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Uniform 8-polytope

Polytope contained by 7-polytope facets

[[File:Gosset 4 21 polytope petrie.svg	100px]]
421	[[File:Gosset 1 42 polytope petrie.svg	100px]]
142	[[File:2 41 polytope petrie.svg	100px]]
241

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets.

Regular 8-polytopes

Regular 8-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with v {p,q,r,s,t,u} 7-polytope facets around each peak.

There are exactly three such convex regular 8-polytopes:

{3,3,3,3,3,3,3} - 8-simplex
{4,3,3,3,3,3,3} - 8-cube
{3,3,3,3,3,3,4} - 8-orthoplex

There are no nonconvex regular 8-polytopes.

Characteristics

The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

Uniform 8-polytopes by fundamental Coxeter groups

Uniform 8-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

#	Coxeter group	Forms
1	A8	[37]
2	BC8	[4,36]
3	D8	[35,1,1]
4	E8	[34,2,1]

Selected regular and uniform 8-polytopes from each family include:

Simplex family: A8 [37] -

135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
1. {37} - 8-simplex or ennea-9-tope or enneazetton -

Hypercube/orthoplex family: B8 [4,36] -

255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
1. {4,36} - 8-cube or octeract-
2. {36,4} - 8-orthoplex or octacross -

Demihypercube D8 family: [35,1,1] -

191 uniform 8-polytopes as permutations of rings in the group diagram, including:
1. {3,35,1} - 8-demicube or demiocteract, 151 - ; also as h{4,36} .
2. {3,3,3,3,3,31,1} - 8-orthoplex, 511 -

E-polytope family E8 family: [34,1,1] -

255 uniform 8-polytopes as permutations of rings in the group diagram, including:
1. {3,3,3,3,32,1} - Thorold Gosset's semiregular 421,
2. {3,34,2} - the uniform 142, ,
3. {3,3,34,1} - the uniform 241,

Uniform prismatic forms

There are many uniform prismatic families, including:

Uniform 8-polytope prism families	#	Coxeter group
1	A7A1	[3,3,3,3,3,3]×[ ]
2	B7A1	[4,3,3,3,3,3]×[ ]
3	D7A1	[34,1,1]×[ ]
4	E7A1	[33,2,1]×[ ]
1	A6I2(p)	[3,3,3,3,3]×[p]
2	B6I2(p)	[4,3,3,3,3]×[p]
3	D6I2(p)	[33,1,1]×[p]
4	E6I2(p)	[3,3,3,3,3]×[p]
1	A6A1A1	[3,3,3,3,3]×[ ]x[ ]
2	B6A1A1	[4,3,3,3,3]×[ ]x[ ]
3	D6A1A1	[33,1,1]×[ ]x[ ]
4	E6A1A1	[3,3,3,3,3]×[ ]x[ ]
1	A5A3	[34]×[3,3]
2	B5A3	[4,33]×[3,3]
3	D5A3	[32,1,1]×[3,3]
4	A5B3	[34]×[4,3]
5	B5B3	[4,33]×[4,3]
6	D5B3	[32,1,1]×[4,3]
7	A5H3	[34]×[5,3]
8	B5H3	[4,33]×[5,3]
9	D5H3	[32,1,1]×[5,3]
1	A5I2(p)A1	[3,3,3]×[p]×[ ]
2	B5I2(p)A1	[4,3,3]×[p]×[ ]
3	D5I2(p)A1	[32,1,1]×[p]×[ ]
1	A5A1A1A1	[3,3,3]×[ ]×[ ]×[ ]
2	B5A1A1A1	[4,3,3]×[ ]×[ ]×[ ]
3	D5A1A1A1	[32,1,1]×[ ]×[ ]×[ ]
1	A4A4	[3,3,3]×[3,3,3]
2	B4A4	[4,3,3]×[3,3,3]
3	D4A4	[31,1,1]×[3,3,3]
4	F4A4	[3,4,3]×[3,3,3]
5	H4A4	[5,3,3]×[3,3,3]
6	B4B4	[4,3,3]×[4,3,3]
7	D4B4	[31,1,1]×[4,3,3]
8	F4B4	[3,4,3]×[4,3,3]
9	H4B4	[5,3,3]×[4,3,3]
10	D4D4	[31,1,1]×[31,1,1]
11	F4D4	[3,4,3]×[31,1,1]
12	H4D4	[5,3,3]×[31,1,1]
13	F4×F4	[3,4,3]×[3,4,3]
14	H4×F4	[5,3,3]×[3,4,3]
15	H4H4	[5,3,3]×[5,3,3]
1	A4A3A1	[3,3,3]×[3,3]×[ ]
2	A4B3A1	[3,3,3]×[4,3]×[ ]
3	A4H3A1	[3,3,3]×[5,3]×[ ]
4	B4A3A1	[4,3,3]×[3,3]×[ ]
5	B4B3A1	[4,3,3]×[4,3]×[ ]
6	B4H3A1	[4,3,3]×[5,3]×[ ]
7	H4A3A1	[5,3,3]×[3,3]×[ ]
8	H4B3A1	[5,3,3]×[4,3]×[ ]
9	H4H3A1	[5,3,3]×[5,3]×[ ]
10	F4A3A1	[3,4,3]×[3,3]×[ ]
11	F4B3A1	[3,4,3]×[4,3]×[ ]
12	F4H3A1	[3,4,3]×[5,3]×[ ]
13	D4A3A1	[31,1,1]×[3,3]×[ ]
14	D4B3A1	[31,1,1]×[4,3]×[ ]
15	D4H3A1	[31,1,1]×[5,3]×[ ]
...
...
...
	A3A3I2(p)	[3,3]×[3,3]×[p]
	B3A3I2(p)	[4,3]×[3,3]×[p]
	H3A3I2(p)	[5,3]×[3,3]×[p]
	B3B3I2(p)	[4,3]×[4,3]×[p]
	H3B3I2(p)	[5,3]×[4,3]×[p]
	H3H3I2(p)	[5,3]×[5,3]×[p]
	A32A12	[3,3]×[3,3]×[ ]×[ ]
	B3A3A12	[4,3]×[3,3]×[ ]×[ ]
	H3A3A12	[5,3]×[3,3]×[ ]×[ ]
	B3B3A12	[4,3]×[4,3]×[ ]×[ ]
	H3B3A12	[5,3]×[4,3]×[ ]×[ ]
	H3H3A12	[5,3]×[5,3]×[ ]×[ ]
1	A3I2(p)I2(q)A1	[3,3]×[p]×[q]×[ ]
2	B3I2(p)I2(q)A1	[4,3]×[p]×[q]×[ ]
3	H3I2(p)I2(q)A1	[5,3]×[p]×[q]×[ ]
1	A3I2(p)A13	[3,3]×[p]×[ ]x[ ]×[ ]
2	B3I2(p)A13	[4,3]×[p]×[ ]x[ ]×[ ]
3	H3I2(p)A13	[5,3]×[p]×[ ]x[ ]×[ ]
1	A3A15	[3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2	B3A15	[4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
3	H3A15	[5,3]×[ ]x[ ]×[ ]x[ ]×[ ]
1	I2(p)I2(q)I2(r)I2(s)	[p]×[q]×[r]×[s]
1	I2(p)I2(q)I2(r)A12	[p]×[q]×[r]×[ ]×[ ]
2	I2(p)I2(q)A14	[p]×[q]×[ ]×[ ]×[ ]×[ ]
1	I2(p)A16	[p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1	A18	[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

