Noble polytope

A noble polytope is a polytope whose vertices and facets are identical under its symmetry group, and is 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 regular, and in particular uniform.

In 3 dimensions
In addition to the regular polyhedra, there are other noble polyhedra. The only convex nonregular noble polyhedra are tetragonal disphenoids and rhombic disphenoids, which are lower-symmetry variants of the tetrahedron. 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.

In addition to the regulars, disphenoids, and crown polyhedra, there are a further 66 known non-exotic noble polyhedra, 2 of which are fissary.

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, tetracontoctachoron, and n-n duoprisms. There also exist noble scaliform polychora, such as the bi-icositetradiminished hexacosichoron.

In higher dimensions
The Birkhoff polytopes yield an n2-dimensional nonuniform convex noble polytope for every $$n \geq 1$$.