Kepler–Poinsot polyhedra

E190163
family of polyhedra non-convex regular polyhedra regular star polyhedra

The Kepler–Poinsot polyhedra are the four regular star polyhedra that extend the concept of Platonic solids into non-convex, self-intersecting forms.

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All labels observed (5)

Label Occurrences
Kepler–Poinsot polyhedra canonical 3
great dodecahedron 1
great icosahedron 1

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 ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Platonic solids → areContrastedWith → Kepler–Poinsot polyhedra ⓘ
Archimedean solids → relatedTo → Kepler–Poinsot polyhedra ⓘ
Kepler–Poinsot polyhedra → consistsOf → Kepler–Poinsot polyhedra self-linksurface differs ⓘ
this entity surface form: great stellated dodecahedron
Kepler–Poinsot polyhedra → consistsOf → Kepler–Poinsot polyhedra self-linksurface differs ⓘ
this entity surface form: great dodecahedron
Kepler–Poinsot polyhedra → consistsOf → Kepler–Poinsot polyhedra self-linksurface differs ⓘ
this entity surface form: great icosahedron
Kepler–Poinsot polyhedra → areDualInPairsWith → Kepler–Poinsot polyhedra self-linksurface differs ⓘ
this entity surface form: great stellated dodecahedron and great icosahedron
The Fifty-Nine Icosahedra → relatedTo → Kepler–Poinsot polyhedra ⓘ