rotation group SO(3)
E174596
Lie group
compact Lie group
connected Lie group
matrix group
non-abelian group
real Lie group
rotation group
simple Lie group
topological group
The rotation group SO(3) is the group of all rotations in three-dimensional space, represented by 3×3 orthogonal matrices with determinant 1, and plays a central role in classical mechanics, quantum mechanics, and geometry.
All labels observed (3)
| Label | Occurrences |
|---|---|
| rotation group SO(3) canonical | 2 |
| SO(3) | 1 |
| SO(3) Lie group | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1535974 — 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: rotation group SO(3) Context triple: [Lorentz group, hasSubgroup, rotation group SO(3)]
-
A.
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.
-
B.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
C.
Sophus
Sophus was the given name of the Norwegian mathematician Sophus Lie, a pioneer in the theory of continuous transformation groups now known as Lie groups.
-
D.
Galilean group
The Galilean group is the mathematical group of spacetime transformations—comprising translations, rotations, and Galilean boosts—that characterize the symmetries of classical Newtonian mechanics.
-
E.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
- 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: rotation group SO(3) Target entity description: The rotation group SO(3) is the group of all rotations in three-dimensional space, represented by 3×3 orthogonal matrices with determinant 1, and plays a central role in classical mechanics, quantum mechanics, and geometry.
-
A.
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.
-
B.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
C.
Sophus
Sophus was the given name of the Norwegian mathematician Sophus Lie, a pioneer in the theory of continuous transformation groups now known as Lie groups.
-
D.
Galilean group
The Galilean group is the mathematical group of spacetime transformations—comprising translations, rotations, and Galilean boosts—that characterize the symmetries of classical Newtonian mechanics.
-
E.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
- F. None of above. chosen
Statements (57)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
compact Lie group ⓘ connected Lie group ⓘ matrix group ⓘ non-abelian group ⓘ real Lie group ⓘ rotation group ⓘ simple Lie group ⓘ topological group ⓘ |
| actsOn | unit sphere S² ⓘ |
| actsTransitivelyOn | unit sphere S² ⓘ |
| appearsIn |
classical mechanics
ⓘ
differential geometry ⓘ quantum mechanics ⓘ representation theory ⓘ rigid body dynamics ⓘ |
| center | {identity matrix} ⓘ |
| containsElementType | 3×3 real matrices ⓘ |
| definedAs | group of all rotations of three-dimensional Euclidean space ⓘ |
| definedBy | set of 3×3 real orthogonal matrices with determinant 1 ⓘ |
| dimension | 3 ⓘ |
| doubleCoveredBy | SU(2) ⓘ |
| fundamentalGroup | ℤ₂ ⓘ |
| groupOperation | matrix multiplication ⓘ |
| hasCoveringMapFrom | unit quaternions ⓘ |
| hasIrreducibleRepresentationsLabeledBy | non-negative integers l = 0,1,2,… ⓘ |
| homotopyType | real projective 3-space RP³ ⓘ |
| identityElement | 3×3 identity matrix ⓘ |
| isNormalSubgroupOf |
orthogonal group O(n)
ⓘ
surface form:
O(3)
|
| isometryGroupOf | oriented Euclidean 3-space fixing the origin ⓘ |
| isomorphicTo | group of orientation-preserving isometries of S² ⓘ |
| isSubgroupOf | O(3) ⓘ |
| LieAlgebra | so(3) ⓘ |
| LieAlgebraDimension | 3 ⓘ |
| LieAlgebraIsomorphicTo | ℝ³ with cross product ⓘ |
| maximalTorus |
special orthogonal group SO(n)
ⓘ
surface form:
SO(2)
|
| overField | real numbers ⓘ |
| parameterization |
Euler angles
ⓘ
axis-angle representation ⓘ unit quaternions modulo ±1 ⓘ |
| property |
center is trivial
ⓘ
compact ⓘ connected ⓘ every element has determinant 1 ⓘ every element is orthogonal ⓘ non-abelian ⓘ simple as a Lie group ⓘ |
| quotientGroupWith | O(3)/SO(3) ≅ ℤ₂ ⓘ |
| rank | 1 ⓘ |
| relatedTo |
angular momentum operators in quantum mechanics
ⓘ
spherical harmonics ⓘ |
| standsFor | special orthogonal group in dimension 3 ⓘ |
| symbol |
rotation group SO(3)
self-link
ⓘ
surface form:
SO(3)
|
| topologicallyHomeomorphicTo | RP³ ⓘ |
| universalCover |
rotation group SU(2)
ⓘ
surface form:
SU(2)
|
| usedToModel |
orientations of a rigid body in 3D space
ⓘ
rotational symmetries of Euclidean 3-space ⓘ |
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: rotation group SO(3) Description of subject: The rotation group SO(3) is the group of all rotations in three-dimensional space, represented by 3×3 orthogonal matrices with determinant 1, and plays a central role in classical mechanics, quantum mechanics, and geometry.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
SO(3)
this entity surface form:
SO(3)
this entity surface form:
SO(3) Lie group