noble

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		{5}	regular	0.951057
sissid		{5/2}	regular	0.587785
gissid		{5/2}	regular	0.535233
ike	20	{3}	regular convex	0.951057
gike	20	{3}	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		{6}	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
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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	20	stiddip		0.874032
twaddip	24	twip	convex	2.732051
ico	24	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	1.618034
gypasp		stappy	scaliform	0.618034
gatodsap		2-starp-blend	scaliform	0.618034
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	3600	tustip	fissary scaliform	0.726543
{n},{n}-dip	2n	{n}-p	convex	...
{n/d},{n/d}-dip	2n	{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	1.798907 i
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	0.707107 i
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 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.


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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	10	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	32	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
pente		pen	regular convex
odipt		odip	convex	0.321797 i
x5/2o5o3o3o		sishi	regular	0.242934 i
pitest		tes	regular convex	0.242934 i
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	...
...


----
 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
...


----
 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	...
...

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Noble Polytopes

---- 3D ----

---- 4D ----

---- 5D ----

---- 6D ----

---- 7D ----

---- 8D ----