The A₈ family

The A8 family has symmetry of order 362880 (9 factorial).

There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings (128+8-1 cases). These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.

See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes.

#	Coxeter-Dynkin diagram	Truncationindices	Johnson name(acronym)	Basepoint	Element counts	7		6		5
t0	8-simplex (ene)	(0,0,0,0,0,0,0,0,1)		9	36	84	126	126	84	36	9
t1	Rectified 8-simplex (rene)	(0,0,0,0,0,0,0,1,1)		18	108	336	630	576	588	252	36
t2	Birectified 8-simplex (brene)	(0,0,0,0,0,0,1,1,1)		18	144	588	1386	2016	1764	756	84
t3	Trirectified 8-simplex (trene)	(0,0,0,0,0,1,1,1,1)								1260	126
t0,1	Truncated 8-simplex (tene)	(0,0,0,0,0,0,0,1,2)								288	72
t0,2	Cantellated 8-simplex (srene)	(0,0,0,0,0,0,1,1,2)								1764	252
t1,2	Bitruncated 8-simplex (batene)	(0,0,0,0,0,0,1,2,2)								1008	252
t0,3	Runcinated 8-simplex (spene)	(0,0,0,0,0,1,1,1,2)								4536	504
t1,3	Bicantellated 8-simplex (sabrene)	(0,0,0,0,0,1,1,2,2)								5292	756
t2,3	Tritruncated 8-simplex (tatene)	(0,0,0,0,0,1,2,2,2)								2016	504
t0,4	Stericated 8-simplex (secane)	(0,0,0,0,1,1,1,1,2)								6300	630
t1,4	Biruncinated 8-simplex (sabpene)	(0,0,0,0,1,1,1,2,2)								11340	1260
t2,4	Tricantellated 8-simplex (satrene)	(0,0,0,0,1,1,2,2,2)								8820	1260
t3,4	Quadritruncated 8-simplex (be)	(0,0,0,0,1,2,2,2,2)								2520	630
t0,5	Pentellated 8-simplex (sotane)	(0,0,0,1,1,1,1,1,2)								5040	504
t1,5	Bistericated 8-simplex (sobcane)	(0,0,0,1,1,1,1,2,2)								12600	1260
t2,5	Triruncinated 8-simplex (satpeb)	(0,0,0,1,1,1,2,2,2)								15120	1680
t0,6	Hexicated 8-simplex (supane)	(0,0,1,1,1,1,1,1,2)								2268	252
t1,6	Bipentellated 8-simplex (sobteb)	(0,0,1,1,1,1,1,2,2)								7560	756
t0,7	Heptellated 8-simplex (soxeb)	(0,1,1,1,1,1,1,1,2)								504	72
t0,1,2	Cantitruncated 8-simplex (grene)	(0,0,0,0,0,0,1,2,3)								2016	504
t0,1,3	Runcitruncated 8-simplex (potane)	(0,0,0,0,0,1,1,2,3)								9828	1512
t0,2,3	Runcicantellated 8-simplex (prene)	(0,0,0,0,0,1,2,2,3)								6804	1512
t1,2,3	Bicantitruncated 8-simplex (gabrene)	(0,0,0,0,0,1,2,3,3)								6048	1512
t0,1,4	Steritruncated 8-simplex (catene)	(0,0,0,0,1,1,1,2,3)								20160	2520
t0,2,4	Stericantellated 8-simplex (crane)	(0,0,0,0,1,1,2,2,3)	2							26460	3780
t1,2,4	Biruncitruncated 8-simplex (biptene)	(0,0,0,0,1,1,2,3,3)								22680	3780
t0,3,4	Steriruncinated 8-simplex (capene)	(0,0,0,0,1,2,2,2,3)								12600	2520
t1,3,4	Biruncicantellated 8-simplex (biprene)	(0,0,0,0,1,2,2,3,3)								18900	3780
t2,3,4	Tricantitruncated 8-simplex (gatrene)	(0,0,0,0,1,2,3,3,3)								10080	2520
t0,1,5	Pentitruncated 8-simplex (tetane)	(0,0,0,1,1,1,1,2,3)								21420	2520
t0,2,5	Penticantellated 8-simplex (turane)	(0,0,0,1,1,1,2,2,3)								42840	5040
t1,2,5	Bisteritruncated 8-simplex (bictane)	(0,0,0,1,1,1,2,3,3)								35280	5040
t0,3,5	Pentiruncinated 8-simplex (topene)	(0,0,0,1,1,2,2,2,3)								37800	5040
t1,3,5	Bistericantellated 8-simplex (bocrane)	(0,0,0,1,1,2,2,3,3)								52920	7560
t2,3,5	Triruncitruncated 8-simplex (toprane)	(0,0,0,1,1,2,3,3,3)								27720	5040
