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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hilbert's third problem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T11215094 — 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: Hilbert's third problem Context triple: [Dehn invariant, relatedTo, Hilbert's third problem]
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A.
Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
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B.
Dehn invariant
The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
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C.
Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
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D.
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry stating that a solid ball in 3‑dimensional space can be decomposed into finitely many non-measurable pieces and reassembled into two identical copies of the original ball, highlighting counterintuitive consequences of the axiom of choice.
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E.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
- 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: Hilbert's third problem Target entity description: 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.
-
A.
Hilbert’s twenty-third problem
Hilbert’s twenty-third problem is one of David Hilbert’s famous list of unsolved problems, focusing on the further development and systematic application of the calculus of variations.
-
B.
Dehn invariant
The Dehn invariant is a mathematical quantity in geometry that helps determine whether two polyhedra are scissors-congruent, playing a key role in the solution of Hilbert’s third problem.
-
C.
Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
-
D.
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry stating that a solid ball in 3‑dimensional space can be decomposed into finitely many non-measurable pieces and reassembled into two identical copies of the original ball, highlighting counterintuitive consequences of the axiom of choice.
-
E.
Hilbert problems
The Hilbert problems are a famous list of 23 unsolved mathematical problems presented by David Hilbert in 1900 that profoundly influenced the development of 20th-century mathematics.
- F. None of above. chosen
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
ⓘ
linked to:
Dehn invariant
|
| 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 ⓘ |
| involvesConcept |
Dehn invariant
ⓘ
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 ⓘ |
| partOf |
Hilbert's list of 23 problems
ⓘ
linked to:
Hilbert problems
Hilbert's problems ⓘ
linked to:
Hilbert problems
|
| publicationContext |
Hilbert's 1900 Paris lecture
ⓘ
linked to:
Hilbert problems
|
| relatedTo |
Banach–Tarski paradox
ⓘ
Hilbert's fourth problem ⓘ scissors congruence problem ⓘ
linked to:
Dehn invariant
|
| solutionBy | Max Dehn ⓘ |
| solutionYear |
1900
ⓘ
1901 ⓘ |
| statedBy | David Hilbert ⓘ |
| status | solved ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hilbert's third problem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.