Hilbert's third problem
E911228
Hilbert's third problem is one of David Hilbert’s famous list of 23 problems, asking whether every polyhedron of a given volume is equidecomposable with any other of the same volume, a question that led to the development of the Dehn invariant and the discovery of counterexamples.
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
Hilbert problem
ⓘ
mathematical problem ⓘ |
| asksAbout |
decomposition of polyhedra into finitely many polyhedral pieces
ⓘ
reassembly of polyhedral pieces by isometries ⓘ |
| assumes | rigid motions and cut-and-paste operations of polyhedra ⓘ |
| category |
problems in geometry
ⓘ
unsolved problems in mathematics (historical) ⓘ |
| clarifiedBy | Dehn's construction of non-equidecomposable polyhedra NERFINISHED ⓘ |
| concernsDimension | three-dimensional Euclidean space ⓘ |
| field |
discrete geometry
ⓘ
geometry ⓘ polyhedral geometry ⓘ |
| hasAnswer | negative ⓘ |
| hasCounterexample |
cube and regular tetrahedron of equal volume
ⓘ
pairs of polyhedra of equal volume that are not equidecomposable ⓘ |
| hasImplication | volume is not a complete invariant for scissors congruence of polyhedra in 3D ⓘ |
| hasNegativeAnswerReason | existence of Dehn invariant independent of volume ⓘ |
| hasNumber | 3 ⓘ |
| historicalSignificance | first Hilbert problem to be solved ⓘ |
| inspiredDevelopment |
study of scissors congruence classes of polyhedra
ⓘ
theory of Dehn invariants ⓘ |
| introducedInvariant | Dehn invariant NERFINISHED ⓘ |
| involvesConcept |
Dehn invariant
NERFINISHED
ⓘ
equidecomposability ⓘ polyhedron ⓘ scissors congruence ⓘ volume ⓘ |
| languageOfOriginalStatement | German ⓘ |
| mainQuestion |
whether any two polyhedra of equal volume are equidecomposable
ⓘ
whether every polyhedron of a given volume is equidecomposable with any other of the same volume ⓘ |
| motivated | systematic study of invariants under polyhedral decomposition ⓘ |
| originalTitle | Drittes Problem NERFINISHED ⓘ |
| partOf |
Hilbert's list of 23 problems
NERFINISHED
ⓘ
Hilbert's problems NERFINISHED ⓘ |
| publicationContext | Hilbert's 1900 Paris lecture NERFINISHED ⓘ |
| relatedTo |
Banach–Tarski paradox
NERFINISHED
ⓘ
Hilbert's fourth problem NERFINISHED ⓘ scissors congruence problem NERFINISHED ⓘ |
| solutionBy | Max Dehn NERFINISHED ⓘ |
| solutionYear |
1900
ⓘ
1901 ⓘ |
| statedBy | David Hilbert NERFINISHED ⓘ |
| status | solved ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.