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

As an example for a non-convex noble polyhedron using different edge sizes is 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	Facets/Vertex	Facet Type	General Classes	Circumradius
pen	5	4	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		8	tet	regular convex	0.707107
dedip	20	4	dip	convex	2.288246
stadidip	20		stiddip		0.874032
twaddip	24		twip	convex	2.732051
ico	24	6	oct	regular convex	1
cont	48	4	tic	convex	3.414214
sirc		8	sroh		1.847759
bidex		6	teddi	scaliform convex	1.618034
affic		8	cotco	fissary	1.414214
girc		8	groh		0.765367
birgax		6	targi	scaliform	0.618034
gic		4	quith		0.585786
hi	120	4	doe	regular convex	3.702459
sriphi		10	sird		3.077684
sitphi		4	tigid	fissary	2.288246
mriphi		10	siddy		1.902113
gahi		20	doe	regular	1.618034
gohi		12	gad	regular
fix		12	ike	regular
sisp		10	pap
griphi		10	giddy		1.175571
idhi		20	ditdid		1
gaghi		12	gad	regular
paphacki		60	pip
sishi		12	sissid	regular
paphicki		60	stip
gitphi		4	quitsissid	fissary	0.874032
graphi		10	gird		0.726543
gofix		12	gike	regular	0.618034
gishi		20	gissid	regular
gashi		12	sissid	regular
gisp		10	starp
gogishi		4	gissid	regular	0.540182
spysp	240	12	peppy	scaliform	1.618034
stodsap		...	2-pap-blend	scaliform	1.618034
gypasp		12	stappy	scaliform	0.618034
gatodsap		...	2-starp-blend	scaliform	0.618034
saffix	600	...	2-sird-fusion	scaliform	3.077684
ex		20	tet	regular convex	1.618034
gaffix		...	2-gird-fusion	scaliform	0.726543
gax		20	tet	regular	0.618034
gastakix	3600	...	tupip	fissary scaliform	3.077684
sistakix	3600	...	tustip	fissary scaliform	0.726543
{n},{n}-dip	2n	4	{n}-p	convex	...
{n/d},{n/d}-dip	2n	4	{n/d}-p	convex	...
...
euclidean honeycombs
chon	∞	8	cube	regular convex	∞
hiph		6	hip	convex
batch		4	toe	convex
tiph		12	trip	convex
...
hyperbolic honeycombs
dih	∞	4	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		8	doe	regular convex	0.899454 i
shexah		8	hexat	regular convex	0.707107 i
squah		6	squat	regular convex	0.707107 i
lamina-truncate x4x3o8o		16	tic	convex	0.594604 i
pechon		20	cube	regular convex	0.555893 i
ikhon		12	ike	regular convex	0.509376 i
x5o4o3o		6	peat	regular convex	0.509376 i
pedhon		20	doe	regular convex	0.427486 i
fipped		12	pepat	regular convex	0.427486 i
x8o4o3o		6	socat	regular convex	0.400236 i
phexah		20	hexat	regular convex	0.381966 i
x5o5o3o		12	pepat	regular convex	0.245719 i
x10o5o3o		12	pedecat	regular convex	0.202376 i
x3oØo3a4oØo4a		∞	co	convex	0
hachon			cube	regular convex
hedhon			doe	regular convex
hihexah			hexat	regular convex
x3oØo3a5oØo5a			id	convex
octh			oct	regular convex
sisquah			squat	regular convex
thon			tet	regular convex
trah			trat	regular convex
x3o5o6o			ike	regular convex	0.465341
sipped			sissid	regular convex	0.475273
x∞o∞o∞o			azazat	regular convex	0.5
...

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 uniform 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	Facets/Vertex	Facet Type	General Classes	Circumradius
hix	6	5	pen	regular convex	0.645497
stadow	10	7	stasc	scaliform	0.623054
pent	10	5	tes	regular convex	1.118034
dot	12	6	rap	convex	0.866025
shadow	14	9	shasc	scaliform	0.674163
ogdow	16	10	ogasc	scaliform	0.629640
hit	16	10	pinnip		0.790569
nat	32	12	garpop		1.224745
tac	32	16	pen	regular convex	0.707107
{n/d}-dow	2n	n+2	{n/d}-sc	scaliform	...
...
euclidean tetracombs
otcypit	∞	5	gippid	convex	∞
hext		24	hex	regular convex
hibbit		9	hiddip	convex
icot		8	ico	regular convex
test		16	tes	regular convex
tribbit		36	triddip	convex
...
hyperbolic tetracombs
hitte	∞	5	hi	regular convex	2.301105 i
shitte		16	hi	regular convex	0.899454 i
o5o3x3o5o		10	rox	convex	0.747674 i
contit		64	cont	convex	0.594604 i
chont		24	chon	regular convex	0.5 i
fipte		120	ex	regular	0.393076 i
fighitte			fipped	regular
tifipte			fix	regular
gohitte			gohi	regular
fatfipte			ikhon	regular
pente		600	pen	regular convex
odipt		288	odip	convex	0.321797 i
sishitte		120	sishi	regular	0.242934 i
pitest		600	tes	regular convex	0.242934 i
phitte		600	hi	regular convex	0.206652 i
x3o4o3o4o		∞	ico	regular convex	0
gishitte			gishi	regular	0.476925
gotfipte			gofix	regular	0.568286
...


----
 6D
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Noble Polypeton	Facet Count	Facets/Vertex	Facet Type	General Classes	Circumradius
hop	7	6	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	8	troct	convex	1
dodoe	24	6	pedoe	convex	1.981679
gaje	27	16	hit		0.816497
ikedip	40	10	trike	convex	1.344997
mo	54	12	hin	convex	1
gee	64	32	hix	regular convex	0.707107
{n},{n},{n}-tip	3n	6	{n},{n}-dippip	convex	...
...
euclidean pentacombs
axh	∞	64	ax	regular convex	∞
traxh		16	brag	convex
gapcyxh		6	gocad	convex
jakoh		54	jak	convex
ramoh		9	ram	convex
...


----
 7D
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Noble Polyexon	Facet Count	Facets/Vertex	Facet Type	General Classes	Circumradius
oca	8	7	hop	regular convex	0.661438
hept	14	7	ax	regular convex	1.322876
he	16	8	bril	convex	1
sissiddow	24	17	stasissiddow	scaliform	0.831254
zee	128	64	hop	regular convex	0.707107
...
euclidean hexacombs
otcyloh	∞	7	gotaf	convex	∞
hepth		128	hept	regular convex
linoh		16	lin	convex
...


----
 8D
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Noble Polyzetton	Facet Count	Facets/Vertex	Facet Type	General Classes	Circumradius
ene	9	8	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	16	tethex	convex	1
icodip	48	12	octico	convex	1.414214
hidip	240	8	dohi	convex	5.236068
ek	256	128	oca	regular convex	0.707107
exdip	1200	...	tetex	convex	2.288246
{n},{n},{n},{n}-quip	4n	8	{n},{n},{n}-tippip	convex	...
...


----
 9D
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Noble Polyyotton	Facet Count	Facets/Vertex	Facet Type	General Classes	Circumradius
day	10	9	ene	regular convex	0.670820
enne	18	9	octo	regular convex	1.5
icoy	20	10	trene	convex	1.118034
vee	512	256	ene	regular convex	0.707107
...

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Noble Polytopes

---- 3D ----

---- 4D ----

---- 5D ----

---- 6D ----

---- 7D ----

---- 8D ----

---- 9D ----