t0,4,5	Pentistericated 8-simplex (tecane)	(0,0,0,1,2,2,2,2,3)								13860	2520
t1,4,5	Bisteriruncinated 8-simplex (bacpane)	(0,0,0,1,2,2,2,3,3)								30240	5040
t0,1,6	Hexitruncated 8-simplex (putene)	(0,0,1,1,1,1,1,2,3)								12096	1512
t0,2,6	Hexicantellated 8-simplex (purene)	(0,0,1,1,1,1,2,2,3)								34020	3780
t1,2,6	Bipentitruncated 8-simplex (bitotene)	(0,0,1,1,1,1,2,3,3)								26460	3780
t0,3,6	Hexiruncinated 8-simplex (pupene)	(0,0,1,1,1,2,2,2,3)								45360	5040
t1,3,6	Bipenticantellated 8-simplex (bitrene)	(0,0,1,1,1,2,2,3,3)								60480	7560
t0,4,6	Hexistericated 8-simplex (pucane)	(0,0,1,1,2,2,2,2,3)								30240	3780
t0,5,6	Hexipentellated 8-simplex (putane)	(0,0,1,2,2,2,2,2,3)								9072	1512
t0,1,7	Heptitruncated 8-simplex (xotane)	(0,1,1,1,1,1,1,2,3)								3276	504
t0,2,7	Hepticantellated 8-simplex (xorene)	(0,1,1,1,1,1,2,2,3)								12852	1512
t0,3,7	Heptiruncinated 8-simplex (xapane)	(0,1,1,1,1,2,2,2,3)								23940	2520
t0,1,2,3	Runcicantitruncated 8-simplex (gapene)	(0,0,0,0,0,1,2,3,4)								12096	3024
t0,1,2,4	Stericantitruncated 8-simplex (cograne)	(0,0,0,0,1,1,2,3,4)								45360	7560
t0,1,3,4	Steriruncitruncated 8-simplex (coptane)	(0,0,0,0,1,2,2,3,4)								34020	7560
t0,2,3,4	Steriruncicantellated 8-simplex (coprene)	(0,0,0,0,1,2,3,3,4)								34020	7560
t1,2,3,4	Biruncicantitruncated 8-simplex (gabpene)	(0,0,0,0,1,2,3,4,4)								30240	7560
t0,1,2,5	Penticantitruncated 8-simplex (tograne)	(0,0,0,1,1,1,2,3,4)								70560	10080
t0,1,3,5	Pentiruncitruncated 8-simplex (taptane)	(0,0,0,1,1,2,2,3,4)								98280	15120
t0,2,3,5	Pentiruncicantellated 8-simplex (taprene)	(0,0,0,1,1,2,3,3,4)								90720	15120
t1,2,3,5	Bistericantitruncated 8-simplex (bocagrane)	(0,0,0,1,1,2,3,4,4)								83160	15120
t0,1,4,5	Pentisteritruncated 8-simplex (tectane)	(0,0,0,1,2,2,2,3,4)								50400	10080
t0,2,4,5	Pentistericantellated 8-simplex (tocrane)	(0,0,0,1,2,2,3,3,4)								83160	15120
t1,2,4,5	Bisteriruncitruncated 8-simplex (bicpotane)	(0,0,0,1,2,2,3,4,4)								68040	15120
t0,3,4,5	Pentisteriruncinated 8-simplex (tecpane)	(0,0,0,1,2,3,3,3,4)								50400	10080
t1,3,4,5	Bisteriruncicantellated 8-simplex (bicprene)	(0,0,0,1,2,3,3,4,4)								75600	15120
t2,3,4,5	Triruncicantitruncated 8-simplex (gatpeb)	(0,0,0,1,2,3,4,4,4)								40320	10080
t0,1,2,6	Hexicantitruncated 8-simplex (pugrane)	(0,0,1,1,1,1,2,3,4)								52920	7560
t0,1,3,6	Hexiruncitruncated 8-simplex (puptane)	(0,0,1,1,1,2,2,3,4)								113400	15120
t0,2,3,6	Hexiruncicantellated 8-simplex (puprene)	(0,0,1,1,1,2,3,3,4)								98280	15120
t1,2,3,6	Bipenticantitruncated 8-simplex (batograne)	(0,0,1,1,1,2,3,4,4)								90720	15120
t0,1,4,6	Hexisteritruncated 8-simplex (puctane)	(0,0,1,1,2,2,2,3,4)								105840	15120
t0,2,4,6	Hexistericantellated 8-simplex (pucrene)	(0,0,1,1,2,2,3,3,4)								158760	22680
t1,2,4,6	Bipentiruncitruncated 8-simplex (batpitane)	(0,0,1,1,2,2,3,4,4)								136080	22680
t0,3,4,6	Hexisteriruncinated 8-simplex (pocapine)	(0,0,1,1,2,3,3,3,4)								90720	15120
t1,3,4,6	Bipentiruncicantellated 8-simplex (bitprop)	(0,0,1,1,2,3,3,4,4)								136080	22680
