Noble polytope

A noble polytope is a polytope that is both isogonal and isotopic, i.e. its vertices are identical under its symmetry group, and so are its facets. The dual of a noble polytope is another noble polytope. A self-dual isogonal or isotopic polytope is also a noble polytope.

All regular polytopes are noble.

In 2 dimensions
All noble polygons are regular, and in particular uniform.

In 3 dimensions
Completely enumerating the set of noble polyhedra is an unsolved problem and an active research topic, with the most recent discoveries in January 2023. Many noble polyhedra have degrees of freedom, i.e. their edge lengths can vary continuously. There are no known non-prismatic nobles with a degree of freedom.

The only convex nonregular noble polyhedra are tetragonal disphenoids and rhombic disphenoids, which are lower-symmetry variants of the regular 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.

In addition to the regulars, disphenoids, and crown polyhedra, there are a further 69 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 (n - 1)2-dimensional nonuniform convex noble polytope for every $$n \geq 3$$.