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.

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All labels observed (5)

Label	Occurrences
Euler characteristic	5
Euler’s polyhedron formula canonical	2
Euler characteristic formula for polyhedra	1
Euler characteristic in topology	1
Euler–Poincaré formula	1

How this entity was disambiguated

This entity first appeared as the object of triple T426767 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Euler’s polyhedron formula
Context triple: [Leonhard Euler, notableWork, Euler’s polyhedron formula]

A. Platonic solids
Platonic solids are the five highly symmetrical, convex polyhedra (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) that have identical regular polygonal faces and are fundamental in geometry and classical philosophy.
B. Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
C. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
D. Conway’s Game of Sprouts
Conway’s Game of Sprouts is a pencil-and-paper topological game in which players alternately connect dots with lines under simple rules, leading to rich combinatorial and mathematical analysis.
E. Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Euler’s polyhedron formula
Target entity description: 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.

A. Platonic solids
Platonic solids are the five highly symmetrical, convex polyhedra (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) that have identical regular polygonal faces and are fundamental in geometry and classical philosophy.
B. Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
C. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
D. Conway’s Game of Sprouts
Conway’s Game of Sprouts is a pencil-and-paper topological game in which players alternately connect dots with lines under simple rules, leading to rich combinatorial and mathematical analysis.
E. Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
F. None of above. chosen

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–Poincaré formula ⓘ linked to: Euler’s polyhedron formula
hasAlternativeName	Euler characteristic formula for polyhedra ⓘ
hasConcept	Euler characteristic ⓘ linked to: Euler’s polyhedron formula
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	algebraic topology ⓘ linked to: K-theory 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 ⓘ

How these facts were elicited