t0,1,5,6	Hexipentitruncated 8-simplex (putatine)	(0,0,1,2,2,2,2,3,4)								41580	7560
t0,2,5,6	Hexipenticantellated 8-simplex (putarene)	(0,0,1,2,2,2,3,3,4)								98280	15120
t1,2,5,6	Bipentisteritruncated 8-simplex (batcotab)	(0,0,1,2,2,2,3,4,4)								75600	15120
t0,3,5,6	Hexipentiruncinated 8-simplex (putapene)	(0,0,1,2,2,3,3,3,4)								98280	15120
t0,4,5,6	Hexipentistericated 8-simplex (putacane)	(0,0,1,2,3,3,3,3,4)								41580	7560
t0,1,2,7	Hepticantitruncated 8-simplex (xograne)	(0,1,1,1,1,1,2,3,4)								18144	3024
t0,1,3,7	Heptiruncitruncated 8-simplex (xaptane)	(0,1,1,1,1,2,2,3,4)								56700	7560
t0,2,3,7	Heptiruncicantellated 8-simplex (xeprane)	(0,1,1,1,1,2,3,3,4)								45360	7560
t0,1,4,7	Heptisteritruncated 8-simplex (xactane)	(0,1,1,1,2,2,2,3,4)								80640	10080
t0,2,4,7	Heptistericantellated 8-simplex (xacrene)	(0,1,1,1,2,2,3,3,4)								113400	15120
t0,3,4,7	Heptisteriruncinated 8-simplex (xocapob)	(0,1,1,1,2,3,3,3,4)								60480	10080
t0,1,5,7	Heptipentitruncated 8-simplex (xotatine)	(0,1,1,2,2,2,2,3,4)								56700	7560
t0,2,5,7	Heptipenticantellated 8-simplex (xotrab)	(0,1,1,2,2,2,3,3,4)								120960	15120
t0,1,6,7	Heptihexitruncated 8-simplex (xupatab)	(0,1,2,2,2,2,2,3,4)								18144	3024
t0,1,2,3,4	Steriruncicantitruncated 8-simplex (gacene)	(0,0,0,0,1,2,3,4,5)								60480	15120
t0,1,2,3,5	Pentiruncicantitruncated 8-simplex (togapene)	(0,0,0,1,1,2,3,4,5)								166320	30240
t0,1,2,4,5	Pentistericantitruncated 8-simplex (tecograne)	(0,0,0,1,2,2,3,4,5)								136080	30240
t0,1,3,4,5	Pentisteriruncitruncated 8-simplex (tecpatane)	(0,0,0,1,2,3,3,4,5)								136080	30240
t0,2,3,4,5	Pentisteriruncicantellated 8-simplex (ticprane)	(0,0,0,1,2,3,4,4,5)								136080	30240
t1,2,3,4,5	Bisteriruncicantitruncated 8-simplex (gobcane)	(0,0,0,1,2,3,4,5,5)								120960	30240
t0,1,2,3,6	Hexiruncicantitruncated 8-simplex (pogapene)	(0,0,1,1,1,2,3,4,5)								181440	30240
t0,1,2,4,6	Hexistericantitruncated 8-simplex (pocagrane)	(0,0,1,1,2,2,3,4,5)								272160	45360
t0,1,3,4,6	Hexisteriruncitruncated 8-simplex (pocpatine)	(0,0,1,1,2,3,3,4,5)								249480	45360
t0,2,3,4,6	Hexisteriruncicantellated 8-simplex (pocpurene)	(0,0,1,1,2,3,4,4,5)								249480	45360
t1,2,3,4,6	Bipentiruncicantitruncated 8-simplex (botagpane)	(0,0,1,1,2,3,4,5,5)								226800	45360
t0,1,2,5,6	Hexipenticantitruncated 8-simplex (potagrene)	(0,0,1,2,2,2,3,4,5)								151200	30240
t0,1,3,5,6	Hexipentiruncitruncated 8-simplex (potaptane)	(0,0,1,2,2,3,3,4,5)								249480	45360
t0,2,3,5,6	Hexipentiruncicantellated 8-simplex (putaprene)	(0,0,1,2,2,3,4,4,5)								226800	45360
t1,2,3,5,6	Bipentistericantitruncated 8-simplex (betcagrane)	(0,0,1,2,2,3,4,5,5)								204120	45360
t0,1,4,5,6	Hexipentisteritruncated 8-simplex (putcatine)	(0,0,1,2,3,3,3,4,5)								151200	30240
t0,2,4,5,6	Hexipentistericantellated 8-simplex (potacrane)	(0,0,1,2,3,3,4,4,5)								249480	45360
t0,3,4,5,6	Hexipentisteriruncinated 8-simplex (potcapane)	(0,0,1,2,3,4,4,4,5)								151200	30240
t0,1,2,3,7	Heptiruncicantitruncated 8-simplex (xigpane)	(0,1,1,1,1,2,3,4,5)								83160	15120
