Noble polyhedron

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.

• 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

• 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.