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[edit | edit source]

For polygons, being noble is equivalent to being regular.

In 3 dimensions[edit | edit source]

Small noble triangular hexecontahedronGreat noble triangular hexecontahedronFirst noble triangular hecatonicosahedronSecond noble triangular hecatonicosahedronThird noble triangular hecatonicosahedronThird noble faceting of icosidodecahedronSecond noble faceting of icosidodecahedronFirst noble faceting of icosidodecahedronNoble tetragonal tetracontoctahedronSecond noble kipiscoidal hecatonicosahedronThird noble kipiscoidal hexecontahedronSeventh noble kipiscoidal hexecontahedronNoble enneagrammic icosahedronFinal stellation of the icosahedronFinal stellation of the rhombic triacontahedronFifth noble stellation of rhombic triacontahedronSixth noble stellation of rhombic triacontahedronSecond noble octagrammic triacontahedronFirst noble octagrammic triacontahedronNoble octagonal triacontahedronNoble octagrammic icositetrahedronFirst noble kipiscoidal hecatonicosahedronFourth noble kipiscoidal hexecontahedronEighth noble kipiscoidal hexecontahedronSeventh noble kipentagrammic hexecontahedronFifth noble kipentagrammic hexecontahedronNoble pentagrammic hexecontahedronEighth noble kipentagrammic hecatonicosahedronFifth noble kipentagrammic hecatonicosahedronSixth noble kipentagrammic hecatonicosahedronFourth noble kipentagrammic hecatonicosahedronNoble pentagrammic tetracontoctahedronNinth noble kipentagrammic hexecontahedronSecond noble kipentagrammic hecatonicosahedronNinth noble kipentagrammic hecatonicosahedronTenth noble kipentagrammic hecatonicosahedronSixth noble kipentagrammic hexecontahedronEighth noble kipentagrammic hexecontahedronNoble pentagonal hexecontahedronSecond noble pentagonal hecatonicosahedronThird noble pentagonal hecatonicosahedronFourth noble pentagonal hecatonicosahedronFirst noble pentagonal hecatonicosahedronNoble pentagonal tetracontoctahedronFifteenth noble kisombreroidal hexecontahedronFirst noble kisombreroidal hecatonicosahedronSeventh noble kisombreroidal hecatonicosahedronFile:63fissary (tioid 8).pngEleventh noble kipentagrammic hecatonicosahedronFourth noble kipentagrammic hexecontahedronFirst noble kipentagrammic hexecontahedronSecond noble kipentagrammic hexecontahedronFirst noble kipentagrammic hecatonicosahedronSeventh noble kipentagrammic hecatonicosahedronThird noble kipentagrammic hecatonicosahedronNoble pentagrammic icositetrahedronThird noble kipentagrammic hexecontahedronThirteenth noble kisombreroidal hexecontahedronSecond noble sombreroidal hexecontahedronFirst noble sombreroidal hexecontahedronFile:65 fissary (sridoid 8 army).pngSixth noble kipiscoidal hexecontahedronFirst noble kipiscoidal hexecontahedronSecond noble kipiscoidal hexecontahedronThird noble kipiscoidal hecatonicosahedronSeventh noble kipiscoidal hecatonicosahedronEighth noble kipiscoidal hecatonicosahedronNoble piscoidal icositetrahedronFifth noble kipiscoidal hexecontahedronTenth noble kipiscoidal hexecontahedronSecond noble piscoidal hexecontahedronFirst noble piscoidal hexecontahedronFourth noble kisombreroidal hecatonicosahedronThird noble kisombreroidal hecatonicosahedronNinth noble kisombreroidal hexecontahedronSixth noble kisombreroidal hecatonicosahedronNinth noble kisombreroidal hecatonicosahedronEighth noble kisombreroidal hecatonicosahedronSecond noble kisombreroidal icositetrahedronThird noble kisombreroidal icositetrahedronSeventeenth noble kisombreroidal hexecontahedronFifth noble kisombreroidal hecatonicosahedronFifth noble kisombreroidal hexecontahedronFourth noble kisombreroidal hexecontahedronNinth noble kipiscoidal hecatonicosahedronTenth noble kipiscoidal hecatonicosahedronNinth noble kipiscoidal hexecontahedronFourth noble kipiscoidal hecatonicosahedronFifth noble kipiscoidal hecatonicosahedronSixth noble kipiscoidal hecatonicosahedronSecond noble kipiscoidal icositetrahedronFourth noble kisombreroidal icositetrahedronSixteenth noble kisombreroidal hexecontahedronSecond noble kisombreroidal hecatonicosahedronSixth noble kisombreroidal hexecontahedronThird noble kisombreroidal hexecontahedronSecond noble kisombreroidal hexecontahedronEighteenth noble kisombreroidal hexecontahedronSeventh noble kisombreroidal hexecontahedronTenth noble kisombreroidal hexecontahedronEleventh noble kisombreroidal hexecontahedronNoble kipentagrammic icositetrahedronFirst noble kipiscoidal icositetrahedronFirst noble kisombreroidal icositetrahedronFourth noble unihexagrammic hexecontahedronFifth noble unihexagrammic hexecontahedronSixth noble unihexagrammic hexecontahedronNoble kiunihexagrammic hexecontahedronFirst noble kisombreroidal hexecontahedronNineteenth noble kisombreroidal hexecontahedronEighth noble kisombreroidal hexecontahedronFourteenth noble kisombreroidal hexecontahedronTwelfth noble kisombreroidal hexecontahedronFirst noble pterogrammic hexecontahedronSecond noble pterogrammic hexecontahedronThird noble pterogrammic hexecontahedronFirst noble kizippergrammic hexecontahedronDitrigonal icosahedronNoble propeller tripodal hexecontahedronSecond noble propellogrammic hexecontahedronFirst noble propellogrammic hexecontahedronSecond noble crossed kignathogrammic hexecontahedronSecond noble unihexagrammic hexecontahedronThird noble unihexagrammic hexecontahedronSeventh noble unihexagrammic hexecontahedronFourth noble pterogrammic hexecontahedronFirst noble crossed kignathogrammic hexecontahedronSecond noble kizippergrammic hexecontahedronSecond noble kipravogrammic hexecontahedronFirst noble kipravogrammic hexecontahedronNoble kicapellogrammic hecatonicosahedronNoble kicapellogrammic hexecontahedronFirst noble unihexagrammic hexecontahedronFirst noble ditrapezoidal hexecontahedronThird noble ditrapezoidal hexecontahedronSecond noble ditrapezoidal hexecontahedronNoble kignathogrammic hexecontahedron
137 non-exotic noble polyhedra that are not regular, disphenoids, or crowns. The purple polyhedra are self-dual, the magenta polyhedra are dual to their enantiomorphs.

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 known non-exotic noble polyhedra, 2 of which are fissary.

The crown polyhedra and disphenoids have degrees of freedom; i.e. their edge length ratios can vary continuously. It has been proven that no other noble polyhedra have this property.[citation needed]

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

In higher dimensions[edit | edit source]

The Birkhoff polytopes yield an (n - 1)2-dimensional nonuniform convex noble polytope for every .