Uniform polyhedron

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a uniform polyhedron has <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a> as <a href="/facts/Face_(geometry)/T2E6BTU4">faces</a> and is <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a>—there is an <a href="/facts/Isometry/K5wX8ah6">isometry</a> mapping any vertex onto any other. It follows that all vertices are <a href="/facts/Congruence_(geometry)/2tKJ6AhB">congruent</a>. Uniform <a href="/facts/Polyhedra/UC0p9FTF">polyhedra</a> may be <a href="/facts/Regular_polyhedron/HJ6oGyug">regular</a> (if also <a href="/facts/Isohedral_figure/ukLVQlK2">face-</a> and <a href="/facts/Isotoxal_figure/p5j7GhwX">edge-transitive</a>), <a href="/facts/Quasiregular_polyhedron/1ukrFhec">quasi-regular</a> (if also edge-transitive but not face-transitive), or <a href="/facts/Semiregular_polyhedron/fHygpjHu">semi-regular</a> (if neither edge- nor face-transitive). The faces and vertices don't need to be <a href="/facts/Convex_polyhedron/2pRHcRgC">convex</a>, so many of the uniform polyhedra are also <a href="/facts/Star_polyhedron/XyeJWj3d">star polyhedra</a>.
There are two infinite classes of uniform polyhedra, together with 75 other polyhedra. They are 2 infinite classes of <a href="/facts/Prism_(geometry)/baIRRkCy">prisms</a> and <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a>, the convex polyhedrons as in 5 <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a> and 13 <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a>—2 <a href="/facts/Quasiregular_polyhedron/1ukrFhec">quasiregular</a> and 11 <a href="/facts/Semiregular_polyhedron/fHygpjHu">semiregular</a>— the non-convex star polyhedra as in 4 <a href="/facts/Kepler%25E2%2580%2593Poinsot_polyhedra/mzUUgBYS">Kepler–Poinsot polyhedra</a> and 53 <a href="/facts/Uniform_star_polyhedra/rSv9XpHQ">uniform star polyhedra</a>—14 <a href="/facts/Quasiregular_polyhedron/1ukrFhec">quasiregular</a> and 39 semiregular. There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the <a href="/facts/Great_disnub_dirhombidodecahedron/WqkxxTdV">great disnub dirhombidodecahedron</a>, Skilling's figure.
<a href="/facts/Dual_polyhedra/rkIHbmAf">Dual polyhedra</a> to uniform polyhedra are <a href="/facts/Face-transitive/ukLVQlK2">face-transitive</a> (isohedral) and have regular <a href="/facts/Vertex_figure/3u2ySK0C">vertex figures</a>, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a <a href="/facts/Catalan_solid/R8D2cP7D">Catalan solid</a>.
The concept of uniform polyhedron is a special case of the concept of <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytope</a>, which also applies to shapes in higher-dimensional (or lower-dimensional) space.

Uniform polyhedron open-in-new

Uniform polyhedron