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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 |
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.
---- 4D ----
As examples for convex noble polychora using different edge sizes there would be some swirlchora like
squap-72,
trap-96,
pap-360,
trap-600.
Also some vertex figures of polytera happen to be noble polychora with different edge sizes,
for instance the non-convex
hit-verf and
nat-verf.
---- 5D ----
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 |
ogdow | 16 | ogasc | scaliform | 0.629640 |
hit | 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 | ||
hibbit | hiddip | 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 | ||
pente | 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 | |
... |
---- 6D ----
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 | ... |
... | ||||
euclidean pentacombs | ||||
axh | ∞ | ax | regular convex | ∞ |
traxh | brag | convex | ||
gapcyxh | gocad | convex | ||
jakoh | jak | convex | ||
ramoh | ram | convex | ||
... |
---- 7D ----
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 |
... | ||||
euclidean hexacombs | ||||
otcyloh | ∞ | gotaf | convex | ∞ |
hepth | hept | regular convex | ||
linoh | lin | convex | ||
... |
---- 8D ----
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 | ... |
... |
---- 9D ----
Noble Polyyotton | Facet Count | Facet Type | General Classes | Circumradius |
day | 10 | ene | regular convex | 0.670820 |
enne | 18 | octo | regular convex | 1.5 |
icoy | 20 | trene | convex | 1.118034 |
vee | 512 | ene | regular convex | 0.707107 |
... |
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