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

For polygons, being noble is equivalent to being regular.

In 3 dimensions

Main article: List of noble polyhedra

The 75 known non-exotic noble polyhedra that are not regular, disphenoids, or crowns. The purple polyhedra are self-dual, the magenta polyhedron is dual to its enantiomorph and the green ones are fissary

Completely enumerating the set of noble polyhedra is an unsolved problem and an active research topic, with the most recent discoveries in March 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 85 known non-exotic noble polyhedra, 2 of which are fissary.

In 4 dimensions

Main article: List of noble polychora

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 $n \geq 3$.