Noble polytope

A noble polytope is a polytope whose vertices and facets are identical under its highest symmetry group, and are therefore isogonal and isotopic. The dual of a noble polytope is another noble polytope.

A self-dual isogonal or isotopic polytope is also a noble polytope.

In 2 dimensions
All noble polygons are uniform and regular.

In 3 dimensions
In addition to the regular polyhedra, there are other noble polyhedra. The only convex nonregular noble polyhedra are tetragonal and rhombic disphenoids. Crown polyhedra are an infinite family of toroidal noble polyhedra with dihedral symmetry. The ditrigonal icosahedron is semi-uniform and, when treated as an abstract polytope, regular. The noble polyhedra have not yet been completely enumerated.

TO DO: enumerate the currently known noble polyhedra

In 4 dimensions
In 2 and 3 dimensions, all noble uniform polytopes are regular. In 4 dimensions, there exist non-regular noble uniform polychora, such as the decachoron and n-nduoprisms. There also exist noble scaliform polychora, such as the bi-icositetradiminished hexacosichoron.