Euler’s polyhedron formula

E54784

mathematical formula result in geometry result in topology topological invariant

Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.

Aliases (3)

Statements (46)

Predicate	Object
instanceOf	mathematical formula → result in geometry → result in topology → topological invariant →
appliesTo	convex polyhedra → simply connected polyhedra homeomorphic to a sphere →
assumes	polyhedron is topologically equivalent to a sphere →
category	polyhedron invariants →
correspondsTo	Euler’s formula for connected planar graphs V − E + F = 2 →
expresses	V − E + F = 2 →
failsFor	certain non-convex polyhedra with holes →
field	geometry → polyhedral combinatorics → topology →
generalizedBy	Euler characteristic of topological spaces → Euler’s polyhedron formula → surface form: "Euler–Poincaré formula"
hasAlternativeName	Euler characteristic formula for polyhedra →
hasConcept	Euler’s polyhedron formula → surface form: "Euler characteristic"
hasConstantTerm	2 →
hasDidacticUse	classic example in discrete geometry → introductory example in topology courses →
hasEquationSide	E (number of edges) → F (number of faces) → V (number of vertices) →
hasEulerCharacteristic	2 →
historicalPeriod	18th century →
holdsFor	Platonic solids → cube → dodecahedron → icosahedron → octahedron → tetrahedron →
implies	V + F = E + 2 →
mathematicalDomain	K-theory → surface form: "algebraic topology" discrete mathematics →
namedAfter	Leonhard Euler →
relatedTo	Jordan curve theorem → planar graphs →
relates	number of edges of a polyhedron → number of faces of a polyhedron → number of vertices of a polyhedron →
usedIn	classification of convex polyhedra → combinatorial topology → computational geometry → graph theory →
usedToCheck	combinatorial validity of polyhedral meshes →

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Euler’s polyhedron formula → generalizedBy → Euler’s polyhedron formula →

this entity surface form: "Euler–Poincaré formula"

Euler’s polyhedron formula → hasConcept → Euler’s polyhedron formula →