({{, (RefLink[EulerCharacteristic, paclet:ref/EulerCharacteristic])[poly] 
gives the Euler characteristic of a poly.}})

OpenersNotesSection(Spacer1)Details

(RefLink[EulerCharacteristic, paclet:ref/EulerCharacteristic]) is also known as Euler number or Euler–Poincaré characteristic.

(RefLink[EulerCharacteristic, paclet:ref/EulerCharacteristic]) is a topological invariant that describes the shape of the polyhedron, regardless of the way it is bent.

The Euler characteristic χ for a polyhedron is given by χ=v-e+f, where v is the number of vertices, e the number of edges and f the number of faces.

A polyhedron with s voids and g tunnels satisfies χ=2 (s-g).

The Euler characteristic for a mesh region is given by χ=(∑_(n=0)^2)(((-1))^n)(RefLink[MeshCellCount, paclet:ref/MeshCellCount])[poly,n].

OpenersPrimaryExamplesSection(Spacer1)Examples  (8)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

Euler characteristic of a polyhedron:

𝒫 = Polyhedron[];

EulerCharacteristic[𝒫]

Region[𝒫]

OpenersExampleSection1(Spacer1)Scope  (4)

(RefLink[EulerCharacteristic, paclet:ref/EulerCharacteristic]) works on polyhedrons:

𝒫 = Polyhedron[];

EulerCharacteristic[𝒫]

Region[𝒫]

(RefLink[Tetrahedron, paclet:ref/Tetrahedron]):

EulerCharacteristic[Tetrahedron[]]

(RefLink[Hexahedron, paclet:ref/Hexahedron]):

EulerCharacteristic[Hexahedron[]]

Polyhedron with holes:

Polyhedron[{{0,0,0},{0,3,0},{3,3,0},{3,0,0},{0,0,3},{0,3,3},{3,3,3},{3,0,3},{1,1,1},{1,2,1},{2,2,1},{2,1,1},{1,1,2},{1,2,2},{2,2,2},{2,1,2}},{{{2,3,4,1},{1,4,8,5},{4,3,7,8},{3,2,6,7},{2,1,5,6},{5,8,7,6}}->{{{10,11,12,9},{9,12,16,13},{12,11,15,16},{11,10,14,15},{10,9,13,14},{13,16,15,14}}}}];

EulerCharacteristic[%]

Polyhedrons with disconnected components:

Polyhedron[{{0,0,0},{1,0,0},{0,1,0},{0,0,1},{1,1,1},{2,1,1},{1,2,1},{1,1,2}},{{{1,2,3},{1,2,4},{2,3,4},{1,3,4}},{{5,6,7},{5,6,8},{6,7,8},{5,7,8}}}]

EulerCharacteristic[%]

(RefLink[EulerCharacteristic, paclet:ref/EulerCharacteristic]) works on mesh regions:

MengerMesh[1,3]

EulerCharacteristic[%]

OpenersExampleSection4(Spacer1)Properties & Relations  (3)

Use (RefLink[EulerCharacteristic, paclet:ref/EulerCharacteristic]) to compute (RefLink[PolyhedronGenus, paclet:ref/PolyhedronGenus]) for a simple polyhedron:

𝒫 = Polyhedron[];

SimplePolyhedronQ[𝒫]

1-EulerCharacteristic[𝒫]/2 == PolyhedronGenus[𝒫]

Euler characteristic of a convex polyhedron equals 2:

𝒫 = Polyhedron[];

ConvexPolyhedronQ[𝒫]

EulerCharacteristic[𝒫] == 2

Euler characteristic of (RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron]) is 2:

UniformPolyhedron[{3,5}]

EulerCharacteristic[%]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[PolyhedronGenus, paclet:ref/PolyhedronGenus])  ▪  (RefLink[Polyhedron, paclet:ref/Polyhedron])  ▪  (RefLink[MeshCellCount, paclet:ref/MeshCellCount])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Polyhedra, paclet:guide/Polyhedra])}})}})