A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same).
They were first studied in any depth by Edmund Hess and Max Brückner in the late 19th century, and later by Branko Grünbaum.
There are several main classes of noble polyhedra: If we allow some of Grünbaum's stranger constructions as polyhedra, then we have two more infinite series of toroids (besides the crown polyhedra mentioned above): We can distinguish between dual structural forms (topologies) on the one hand, and dual geometrical arrangements when reciprocated about a concentric sphere, on the other.
Many are also self-dual: In 2008, Robert Webb discovered a new noble polyhedron, a faceting of the snub cube.
[2] This was the first new class of noble polyhedra (with chiral octahedral symmetry) to be discovered since Brückner's work over a century before.