📂Geometry

Euler Characteristics in Geometry

Definition

Simple Definition

Let’s assume we are given a shape. Let’s call the number of vertices $V$, the number of edges $E$, and the number of faces $F$. The Euler characteristic $\chi$ of this shape is defined as follows.

$\chi := V - E + F$

Complex Definition¹

For a surface $M$ and its region $\mathscr{R}$, the term $\chi(\mathscr{R}) \in \mathbb{Z}$ that satisfies the Gauss-Bonnet theorem is called the Euler characteristic of $\mathscr{R}$.

$$ \sum_{i=1}^{n} \int_{C_{i}}K_{g}ds + \iint_{R} K dA + \sum\theta_{i} = 2\pi \chi(\mathscr{R}) $$

See Also

Euler’s Formula in Graph Theory

The original Euler characteristic is most famous in graph theory, where the Euler polyhedron theorem or Euler formula is a theorem of graph theory that states $\chi = 2$ for connected planar graphs.

Euler Characteristic in Geometry

It is defined as an integer that satisfies the equation of the Gauss-Bonnet theorem.

Euler Characteristic in Algebraic Topology

It is defined as the alternating sum of the Betti numbers of each dimension.