E8 lattice
E656669
even unimodular lattice
integral lattice
lattice
positive-definite lattice
root lattice
sphere packing
The E8 lattice is an eight-dimensional, highly symmetric even unimodular lattice that plays a central role in Lie theory, sphere packing, and string theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| E8 lattice canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338359 — 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: E8 lattice Context triple: [Leech lattice, relatedTo, E8 lattice]
-
A.
Leech lattice
The Leech lattice is a highly symmetric 24-dimensional lattice in Euclidean space, notable for its dense sphere packing and deep connections to sporadic simple groups and modular forms.
-
B.
Clebsch
Clebsch is a German surname most notably associated with mathematician Alfred Clebsch, known for his contributions to algebraic geometry and invariant theory.
-
C.
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.
-
D.
Regular Polytopes
"Regular Polytopes" is a classic mathematical monograph by H. S. M. Coxeter that systematically develops the theory and classification of highly symmetric polytopes in various dimensions.
-
E.
Conway–Norton collaboration
The Conway–Norton collaboration was a joint mathematical effort, led by John Conway and Simon Norton, that played a key role in developing the theory of monstrous moonshine and the construction of the Monster group.
- 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: E8 lattice Target entity description: The E8 lattice is an eight-dimensional, highly symmetric even unimodular lattice that plays a central role in Lie theory, sphere packing, and string theory.
-
A.
Leech lattice
The Leech lattice is a highly symmetric 24-dimensional lattice in Euclidean space, notable for its dense sphere packing and deep connections to sporadic simple groups and modular forms.
-
B.
Clebsch
Clebsch is a German surname most notably associated with mathematician Alfred Clebsch, known for his contributions to algebraic geometry and invariant theory.
-
C.
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.
-
D.
Regular Polytopes
"Regular Polytopes" is a classic mathematical monograph by H. S. M. Coxeter that systematically develops the theory and classification of highly symmetric polytopes in various dimensions.
-
E.
Conway–Norton collaboration
The Conway–Norton collaboration was a joint mathematical effort, led by John Conway and Simon Norton, that played a key role in developing the theory of monstrous moonshine and the construction of the Monster group.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
even unimodular lattice
ⓘ
integral lattice ⓘ lattice ⓘ positive-definite lattice ⓘ root lattice ⓘ sphere packing ⓘ |
| associatedRootSystem | E8 root system ⓘ |
| automorphismGroup | Weyl group of type E8 NERFINISHED ⓘ |
| determinant | 1 ⓘ |
| dimension | 8 ⓘ |
| directSumWithE8AndD16Forms | Niemeier lattices NERFINISHED ⓘ |
| directSumWithItselfForms | E8⊕E8 lattice ⓘ |
| discriminant | 1 ⓘ |
| givesOptimalSpherePackingInDimension | 8 ⓘ |
| hasCoxeterNumber | 30 ⓘ |
| hasDynkinType | E8 ⓘ |
| hasNoRootsOfNorm | 1 ⓘ |
| hasRootsOfNorm | 2 ⓘ |
| hasRootSystemSize | 240 ⓘ |
| hasThetaSeries | weight4 modular form ⓘ |
| isBuildingBlockOf | Leech lattice NERFINISHED ⓘ |
| isEven | true ⓘ |
| isEvenUnimodular | true ⓘ |
| isEvenUnimodularPositiveDefiniteOfRankLessThan | 24 ⓘ |
| isGeneratedBy | E8 root system NERFINISHED ⓘ |
| isHighlySymmetric | true ⓘ |
| isIntegral | true ⓘ |
| isPositiveDefinite | true ⓘ |
| isRootLatticeOf | E8 Lie algebra NERFINISHED ⓘ |
| isSelfDual | true ⓘ |
| isUnimodular | true ⓘ |
| isUniqueEvenUnimodularLatticeOfRank | 8 ⓘ |
| kissingNumber | 240 ⓘ |
| minimalNorm | 2 ⓘ |
| numberOfMinimalVectors | 240 ⓘ |
| rank | 8 ⓘ |
| relatedTo |
Lie algebra of type E8
ⓘ
exceptional Lie group E8 NERFINISHED ⓘ |
| spherePackingDensity | highestKnownInDimension8 ⓘ |
| thetaSeriesIsModularFormFor | SL2(Z) ⓘ |
| usedIn |
E8×E8 heterotic string
NERFINISHED
ⓘ
Lie theory ⓘ conformal field theory ⓘ heterotic string theory NERFINISHED ⓘ lattice vertex operator algebras ⓘ sphere packing theory ⓘ string theory ⓘ |
| WeylGroup | Weyl group of type E8 NERFINISHED ⓘ |
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: E8 lattice Description of subject: The E8 lattice is an eight-dimensional, highly symmetric even unimodular lattice that plays a central role in Lie theory, sphere packing, and string theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.