Euclidean group
E121354
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
All labels observed (9)
How this entity was disambiguated
This entity first appeared as the object of triple T1056977 — 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: Euclidean group Context triple: [Euclidean space, hasSymmetryGroup, Euclidean group]
-
A.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
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B.
Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
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C.
Euclidean space
Euclidean space is the standard flat, n-dimensional geometric setting of classical geometry and vector calculus, characterized by straight lines, right angles, and the usual distance and dot product.
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D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
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E.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
- 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: Euclidean group Target entity description: The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
A.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
-
B.
Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
-
C.
Euclidean space
Euclidean space is the standard flat, n-dimensional geometric setting of classical geometry and vector calculus, characterized by straight lines, right angles, and the usual distance and dot product.
-
D.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
-
E.
Weyl group
A Weyl group is a finite reflection group associated with a root system that encodes the symmetries of Lie algebras and Lie groups in representation theory and geometry.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
isometry group ⓘ mathematical group ⓘ topological group ⓘ |
| actsOn | Euclidean space ⓘ |
| alsoKnownAs |
group of Euclidean isometries
ⓘ
group of rigid motions ⓘ |
| contains | all rigid motions of Euclidean space ⓘ |
| dimensionAsLieGroup | n(n+1)/2 ⓘ |
| field | mathematics ⓘ |
| groupOperation | composition of transformations ⓘ |
| hasComponent |
reflections
ⓘ
rotations ⓘ translations ⓘ |
| hasConnectedComponentOfIdentity |
Euclidean group
self-linksurface differs
ⓘ
surface form:
orientation-preserving Euclidean group
|
| hasGeneralElementForm | x ↦ Rx + t with R in O(n) and t in R^n ⓘ |
| hasNotation |
E(n)
ⓘ
ISO(n) ⓘ Euclidean group self-linksurface differs ⓘ
surface form:
Isom(R^n)
|
| hasOrientationPreservingSubgroupNotation |
E^+(n)
ⓘ
SE(n) ⓘ |
| hasProperty | acts transitively on Euclidean space ⓘ |
| hasSubgroup |
Euclidean group
self-linksurface differs
ⓘ
surface form:
orientation-preserving Euclidean group
orthogonal group O(n) ⓘ orthogonal group O(n) ⓘ
surface form:
special orthogonal group SO(n)
translation group of R^n ⓘ |
| identityElement | identity isometry ⓘ |
| inverseElement | inverse isometry ⓘ |
| isConnected | false ⓘ |
| isHomogeneousSpaceFor | Euclidean space as E(n)/O(n) ⓘ |
| isNoncompact | true ⓘ |
| isometryType | distance-preserving transformations ⓘ |
| isSemidirectProductOf |
orthogonal group O(n)
ⓘ
translation group of R^n ⓘ |
| parameterizedBy | dimension n of Euclidean space ⓘ |
| preserves |
Euclidean distance
ⓘ
angles ⓘ inner product up to orthogonality ⓘ orientation (for orientation-preserving subgroup) ⓘ |
| relatedTo |
Galilean group
ⓘ
Poincaré group ⓘ |
| subfield |
Lie theory
ⓘ
geometry ⓘ group theory ⓘ |
| usedIn |
classical mechanics
ⓘ
computer graphics ⓘ computer vision ⓘ crystallography ⓘ rigid body kinematics ⓘ robotics ⓘ |
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: Euclidean group Description of subject: The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Isom(R^n)
this entity surface form:
orientation-preserving Euclidean group
this entity surface form:
orientation-preserving Euclidean group
this entity surface form:
Euclidean group of dimension n
this entity surface form:
orientation-preserving Euclidean group E^+(n)
this entity surface form:
Euclidean group E(2)
this entity surface form:
Euclidean group E(3)
this entity surface form:
Euclidean group in three dimensions
this entity surface form:
Euclidean isometries