# EulerCharacteristic EulerCharacteristic[poly]

gives the Euler characteristic of a poly.

# Details • EulerCharacteristic is also known as Euler number or EulerPoincaré characteristic.
• EulerCharacteristic is a topological invariant that describes the shape of the polyhedron, regardless of the way it is bent.
• The Euler characteristic for a polyhedron is given by , where is the number of vertices, the number of edges and the number of faces.
• A polyhedron with voids and tunnels satisfies .
• The Euler characteristic for a mesh region is given by χ= (-1)nMeshCellCount[poly,n].

# Examples

open allclose all

## Basic Examples(1)

Euler characteristic of a polyhedron:

## Scope(4)

EulerCharacteristic works on polyhedrons:

Polyhedron with holes:

Polyhedrons with disconnected components:

EulerCharacteristic works on mesh regions:

## Properties & Relations(3)

Use EulerCharacteristic to compute PolyhedronGenus for a simple polyhedron:

Euler characteristic of a convex polyhedron equals 2:

Euler characteristic of UniformPolyhedron is 2: