A noble polyhedron is one which is isohedral (all faces the same) and isogonal

Isogonal

Isogonal is a mathematical term which means "having similar angles". It occurs in several contexts:*Isogonal polygon, polyhedron, polytope or tiling.*Isogonal trajectory in curve theory.*Isogonal conjugate in triangle geometry....

 (all vertices the same). They were first studied in any depth by Hess and Bruckner around the turn of the century (ca. 1900), and later by Grünbaum

Branko Grünbaum

Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....

.

Classes of noble polyhedra

There are four main classes of noble polyhedra:

• Regular polyhedra
  Regular polyhedron
  A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...
  
  are also noble.
• Disphenoid tetrahedra. These and the platonic solids
  Platonic solid
  In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
  
   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

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

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

• The 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.