Platonic solids
E36442
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.
Aliases (5)
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
class of polyhedra
→
geometric solids → mathematical concept → |
| areAll |
convex regular polyhedra
→
edge-transitive polyhedra → examples of regular maps on the sphere → face-transitive polyhedra → finite polyhedra → isogonal polyhedra → isohedral polyhedra → isotoxal polyhedra → vertex-transitive polyhedra → |
| areContrastedWith |
Archimedean solids
→
Kepler–Poinsot polyhedra → |
| areSubsetOf |
convex polyhedra
→
regular polyhedra → |
| associatedWith |
Plato
→
|
| classificationCriterion |
regularity of faces and vertices
→
|
| describedIn |
Book XIII of Euclid's Elements
→
Euclid's Elements → |
| dualPair |
cube–octahedron
→
dodecahedron–icosahedron → tetrahedron–tetrahedron → |
| edgeProperty |
same number of faces meet at each edge
→
|
| existIn |
three-dimensional Euclidean space
→
|
| faceType |
congruent regular polygons
→
|
| hasMember |
cube
→
dodecahedron → icosahedron → octahedron → tetrahedron → |
| hasProperty |
convex
→
highly symmetrical → regular polyhedra → |
| haveDualityProperty |
each has a dual Platonic solid
→
|
| haveHistoricalOrigin |
ancient Greek mathematics
→
|
| numberOfElements |
5
→
|
| philosophicalRole |
linked to classical elements in Platonism
→
|
| studiedBy |
Euclid
→
|
| symmetryGroupType |
finite rotation groups
→
|
| topology |
homeomorphic to the sphere
→
|
| uniquenessProperty |
only five convex regular polyhedra exist in 3D Euclidean space
→
|
| usedIn |
architecture
→
art → chemistry → classical philosophy → crystallography → geometry → group theory → |
| vertexProperty |
same number of faces meet at each vertex
→
|
Referenced by (9)
| Subject (surface form when different) | Predicate |
|---|---|
|
Platonic solids
("cube–octahedron")
→
Platonic solids ("dodecahedron–icosahedron") → |
dualPair |
|
Harmonices Mundi
→
|
explores |
|
Platonic solids
("icosahedron")
→
|
hasMember |
|
Plato
→
|
hasPhilosophicalConcept |
|
Euler’s polyhedron formula
→
|
holdsFor |
|
Felix Klein
("Lectures on the Icosahedron")
→
|
notableWork |
|
Mysterium Cosmographicum
→
|
usesConcept |
|
Piero della Francesca
("De quinque corporibus regularibus")
→
|
wrote |