# Noble polytope

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A noble polytope is a polytope that is both isogonal and isotopic, i.e. its vertices are identical under its symmetry group, and so are its facets. The dual of a noble polytope is another noble polytope. A self-dual isogonal or isotopic polytope is also a noble polytope.

All regular polytopes are noble.

## In 2 dimensions

For polygons, being noble is equivalent to being regular.

## In 3 dimensions

The only convex nonregular noble polyhedra are tetragonal disphenoids and rhombic disphenoids, which are lower-symmetry variants of the regular tetrahedron. Crown polyhedra (stephanoids) 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.

In addition to the regulars, disphenoids, and crown polyhedra, there are a further 137 non-exotic noble polyhedra, 2 of which are fissary. The list has been proven to be complete.

The crown polyhedra and disphenoids have more than one degree of freedom; i.e. their edge length ratios, aside from scaling, can vary continuously. It has been proven that no other noble polyhedra have this property.[citation needed]

## In 4 dimensions

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, tetracontoctachoron, and n-n duoprisms. There also exist noble scaliform polychora, such as the bi-icositetradiminished hexacosichoron.

## In higher dimensions

The Birkhoff polytopes yield an (n - 1)2-dimensional nonuniform convex noble polytope for every ${\displaystyle n\geq 3}$.

Both the prism product and direct sum of a noble n-polytope with itself will be a noble 2n-polytope.