t0,1,2,4,7	Heptistericantitruncated 8-simplex (xecagrane)	(0,1,1,1,2,2,3,4,5)								196560	30240
t0,1,3,4,7	Heptisteriruncitruncated 8-simplex (xucaptane)	(0,1,1,1,2,3,3,4,5)								166320	30240
t0,2,3,4,7	Heptisteriruncicantellated 8-simplex (xecaprane)	(0,1,1,1,2,3,4,4,5)								166320	30240
t0,1,2,5,7	Heptipenticantitruncated 8-simplex (xotagrane)	(0,1,1,2,2,2,3,4,5)								196560	30240
t0,1,3,5,7	Heptipentiruncitruncated 8-simplex (xitaptene)	(0,1,1,2,2,3,3,4,5)								294840	45360
t0,2,3,5,7	Heptipentiruncicantellated 8-simplex (xataprane)	(0,1,1,2,2,3,4,4,5)								272160	45360
t0,1,4,5,7	Heptipentisteritruncated 8-simplex (xotcatene)	(0,1,1,2,3,3,3,4,5)								166320	30240
t0,1,2,6,7	Heptihexicantitruncated 8-simplex (xopugrane)	(0,1,2,2,2,2,3,4,5)								83160	15120
t0,1,3,6,7	Heptihexiruncitruncated 8-simplex (xopupatane)	(0,1,2,2,2,3,3,4,5)								196560	30240
t0,1,2,3,4,5	Pentisteriruncicantitruncated 8-simplex (gotane)	(0,0,0,1,2,3,4,5,6)								241920	60480
t0,1,2,3,4,6	Hexisteriruncicantitruncated 8-simplex (pogacane)	(0,0,1,1,2,3,4,5,6)								453600	90720
t0,1,2,3,5,6	Hexipentiruncicantitruncated 8-simplex (potegpane)	(0,0,1,2,2,3,4,5,6)								408240	90720
t0,1,2,4,5,6	Hexipentistericantitruncated 8-simplex (potacagrane)	(0,0,1,2,3,3,4,5,6)								408240	90720
t0,1,3,4,5,6	Hexipentisteriruncitruncated 8-simplex (poticaptine)	(0,0,1,2,3,4,4,5,6)								408240	90720
t0,2,3,4,5,6	Hexipentisteriruncicantellated 8-simplex (poticoprane)	(0,0,1,2,3,4,5,5,6)								408240	90720
t1,2,3,4,5,6	Bipentisteriruncicantitruncated 8-simplex (gobteb)	(0,0,1,2,3,4,5,6,6)								362880	90720
t0,1,2,3,4,7	Heptisteriruncicantitruncated 8-simplex (xogacane)	(0,1,1,1,2,3,4,5,6)								302400	60480
t0,1,2,3,5,7	Heptipentiruncicantitruncated 8-simplex (xotagapane)	(0,1,1,2,2,3,4,5,6)								498960	90720
t0,1,2,4,5,7	Heptipentistericantitruncated 8-simplex (xotcagrane)	(0,1,1,2,3,3,4,5,6)								453600	90720
t0,1,3,4,5,7	Heptipentisteriruncitruncated 8-simplex (xotacaptane)	(0,1,1,2,3,4,4,5,6)								453600	90720
t0,2,3,4,5,7	Heptipentisteriruncicantellated 8-simplex (xotacaparb)	(0,1,1,2,3,4,5,5,6)								453600	90720
t0,1,2,3,6,7	Heptihexiruncicantitruncated 8-simplex (xupogapene)	(0,1,2,2,2,3,4,5,6)								302400	60480
t0,1,2,4,6,7	Heptihexistericantitruncated 8-simplex (xupcagrene)	(0,1,2,2,3,3,4,5,6)								498960	90720
t0,1,3,4,6,7	Heptihexisteriruncitruncated 8-simplex (xupacputob)	(0,1,2,2,3,4,4,5,6)								453600	90720
t0,1,2,5,6,7	Heptihexipenticantitruncated 8-simplex (xuptagrab)	(0,1,2,3,3,3,4,5,6)								302400	60480
t0,1,2,3,4,5,6	Hexipentisteriruncicantitruncated 8-simplex (gupane)	(0,0,1,2,3,4,5,6,7)								725760	181440
t0,1,2,3,4,5,7	Heptipentisteriruncicantitruncated 8-simplex (xogtane)	(0,1,1,2,3,4,5,6,7)								816480	181440
t0,1,2,3,4,6,7	Heptihexisteriruncicantitruncated 8-simplex (xupogacane)	(0,1,2,2,3,4,5,6,7)								816480	181440
t0,1,2,3,5,6,7	Heptihexipentiruncicantitruncated 8-simplex (xuptagapene)	(0,1,2,3,3,4,5,6,7)								816480	181440
t0,1,2,3,4,5,6,7	Omnitruncated 8-simplex (goxeb)	(0,1,2,3,4,5,6,7,8)								1451520	362880

The B₈ family

The B8 family has symmetry of order 10321920 (8 factorial x 28). There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

See also a list of B8 polytopes for symmetric Coxeter plane graphs of these polytopes.

#	Coxeter-Dynkin diagram	Schläflisymbol	Name	Element counts	7		6		5
t0{36,4}	8-orthoplexDiacosipentacontahexazetton (ek)	256	1024	1792	1792	1120	448	112	16
t1{36,4}	Rectified 8-orthoplexRectified diacosipentacontahexazetton (rek)	272	3072	8960	12544	10080	4928	1344	112
t2{36,4}	Birectified 8-orthoplexBirectified diacosipentacontahexazetton (bark)	272	3184	16128	34048	36960	22400	6720	448
t3{36,4}	Trirectified 8-orthoplexTrirectified diacosipentacontahexazetton (tark)	272	3184	16576	48384	71680	53760	17920	1120
t3{4,36}	Trirectified 8-cubeTrirectified octeract (tro)	272	3184	16576	47712	80640	71680	26880	1792
t2{4,36}	Birectified 8-cubeBirectified octeract (bro)	272	3184	14784	36960	55552	50176	21504	1792
t1{4,36}	Rectified 8-cubeRectified octeract (recto)	272	2160	7616	15456	19712	16128	7168	1024
t0{4,36}	8-cubeOcteract (octo)	16	112	448	1120	1792	1792	1024	256
t0,1{36,4}	Truncated 8-orthoplexTruncated diacosipentacontahexazetton (tek)							1456	224
t0,2{36,4}	Cantellated 8-orthoplexSmall rhombated diacosipentacontahexazetton (srek)							14784	1344
t1,2{36,4}	Bitruncated 8-orthoplexBitruncated diacosipentacontahexazetton (batek)							8064	1344
t0,3{36,4}	Runcinated 8-orthoplexSmall prismated diacosipentacontahexazetton (spek)							60480	4480
t1,3{36,4}	Bicantellated 8-orthoplexSmall birhombated diacosipentacontahexazetton (sabork)							67200	6720
t2,3{36,4}	Tritruncated 8-orthoplexTritruncated diacosipentacontahexazetton (tatek)							24640	4480
t0,4{36,4}	Stericated 8-orthoplexSmall cellated diacosipentacontahexazetton (scak)							125440	8960
t1,4{36,4}	Biruncinated 8-orthoplexSmall biprismated diacosipentacontahexazetton (sabpek)							215040	17920
t2,4{36,4}	Tricantellated 8-orthoplexSmall trirhombated diacosipentacontahexazetton (satrek)							161280	17920
t3,4{4,36}	Quadritruncated 8-cubeOcteractidiacosipentacontahexazetton (oke)							44800	8960
t0,5{36,4}	Pentellated 8-orthoplexSmall terated diacosipentacontahexazetton (setek)							134400	10752
t1,5{36,4}	Bistericated 8-orthoplexSmall bicellated diacosipentacontahexazetton (sibcak)							322560	26880
t2,5{4,36}	Triruncinated 8-cubeSmall triprismato-octeractidiacosipentacontahexazetton (sitpoke)							376320	35840
t2,4{4,36}	Tricantellated 8-cubeSmall trirhombated octeract (satro)							215040	26880
t2,3{4,36}	Tritruncated 8-cubeTritruncated octeract (tato)							48384	10752
t0,6{36,4}	Hexicated 8-orthoplexSmall petated diacosipentacontahexazetton (supek)							64512	7168
t1,6{4,36}	Bipentellated 8-cubeSmall biteri-octeractidiacosipentacontahexazetton (sabtoke)							215040	21504
t1,5{4,36}	Bistericated 8-cubeSmall bicellated octeract (sobco)							358400	35840
t1,4{4,36}	Biruncinated 8-cubeSmall biprismated octeract (sabepo)							322560	35840
t1,3{4,36}	Bicantellated 8-cubeSmall birhombated octeract (subro)							150528	21504
t1,2{4,36}	Bitruncated 8-cubeBitruncated octeract (bato)							28672	7168
t0,7{4,36}	Heptellated 8-cubeSmall exi-octeractidiacosipentacontahexazetton (saxoke)							14336	2048
t0,6{4,36}	Hexicated 8-cubeSmall petated octeract (supo)							64512	7168
t0,5{4,36}	Pentellated 8-cubeSmall terated octeract (soto)							143360	14336
t0,4{4,36}	Stericated 8-cubeSmall cellated octeract (soco)							179200	17920
t0,3{4,36}	Runcinated 8-cubeSmall prismated octeract (sopo)							129024	14336
t0,2{4,36}	Cantellated 8-cubeSmall rhombated octeract (soro)							50176	7168
t0,1{4,36}	Truncated 8-cubeTruncated octeract (tocto)							8192	2048
t0,1,2{36,4}	Cantitruncated 8-orthoplexGreat rhombated diacosipentacontahexazetton							16128	2688
