Noble polyhedron

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 Hess and Bruckner in late 19th century, and later by Grünbaum.

There are four 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:

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. Where the distinction is not made below, the term 'dual' covers both kinds.

The dual of a noble polyhedron is also noble. Many are also self-dual:

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