Dehn invariant
E265415
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Dehn invariant canonical | 5 |
| Dehn's construction of non-equidecomposable polyhedra | 1 |
| scissors congruence problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2416881 — 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: Dehn invariant Context triple: [Max Dehn, notableConcept, Dehn invariant]
-
A.
Euler’s polyhedron formula
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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B.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
-
C.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
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D.
Steinmetz solid
The Steinmetz solid is a three-dimensional geometric shape formed by the intersection of two or more cylinders at right angles, often studied in calculus and solid geometry for its interesting volume and symmetry properties.
-
E.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
- 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: Dehn invariant Target entity description: 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.
-
A.
Euler’s polyhedron formula
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.
-
B.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
-
C.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
D.
Steinmetz solid
The Steinmetz solid is a three-dimensional geometric shape formed by the intersection of two or more cylinders at right angles, often studied in calculus and solid geometry for its interesting volume and symmetry properties.
-
E.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
geometric invariant
ⓘ
mathematical quantity ⓘ polyhedral invariant ⓘ |
| appliesTo | polyhedra in three-dimensional Euclidean space ⓘ |
| assumes | polyhedron has well-defined dihedral angles at each edge ⓘ |
| category | invariants in geometric topology ⓘ |
| definedOn | edges of a polyhedron ⓘ |
| dependsOn |
dihedral angles
ⓘ
edge lengths ⓘ |
| domain | polyhedra with finite number of planar faces ⓘ |
| example | regular tetrahedron and cube of equal volume have different Dehn invariants ⓘ |
| field |
geometry
ⓘ
metric geometry ⓘ polyhedral geometry ⓘ |
| generalizationOf | additive angle-length invariants for polygons in the plane ⓘ |
| hasComponent | tensor product of real numbers with reals modulo πℚ ⓘ |
| hasNotation | D(P) ⓘ |
| implies |
if two polyhedra are scissors-congruent then they have equal Dehn invariant
ⓘ
if two polyhedra have different Dehn invariants then they are not scissors-congruent ⓘ |
| influenced | later work on scissors congruence in higher dimensions ⓘ |
| introducedBy | Max Dehn ⓘ |
| introducedIn | 1900s ⓘ |
| invariantUnder |
isometries of Euclidean 3-space
ⓘ
rigid motions of ℝ³ ⓘ |
| mathematicalExpression | D(P)=∑_e ℓ(e) ⊗ (θ(e) mod πℚ) ⓘ |
| namedAfter | Max Dehn ⓘ |
| property |
additive under dissection of polyhedra
ⓘ
invariant under scissors congruence ⓘ |
| relatedConcept |
equidecomposability of polyhedra
ⓘ
scissors congruence ⓘ volume of polyhedra ⓘ |
| relatedTo |
Bloch group
ⓘ
linked to:
Milnor K-theory
Hadwiger invariants ⓘ Hilbert's problems ⓘ
linked to:
Hilbert problems
Hilbert's third problem ⓘ K-theory of fields ⓘ scissors congruence groups ⓘ |
| solves | Hilbert's third problem ⓘ |
| takesValuesIn | ℝ ⊗ℚ (ℝ/πℚ) ⓘ |
| usedFor |
deciding scissors congruence of polyhedra
ⓘ
distinguishing non-scissors-congruent polyhedra of equal volume ⓘ |
| usedIn | proof that a cube and a regular tetrahedron of equal volume are not scissors-congruent ⓘ |
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: Dehn invariant Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.