t0,1,3{36,4}	Runcitruncated 8-orthoplexPrismatotruncated diacosipentacontahexazetton							127680	13440
t0,2,3{36,4}	Runcicantellated 8-orthoplexPrismatorhombated diacosipentacontahexazetton							80640	13440
t1,2,3{36,4}	Bicantitruncated 8-orthoplexGreat birhombated diacosipentacontahexazetton							73920	13440
t0,1,4{36,4}	Steritruncated 8-orthoplexCellitruncated diacosipentacontahexazetton							394240	35840
t0,2,4{36,4}	Stericantellated 8-orthoplexCellirhombated diacosipentacontahexazetton							483840	53760
t1,2,4{36,4}	Biruncitruncated 8-orthoplexBiprismatotruncated diacosipentacontahexazetton							430080	53760
t0,3,4{36,4}	Steriruncinated 8-orthoplexCelliprismated diacosipentacontahexazetton							215040	35840
t1,3,4{36,4}	Biruncicantellated 8-orthoplexBiprismatorhombated diacosipentacontahexazetton							322560	53760
t2,3,4{36,4}	Tricantitruncated 8-orthoplexGreat trirhombated diacosipentacontahexazetton							179200	35840
t0,1,5{36,4}	Pentitruncated 8-orthoplexTeritruncated diacosipentacontahexazetton							564480	53760
t0,2,5{36,4}	Penticantellated 8-orthoplexTerirhombated diacosipentacontahexazetton							1075200	107520
t1,2,5{36,4}	Bisteritruncated 8-orthoplexBicellitruncated diacosipentacontahexazetton							913920	107520
t0,3,5{36,4}	Pentiruncinated 8-orthoplexTeriprismated diacosipentacontahexazetton							913920	107520
t1,3,5{36,4}	Bistericantellated 8-orthoplexBicellirhombated diacosipentacontahexazetton							1290240	161280
t2,3,5{36,4}	Triruncitruncated 8-orthoplexTriprismatotruncated diacosipentacontahexazetton							698880	107520
t0,4,5{36,4}	Pentistericated 8-orthoplexTericellated diacosipentacontahexazetton							322560	53760
t1,4,5{36,4}	Bisteriruncinated 8-orthoplexBicelliprismated diacosipentacontahexazetton							698880	107520
t2,3,5{4,36}	Triruncitruncated 8-cubeTriprismatotruncated octeract							645120	107520
t2,3,4{4,36}	Tricantitruncated 8-cubeGreat trirhombated octeract							241920	53760
t0,1,6{36,4}	Hexitruncated 8-orthoplexPetitruncated diacosipentacontahexazetton							344064	43008
t0,2,6{36,4}	Hexicantellated 8-orthoplexPetirhombated diacosipentacontahexazetton							967680	107520
t1,2,6{36,4}	Bipentitruncated 8-orthoplexBiteritruncated diacosipentacontahexazetton							752640	107520
t0,3,6{36,4}	Hexiruncinated 8-orthoplexPetiprismated diacosipentacontahexazetton							1290240	143360
t1,3,6{36,4}	Bipenticantellated 8-orthoplexBiterirhombated diacosipentacontahexazetton							1720320	215040
t1,4,5{4,36}	Bisteriruncinated 8-cubeBicelliprismated octeract							860160	143360
t0,4,6{36,4}	Hexistericated 8-orthoplexPeticellated diacosipentacontahexazetton							860160	107520
t1,3,6{4,36}	Bipenticantellated 8-cubeBiterirhombated octeract							1720320	215040
t1,3,5{4,36}	Bistericantellated 8-cubeBicellirhombated octeract							1505280	215040
t1,3,4{4,36}	Biruncicantellated 8-cubeBiprismatorhombated octeract							537600	107520
t0,5,6{36,4}	Hexipentellated 8-orthoplexPetiterated diacosipentacontahexazetton							258048	43008
t1,2,6{4,36}	Bipentitruncated 8-cubeBiteritruncated octeract							752640	107520
t1,2,5{4,36}	Bisteritruncated 8-cubeBicellitruncated octeract							1003520	143360
t1,2,4{4,36}	Biruncitruncated 8-cubeBiprismatotruncated octeract							645120	107520
t1,2,3{4,36}	Bicantitruncated 8-cubeGreat birhombated octeract							172032	43008
t0,1,7{36,4}	Heptitruncated 8-orthoplexExitruncated diacosipentacontahexazetton							93184	14336
t0,2,7{36,4}	Hepticantellated 8-orthoplexExirhombated diacosipentacontahexazetton							365568	43008
t0,5,6{4,36}	Hexipentellated 8-cubePetiterated octeract							258048	43008
t0,3,7{36,4}	Heptiruncinated 8-orthoplexExiprismated diacosipentacontahexazetton							680960	71680
t0,4,6{4,36}	Hexistericated 8-cubePeticellated octeract							860160	107520
t0,4,5{4,36}	Pentistericated 8-cubeTericellated octeract							394240	71680
