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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.
Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both.
This list includes these:
It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered one more, by relaxing the condition that only two faces may meet at an edge. Some authors do not count it as a uniform polyhedron, because some pairs of edges coincide.
Not included are:
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:
The convex forms are listed in order of degree of vertex configurations from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.
(*) : The great disnub dirhombidodecahedron has 120 of its 240 edges shared by four faces. If these 120 edges counted as two pairs of coincident edges, each shared by two faces, then there are a total of 360 edges, and the Euler characteristic becomes -88. Because of this edge-degeneracy, it is not always considered to be a uniform polyhedron.
Icosidodecahedron, Vertex figure, Geometry, Uniform polyhedron, Pentagon
Topology, Calculus, Euclid, Projective geometry, Algebraic geometry
Leonardo da Vinci, Tetrahedron, Dodecahedron, Octahedron, Icosahedron
Euclidean geometry, Pentagon, Hexagram, Edge (geometry), Schläfli symbol
Tesseract, Polygon, Polyhedron, 4-polytope, MathWorld
Authority control, University of Ottawa, Order of Saint Benedict, College of Saint Benedict and Saint John's University, Cambridge University Press
Geometry, Golden ratio, MathWorld, Uniform star polyhedron, Euler characteristic
Geometry, Great stellated truncated dodecahedron, Small ditrigonal dodecicosidodecahedron, Small dodecicosahedron, MathWorld
Geometry, Truncated cuboctahedron, MathWorld, Uniform star polyhedron, Euler characteristic