Kepler–Poinsot polyhedra
E190163
The Kepler–Poinsot polyhedra are the four regular star polyhedra that extend the concept of Platonic solids into non-convex, self-intersecting forms.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Kepler–Poinsot polyhedra canonical | 3 |
| great dodecahedron | 1 |
| great icosahedron | 1 |
| great stellated dodecahedron | 1 |
| great stellated dodecahedron and great icosahedron | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1686051 — 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: Kepler–Poinsot polyhedra Context triple: [Platonic solids, areContrastedWith, Kepler–Poinsot polyhedra]
-
A.
Platonic solids
Platonic solids are the five highly symmetrical, convex polyhedra (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) that have identical regular polygonal faces and are fundamental in geometry and classical philosophy.
-
B.
Archimedean solids
Archimedean solids are a set of thirteen highly symmetric, semi-regular convex polyhedra characterized by identical vertices and faces composed of more than one type of regular polygon.
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C.
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.
-
D.
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.
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E.
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.
- 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: Kepler–Poinsot polyhedra Target entity description: The Kepler–Poinsot polyhedra are the four regular star polyhedra that extend the concept of Platonic solids into non-convex, self-intersecting forms.
-
A.
Platonic solids
Platonic solids are the five highly symmetrical, convex polyhedra (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) that have identical regular polygonal faces and are fundamental in geometry and classical philosophy.
-
B.
Archimedean solids
Archimedean solids are a set of thirteen highly symmetric, semi-regular convex polyhedra characterized by identical vertices and faces composed of more than one type of regular polygon.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
family of polyhedra
ⓘ
non-convex regular polyhedra ⓘ regular star polyhedra ⓘ |
| are |
non-convex
ⓘ
regular ⓘ self-intersecting ⓘ star polyhedra ⓘ |
| areDefinedBy | regularity plus non-convex self-intersecting structure ⓘ |
| areDualInPairsWith |
Kepler–Poinsot polyhedra
self-linksurface differs
ⓘ
surface form:
great stellated dodecahedron and great icosahedron
small stellated dodecahedron and great dodecahedron ⓘ |
| areExamplesOf | regular maps on the sphere with self-intersections ⓘ |
| areModeledIn | mathematical visualization and art ⓘ |
| areRegularIn |
Coxeter’s sense of regularity
ⓘ
Schläfli’s sense of regularity ⓘ |
| areRelatedTo |
dodecahedron
ⓘ
icosahedron ⓘ |
| areRepresentedIn |
Regular Polytopes
ⓘ
surface form:
Coxeter’s "Regular Polytopes"
Schläfli’s theory of polytopes ⓘ |
| areSometimesCalled | regular star polyhedra ⓘ |
| areSometimesClassifiedAs | non-convex uniform polyhedra ⓘ |
| areUsedIn |
the classification of regular polytopes
ⓘ
the study of polyhedral symmetry ⓘ |
| consistsOf |
Kepler–Poinsot polyhedra
self-linksurface differs
ⓘ
surface form:
great dodecahedron
Kepler–Poinsot polyhedra self-linksurface differs ⓘ
surface form:
great icosahedron
Kepler–Poinsot polyhedra self-linksurface differs ⓘ
surface form:
great stellated dodecahedron
small stellated dodecahedron ⓘ |
| extendConceptOf | Platonic solids ⓘ |
| generalize | convex regular polyhedra ⓘ |
| haveCharacteristic |
each edge belongs to the same number of faces
ⓘ
each vertex has congruent surroundings ⓘ |
| haveEulerCharacteristic | non-standard when faces and vertices are counted naively ⓘ |
| haveProperty |
edge-transitive
ⓘ
face-transitive ⓘ faces are regular polygons ⓘ vertex figures are regular ⓘ vertex-transitive ⓘ |
| haveSchlafliSymbol |
{3,5/2} for the great icosahedron
ⓘ
{5,5/2} for the great dodecahedron ⓘ {5/2,3} for the great stellated dodecahedron ⓘ {5/2,5} for the small stellated dodecahedron ⓘ |
| haveSymmetryGroup | icosahedral symmetry ⓘ |
| numberOfElements | 4 ⓘ |
| shareSchlafliSymbolsWith | Platonic solids ⓘ |
| use | star polygons as faces or vertex figures ⓘ |
| wereStudiedBy |
Johannes Kepler
ⓘ
Louis Poinsot ⓘ |
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: Kepler–Poinsot polyhedra Description of subject: The Kepler–Poinsot polyhedra are the four regular star polyhedra that extend the concept of Platonic solids into non-convex, self-intersecting forms.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
great stellated dodecahedron
this entity surface form:
great dodecahedron
this entity surface form:
great icosahedron
this entity surface form:
great stellated dodecahedron and great icosahedron