### Noble Polytopes

Noble polytopes are defined to have both, vertices and facets are all alike each (also known as isogonal and isohedral respectively – although the stem -hedr- here seems to be a missnomer with respect to higher dimensions, thence nowadays authors rather would like to use isotopal for the latter instead). It is clear therefrom, that the dual of a noble polytope always is a noble polytope as well. In terms of incidence matrices those define to have just a single row within both the top-most and the bottom-most blocks. – Thus especially any regular polytope is a noble polytope as well. But there are others too.

Within the following listing we restrict further to at least scaliform ones in addition. – Ones neither marked regular nor scaliform would be uniform.

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3D
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 Noble Polyhedron Facet Count Facet Type General Classes Circumradius tet 4 {3} regular convex 0.612372 cube 6 {4} regular convex 0.866025 oct 8 {3} regular convex 0.707107 doe 12 {5} regular convex 1.401259 gad regular 0.951057 sissid {5/2} regular 0.587785 gissid regular 0.535233 ike 20 {3} regular convex 0.951057 gike regular 0.587785 euclidean tilings trat ∞ {3} regular convex ∞ squat {4} regular convex hexat {6} regular convex hyperbolic tilings hetrat ∞ {3} regular convex 0.873057 i otrat regular convex 0.594604 i aztrat regular convex 0 i pesquat {4} regular convex 0.747674 i hisquat regular convex 0.5 i osquat regular convex 0.321797 i asquat regular convex 0 i peat {5} regular convex 0.899454 i pepat regular convex 0.528686 i depat regular convex 0.206652 i shexat {6} regular convex 0.707107 i hihexat regular convex 0.353553 i heat {7} regular convex 1.742610 i ocat {8} regular convex 1.345608 i socat regular convex 0.594604 i ococat regular convex 0.227545 i pedecat {10} regular convex 0.154508 i azat {∞} regular convex 0.866025 i squazat regular convex 0.5 i azazat regular convex 0 i ...

As an example for a non-convex noble polyhedron using different edge sizes ditti (a {6,6} modwrap within doe) or a {9,3} modwrap within srid might serve.

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4D
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 Noble Polychoron Facet Count Facet Type General Classes Circumradius pen 5 tet regular convex 0.632456 triddip 6 trip convex 0.816497 tes 8 cube regular convex 1 deca 10 tut convex 1.414214 pedip pip convex 1.203002 stardip stip 0.743496 hiddip 12 hip convex 1.414214 hedip 14 hep convex 1.629715 odip 16 op convex 1.847759 stodip stop 0.765367 hex tet regular convex 0.707107 dedip 20 dip convex 2.288246 staddip stiddip 0.874032 twaddip 24 twip convex 2.732051 ico oct regular convex 1 cont 48 tic convex 3.414214 sirc sroh 1.847759 bidex teddi scaliform convex 1.618034 affic cotco fissary 1.414214 girc groh 0.765367 birgax targi scaliform 0.618034 gic quith 0.585786 hi 120 doe regular convex 3.702459 sriphi sird 3.077684 sitphi tigid fissary 2.288246 mriphi siddy 1.902113 gahi doe regular 1.618034 gohi gad regular fix ike regular sisp pap griphi giddy 1.175571 idhi ditdid 1 gaghi gad regular paphacki pip sishi sissid regular paphicki stip gitphi quitsissid fissary 0.874032 graphi gird 0.726543 gofix gike regular 0.618034 gishi gissid regular gashi sissid regular gisp starp gogishi gissid regular 0.540182 spysp 240 peppy scaliform 1.618034 stodsap 2-pap-blend scaliform gypasp stappy scaliform 0.618034 gatodsap 2-starp-blend scaliform saffix 600 2-sird-fusion scaliform 3.077684 ex tet regular convex 1.618034 gaffix 2-gird-fusion scaliform 0.726543 gax tet regular 0.618034 gastakix 3600 tupip fissary scaliform 3.077684 sistakix tustip fissary scaliform 0.726543 {n},{n}-dip 2n {n}-p convex ... {n/d},{n/d}-dip {n/d}-p convex ... ... euclidean honeycombs chon ∞ cube regular convex ∞ hiph hip convex batch toe convex tiph trip convex ... hyperbolic honeycombs dih ∞ tid convex 2.910693 i x3x4x3x4*a girco convex 2.197368 i ditrah toxat convex 2.075910 i x3x5x3x5*a grid convex 1.798907 i diddoh ti convex dish tosquat convex 1.553774 i hexah hexat regular convex 1.414214 i doehon doe regular convex 0.899454 i shexah hexat regular convex 0.707107 i squah squat regular convex lamina-truncate x4x3o8o tic convex 0.594604 i pechon cube regular convex 0.555893 i ikhon ike regular convex 0.509376 i pedhon doe regular convex 0.427486 i x8o4o3o socat regular convex 0.400236 i phexah hexat regular convex 0.381966 i x10o5o3o pedecat regular convex 0.202376 i hachon cube regular convex 0 i hedhon doe regular convex hihexah hexat regular convex octh oct regular convex sisquah squat regular convex thon tet regular convex trah trat regular convex ...

