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Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, 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 Catalan solid .
This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide. Not included are: The uniform polyhedron compounds. 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter); The uniform tilings (infinite polyhedra)
The duals of the uniform polyhedra have irregular faces but are face-transitive, and every vertex figure is a regular polygon. A uniform polyhedron has the same symmetry orbits as its dual, with the faces and vertices simply swapped over. The duals of the convex Archimedean polyhedra are sometimes called the Catalan solids.
There are many relations among the uniform polyhedra. [1] [2] [3] Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron.Others share the same vertices and edges as other polyhedron.
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric ...
The illustration here shows the vertex figure (red) of the cuboctahedron being used to derive the corresponding face (blue) of the rhombic dodecahedron.. For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction. [2]
A 4-polytope is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra. The faces of a uniform 4-polytope must be regular. A 4-polytope is scaliform if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex ...
This category was created to reference the full set of 75 nonprismatic uniform polyhedra, as well as prismatic forms. It is a subset of Category:Polyhedra.. It is a union of 5 Platonic solids, 4 Kepler–Poinsot solids, 13 Archimedean solids, and the infinite prismatic sets in Prismatoid polyhedra, and adds 53 non-convex, non-regular uniform polyhedra.