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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.


---- 3D ----

Noble Polyhedron Facet Count Facet Type General Classes Circumradius
tet4{3}regular convex0.612372
cube6{4}regular convex0.866025
oct8{3}regular convex0.707107
doe12{5}regular convex1.401259
gadregular0.951057
sissid{5/2}regular0.587785
gissidregular0.535233
ike20{3}regular convex0.951057
gikeregular0.587785
euclidean tilings
trat{3}regular convex
squat{4}regular convex
hexat{6}regular convex
hyperbolic tilings
hetrat{3}regular convex0.873057 i
otratregular convex0.594604 i
aztratregular convex0 i
pesquat{4}regular convex0.747674 i
hisquatregular convex0.5 i
osquatregular convex0.321797 i
asquatregular convex0 i
peat{5}regular convex0.899454 i
pepatregular convex0.528686 i
depatregular convex0.206652 i
shexat{6}regular convex0.707107 i
hihexatregular convex0.353553 i
heat{7}regular convex1.742610 i
ocat{8}regular convex1.345608 i
socatregular convex0.594604 i
ococatregular convex0.227545 i
pedecat{10}regular convex0.154508 i
azat{∞}regular convex0.866025 i
squazatregular convex0.5 i
azazatregular convex0 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.


---- 4D ----

Noble Polychoron Facet Count Facet Type General Classes Circumradius
pen5tetregular convex0.632456
triddip6tripconvex0.816497
tes8cuberegular convex1
deca10tutconvex1.414214
pedippipconvex1.203002
stardipstip 0.743496
hiddip12hipconvex1.414214
hedip14hepconvex1.629715
odip16opconvex1.847759
stodipstop 0.765367
hextetregular convex0.707107
dedip20dipconvex2.288246
staddipstiddip 0.874032
twaddip24twipconvex2.732051
icooctregular convex1
cont48ticconvex3.414214
sircsroh 1.847759
bidexteddiscaliform convex1.618034
afficcotcofissary1.414214
gircgroh 0.765367
birgaxtargiscaliform0.618034
gicquith 0.585786
hi120doeregular convex3.702459
sriphisird 3.077684
sitphitigidfissary2.288246
mriphisiddy 1.902113
gahidoeregular1.618034
gohigadregular
fixikeregular
sisppap 
griphigiddy 1.175571
idhiditdid 1
gaghigadregular
paphackipip 
sishisissidregular
paphickistip 
gitphiquitsissidfissary0.874032
graphigird 0.726543
gofixgikeregular0.618034
gishigissidregular
gashisissidregular
gispstarp 
gogishigissidregular0.540182
spysp240peppyscaliform1.618034
stodsap2-pap-blendscaliform
gypaspstappyscaliform0.618034
gatodsap2-starp-blendscaliform
saffix6002-sird-fusionscaliform3.077684
extetregular convex1.618034
gaffix2-gird-fusionscaliform0.726543
gaxtetregular0.618034
gastakix3600tupipfissary scaliform3.077684
sistakixtustipfissary scaliform0.726543
{n},{n}-dip2n{n}-pconvex...
{n/d},{n/d}-dip{n/d}-pconvex...
...
euclidean honeycombs
choncuberegular convex
hiphhipconvex
batchtoeconvex
tiphtripconvex
...
hyperbolic honeycombs
dihtidconvex2.910693 i
x3x4x3x4*agircoconvex2.197368 i
ditrahtoxatconvex2.075910 i
x3x5x3x5*agridconvex1.798907 i
diddohticonvex
dishtosquatconvex1.553774 i
hexahhexatregular convex1.414214 i
doehondoeregular convex0.899454 i
shexahhexatregular convex0.707107 i
squahsquatregular convex
lamina-truncate x4x3o8oticconvex0.594604 i
pechoncuberegular convex0.555893 i
ikhonikeregular convex0.509376 i
pedhondoeregular convex0.427486 i
x8o4o3osocatregular convex0.400236 i
phexahhexatregular convex0.381966 i
x10o5o3opedecatregular convex0.202376 i
hachoncuberegular convex0 i
hedhondoeregular convex
hihexahhexatregular convex
octhoctregular convex
sisquahsquatregular convex
thontetregular convex
trahtratregular convex
...

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


---- 5D ----

Noble Polyteron Facet Count Facet Type General Classes Circumradius
hix6penregular convex0.645497
stadow10stascscaliform0.623054
penttesregular convex1.118034
dot12rapconvex0.866025
shadow14shascscaliform0.674163
hit16pinnip 0.790569
nat32garpop 1.224745
tacpenregular convex0.707107
{n/d}-dow2n{n/d}-scscaliform...
...
euclidean tetracombs
otcypitgippidconvex
hexthexregular convex
icoticoregular convex
testtesregular convex
tribbittriddipconvex
...
hyperbolic tetracombs
hittehiregular convex2.301105 i
shittehiregular convex0.899454 i
o5o3x3o5oroxconvex0.747674 i
contitcontconvex0.594604 i
chontchonregular convex0.5 i
x3o3o5o5/2oexregular0.393076 i
x3o5o5/2o5ofixregular
x5o5/2o5o3ogohiregular
pennitpenregular convex
odiptodipconvex0.321797 i
x5/2o5o3o3osishiregular0.242934 i
pitesttesregular convex
phittehiregular convex0.206652 i
x3o4o3o4oicoregular convex0 i
...

---- 6D ----

Noble Polypeton Facet Count Facet Type General Classes Circumradius
hop7hixregular convex0.654654
tetdip8tratetconvex0.866025
trittip9tratripconvex1
ax12pentregular convex1.224745
fe14bittixconvex1.732051
octdip16troctconvex1
dodoe24pedoeconvex1.981679
gaje27hit 0.816497
ikedip40trikeconvex1.344997
mo54hinconvex1
gee64hixregular convex0.707107
{n},{n},{n}-tip3n{n},{n}-dippipconvex...
...

---- 7D ----

Noble Polyexon Facet Count Facet Type General Classes Circumradius
oca8hopregular convex0.661438
hept14axregular convex1.322876
he16brilconvex1
sissiddow24stasissiddowscaliform0.831254
zee128hopregular convex0.707107
...

---- 8D ----

Noble Polyzetton Facet Count Facet Type General Classes Circumradius
ene9ocaregular convex0.666667
pendip10tetpenconvex1.054093
triquip12tratratripconvex1.154701
octo16heptregular convex1.414214
be18tattocconvex2
hexdip32tethexconvex1
icodip48octicoconvex1.414214
hidip240dohiconvex5.236068
ek256ocaregular convex0.707107
exdip1200tetexconvex2.288246
{n},{n},{n},{n}-quip4n{n},{n},{n}-tippipconvex...
...


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