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.

Classes of noble polyhedra

There are four main classes of noble polyhedra:

• The nine regular polyhedra are also noble.
• Disphenoid tetrahedra. These and the Platonic solids are the only convex noble polyhedra.
• Crown polyhedra or Stephanoids. An infinite series of toroids.
• A variety of miscellaneous examples. It is not known whether there are finitely many, and if so how many might remain to be discovered.

If we allow some of Grünbaum's stranger constructions as polyhedra, then we have two more infinite series of toroids:

• Wreath polyhedra. These have triangular faces in coplanar pairs which share an edge.
• V-faced polyhedra. These have vertices in coincident pairs, and degenerate faces.

Duality of noble polyhedra

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:

• The nine regular polyhedra form dual pairs, with the tetrahedron being self-dual.
• The disphenoid tetrahedra are all topologically identical. Geometrically they come in dual pairs – one elongated, and one correspondingly squashed.
• A crown polyhedron is topologically self-dual. It does not seem to be known whether any geometrically self-dual examples exist.
• The wreath and V-faced polyhedra are dual to each other.

References

• Grünbaum, B.; Polyhedra with hollow faces, Proc. NATO-ASI Conf. on polytopes: abstract, convex and computational, Toronto 1983, Ed. Bisztriczky, T. Et Al., Kluwer Academic (1994), pp. 43–70.
• Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and Computational Geometry: The Goodman-Pollack Festschrift. B. Aronov, S. Basu, J. Pach, and Sharir, M., eds. Springer, New York 2003, pp. 461–488.