Regular polyhedron
polyhedron with regular congruent polygons as faces
A regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra.
The regular polyhedra
There are five convex regular polyhedra, known as the Platonic solids; four regular star polyhedra, the Kepler–Poinsot polyhedra; and five regular compounds of regular polyhedra:
Platonic solids
Kepler–Poinsot polyhedra
Regular compounds
Characteristics
Equivalent properties
The property of having a similar arrangement of faces around each vertex can be replaced by any of the following equivalent conditions in the definition:
The vertices of a convex regular polyhedron all lie on a sphere.
All the dihedral angles of the polyhedron are equal
All the vertex figures of the polyhedron are regular polygons.
All the solid angles of the polyhedron are congruent.
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This entry incorporates text from “Regular polyhedron” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.