A noble polytope is a polytope whose vertices and facets are identical under its highest symmetry group, and is therefore isogonal and isotopic. The dual of a noble polytope is another noble polytope.
A self-dual isogonal or isotopic polytope is also a noble polytope.
In 2 dimensions[edit | edit source]
In 3 dimensions[edit | edit source]
In addition to the regular polyhedra, there are other noble polyhedra. The only convex nonregular noble polyhedra are tetragonal disphenoids and rhombic disphenoids, which are lower-symmetry variants of the tetrahedron. Crown polyhedra are an infinite family of toroidal noble polyhedra with dihedral symmetry. The ditrigonal icosahedron is semi-uniform and, when treated as an abstract polytope, regular. The noble polyhedra have not yet been completely enumerated.
In 4 dimensions[edit | edit source]
In 2 and 3 dimensions, all noble uniform polytopes are regular. In 4 dimensions, there exist non-regular noble uniform polychora, such as the decachoron and n-n duoprisms. There also exist noble scaliform polychora, such as the bi-icositetradiminished hexacosichoron.
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