Smale’s paradox
E398343
Smale’s paradox is a result in differential topology showing that a sphere can be turned inside out in three-dimensional space through smooth deformations without tearing or creasing, challenging intuitive notions of geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Smale’s paradox canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3910504 — 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: Smale’s paradox Context triple: [Stephen Smale, notableWork, Smale’s paradox]
-
A.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
B.
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.
-
C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
D.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
-
E.
Dehn surgery
Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
- 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: Smale’s paradox Target entity description: Smale’s paradox is a result in differential topology showing that a sphere can be turned inside out in three-dimensional space through smooth deformations without tearing or creasing, challenging intuitive notions of geometry.
-
A.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
B.
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.
-
C.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
D.
Theorema Egregium
Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
-
E.
Dehn surgery
Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in differential topology ⓘ |
| allows | temporary self-intersections of the sphere ⓘ |
| ambientSpace | three-dimensional Euclidean space ⓘ |
| category | smooth category ⓘ |
| clarifies | difference between topological and geometric intuition ⓘ |
| contrastsWith | intuitive rigidity of the sphere in three dimensions ⓘ |
| dimensionOfAmbientSpace | 3 ⓘ |
| dimensionOfSurface | 2 ⓘ |
| field |
differential topology
ⓘ
geometric topology ⓘ |
| forbids |
creating sharp creases
ⓘ
pinching the surface to a point and cutting ⓘ tearing of the surface ⓘ |
| hasConsequence | classification of immersions of S^2 in R^3 up to regular homotopy ⓘ |
| hasVisualization |
computer-generated sphere eversions
ⓘ
sphere eversion movies ⓘ |
| implies | any two immersions of S^2 in R^3 are regularly homotopic ⓘ |
| inspired | research on explicit sphere eversions ⓘ |
| involvesConcept |
homotopy of immersions
ⓘ
immersed surfaces ⓘ regular homotopy ⓘ self-intersection ⓘ smooth deformation ⓘ sphere eversion ⓘ |
| involvesObject | 2-sphere ⓘ |
| isAbout | possibility of turning a sphere inside out smoothly ⓘ |
| isCounterintuitive | true ⓘ |
| mainClaim |
a 2-sphere can be turned inside out in three-dimensional space by a smooth regular homotopy
ⓘ
a sphere eversion is possible without tearing or creating creases ⓘ |
| namedAfter | Stephen Smale ⓘ |
| nonIntuitiveAspect | inside and outside of a sphere are not topologically distinguished by smooth immersions in R^3 ⓘ |
| provedBy | Stephen Smale ⓘ |
| relatedConcept | regular homotopy classes of immersions of S^2 in R^3 ⓘ |
| relatedResult | Smale–Hirsch immersion theorem ⓘ |
| relatedTo | eversion of higher-dimensional spheres ⓘ |
| requires |
smooth structure on Euclidean 3-space
ⓘ
smooth structure on the sphere ⓘ |
| status | mathematically proven ⓘ |
| usesMethod |
differential-topological techniques
ⓘ
homotopy-theoretic arguments ⓘ |
| yearProved | 1958 ⓘ |
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: Smale’s paradox Description of subject: Smale’s paradox is a result in differential topology showing that a sphere can be turned inside out in three-dimensional space through smooth deformations without tearing or creasing, challenging intuitive notions of geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.