-Ancient Greek:Its value in Ancient Greek was an aspirated velar stop .-Koine Greek:...

).

The Euler characteristic was originally defined for polyhedra

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 and used to prove various theorems about them, including the classification of the Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s.