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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 ...
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:
Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra: five convex and four star polyhedra. The five convex examples have been known since antiquity and are called the Platonic solids. These are the triangular pyramid or tetrahedron, cube, octahedron, dodecahedron and icosahedron:
A regular polyhedron with Schläfli symbol {p, q}, Coxeter diagrams , has a regular face type {p}, and regular vertex figure {q}. A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.
Regular polyhedron. Platonic solid: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron; Regular spherical polyhedron. Dihedron, Hosohedron; Kepler–Poinsot polyhedron (Regular star polyhedra) Small stellated dodecahedron, Great stellated dodecahedron, Great icosahedron, Great dodecahedron; Abstract regular polyhedra (Projective polyhedron)
Regular polytope examples A regular pentagon is a polygon, a two-dimensional polytope with 5 edges, represented by Schläfli symbol {5}.: A regular dodecahedron is a polyhedron, a three-dimensional polytope, with 12 pentagonal faces, represented by Schläfli symbol {5,3}.