Noble polytope

A noble polytope is a polytope whose vertices and facets are identical under its highest 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 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 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 60 known noble polyhedra:

1 dual pair with 24 vertices, 60 edges, and 24 pentagonal faces,

2 chiral self-duals with 24 vertices, 60 edges, and 24 pentagonal faces,

2 stellations of the icosahedron with 60 vertices, 90 edges, and 20 enneagrammic faces, and their duals, which are:

2 facetings of the dodecahedron with 20 vertices, 90 edges, and 60 triangular faces.

1 self-dual with 20 vertices, 60 edges, and 20 faces: the ditrigonal icosahedron, being both an icosahedron stellation and a dodecahedron faceting,

3 stellations of the rhombic triacontahedron with 60 vertices, 120 edges, and 30 octagrammic faces, and their duals, which are:

3 facetings of the icosidodecahedron with 30 vertices, 120 edges, and 60 crossed quadrilateral faces.

3 stellations of the rhombic triacontahedron with 120 vertices, 180 edges, and 30 dodecagrammic faces, and their duals, which are:

3 facetings of the icosidodecahedron with 30 vertices, 180 edges, and 120 triangular faces.

3 non-chiral dual pairs with 60 vertices, 150 edges, 60 pentagonal faces,

5 chiral dual pairs with 60 vertices, 150 edges, 60 pentagonal faces,

7 non-chiral dual pairs with 60 vertices, 180 edges, 60 hexagonal faces,

3 non-chiral self-duals with 60 vertices, 180 edges, 60 hexagonal faces,

2 chiral dual pairs with with 60 vertices, 180 edges, 60 hexagonal faces,

2 chiral self-duals with 60 vertices, 180 edges, 60 hexagonal faces.

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