As examples for convex noble polychora using different edge sizes there would be squap-72, trap-96, pap-360, trap-600.

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5D
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 Noble Polyteron Facet Count Facet Type General Classes Circumradius hix 6 pen regular convex 0.645497 stadow 10 stasc scaliform 0.623054 pent tes regular convex 1.118034 dot 12 rap convex 0.866025 shadow 14 shasc scaliform 0.674163 hit 16 pinnip 0.790569 nat 32 garpop 1.224745 tac pen regular convex 0.707107 {n/d}-dow 2n {n/d}-sc scaliform ... ... euclidean tetracombs otcypit ∞ gippid convex ∞ hext hex regular convex icot ico regular convex test tes regular convex tribbit triddip convex ... hyperbolic tetracombs hitte ∞ hi regular convex 2.301105 i shitte hi regular convex 0.899454 i o5o3x3o5o rox convex 0.747674 i contit cont convex 0.594604 i chont chon regular convex 0.5 i x3o3o5o5/2o ex regular 0.393076 i x3o5o5/2o5o fix regular x5o5/2o5o3o gohi regular pennit pen regular convex odipt odip convex 0.321797 i x5/2o5o3o3o sishi regular 0.242934 i pitest tes regular convex phitte hi regular convex 0.206652 i x3o4o3o4o ico regular convex 0 i ...

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6D
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 Noble Polypeton Facet Count Facet Type General Classes Circumradius hop 7 hix regular convex 0.654654 tetdip 8 tratet convex 0.866025 trittip 9 tratrip convex 1 ax 12 pent regular convex 1.224745 fe 14 bittix convex 1.732051 octdip 16 troct convex 1 dodoe 24 pedoe convex 1.981679 gaje 27 hit 0.816497 ikedip 40 trike convex 1.344997 mo 54 hin convex 1 gee 64 hix regular convex 0.707107 {n},{n},{n}-tip 3n {n},{n}-dippip convex ... ...

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7D
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 Noble Polyexon Facet Count Facet Type General Classes Circumradius oca 8 hop regular convex 0.661438 hept 14 ax regular convex 1.322876 he 16 bril convex 1 sissiddow 24 stasissiddow scaliform 0.831254 zee 128 hop regular convex 0.707107 ...

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8D
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 Noble Polyzetton Facet Count Facet Type General Classes Circumradius ene 9 oca regular convex 0.666667 pendip 10 tetpen convex 1.054093 triquip 12 tratratrip convex 1.154701 octo 16 hept regular convex 1.414214 be 18 tattoc convex 2 hexdip 32 tethex convex 1 icodip 48 octico convex 1.414214 hidip 240 dohi convex 5.236068 ek 256 oca regular convex 0.707107 exdip 1200 tetex convex 2.288246 {n},{n},{n},{n}-quip 4n {n},{n},{n}-tippip convex ... ...

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