t0,3,7{4,36}	Heptiruncinated 8-cubeExiprismated octeract							680960	71680
t0,3,6{4,36}	Hexiruncinated 8-cubePetiprismated octeract							1290240	143360
t0,3,5{4,36}	Pentiruncinated 8-cubeTeriprismated octeract							1075200	143360
t0,3,4{4,36}	Steriruncinated 8-cubeCelliprismated octeract							358400	71680
t0,2,7{4,36}	Hepticantellated 8-cubeExirhombated octeract							365568	43008
t0,2,6{4,36}	Hexicantellated 8-cubePetirhombated octeract							967680	107520
t0,2,5{4,36}	Penticantellated 8-cubeTerirhombated octeract							1218560	143360
t0,2,4{4,36}	Stericantellated 8-cubeCellirhombated octeract							752640	107520
t0,2,3{4,36}	Runcicantellated 8-cubePrismatorhombated octeract							193536	43008
t0,1,7{4,36}	Heptitruncated 8-cubeExitruncated octeract							93184	14336
t0,1,6{4,36}	Hexitruncated 8-cubePetitruncated octeract							344064	43008
t0,1,5{4,36}	Pentitruncated 8-cubeTeritruncated octeract							609280	71680
t0,1,4{4,36}	Steritruncated 8-cubeCellitruncated octeract							573440	71680
t0,1,3{4,36}	Runcitruncated 8-cubePrismatotruncated octeract							279552	43008
t0,1,2{4,36}	Cantitruncated 8-cubeGreat rhombated octeract							57344	14336
t0,1,2,3{36,4}	Runcicantitruncated 8-orthoplexGreat prismated diacosipentacontahexazetton							147840	26880
t0,1,2,4{36,4}	Stericantitruncated 8-orthoplexCelligreatorhombated diacosipentacontahexazetton							860160	107520
t0,1,3,4{36,4}	Steriruncitruncated 8-orthoplexCelliprismatotruncated diacosipentacontahexazetton							591360	107520
t0,2,3,4{36,4}	Steriruncicantellated 8-orthoplexCelliprismatorhombated diacosipentacontahexazetton							591360	107520
t1,2,3,4{36,4}	Biruncicantitruncated 8-orthoplexGreat biprismated diacosipentacontahexazetton							537600	107520
t0,1,2,5{36,4}	Penticantitruncated 8-orthoplexTerigreatorhombated diacosipentacontahexazetton							1827840	215040
t0,1,3,5{36,4}	Pentiruncitruncated 8-orthoplexTeriprismatotruncated diacosipentacontahexazetton							2419200	322560
t0,2,3,5{36,4}	Pentiruncicantellated 8-orthoplexTeriprismatorhombated diacosipentacontahexazetton							2257920	322560
t1,2,3,5{36,4}	Bistericantitruncated 8-orthoplexBicelligreatorhombated diacosipentacontahexazetton							2096640	322560
t0,1,4,5{36,4}	Pentisteritruncated 8-orthoplexTericellitruncated diacosipentacontahexazetton							1182720	215040
t0,2,4,5{36,4}	Pentistericantellated 8-orthoplexTericellirhombated diacosipentacontahexazetton							1935360	322560
t1,2,4,5{36,4}	Bisteriruncitruncated 8-orthoplexBicelliprismatotruncated diacosipentacontahexazetton							1612800	322560
t0,3,4,5{36,4}	Pentisteriruncinated 8-orthoplexTericelliprismated diacosipentacontahexazetton							1182720	215040
t1,3,4,5{36,4}	Bisteriruncicantellated 8-orthoplexBicelliprismatorhombated diacosipentacontahexazetton							1774080	322560
t2,3,4,5{4,36}	Triruncicantitruncated 8-cubeGreat triprismato-octeractidiacosipentacontahexazetton							967680	215040
t0,1,2,6{36,4}	Hexicantitruncated 8-orthoplexPetigreatorhombated diacosipentacontahexazetton							1505280	215040
t0,1,3,6{36,4}	Hexiruncitruncated 8-orthoplexPetiprismatotruncated diacosipentacontahexazetton							3225600	430080
t0,2,3,6{36,4}	Hexiruncicantellated 8-orthoplexPetiprismatorhombated diacosipentacontahexazetton							2795520	430080
t1,2,3,6{36,4}	Bipenticantitruncated 8-orthoplexBiterigreatorhombated diacosipentacontahexazetton							2580480	430080
t0,1,4,6{36,4}	Hexisteritruncated 8-orthoplexPeticellitruncated diacosipentacontahexazetton							3010560	430080
t0,2,4,6{36,4}	Hexistericantellated 8-orthoplexPeticellirhombated diacosipentacontahexazetton							4515840	645120
t1,2,4,6{36,4}	Bipentiruncitruncated 8-orthoplexBiteriprismatotruncated diacosipentacontahexazetton							3870720	645120
t0,3,4,6{36,4}	Hexisteriruncinated 8-orthoplexPeticelliprismated diacosipentacontahexazetton							2580480	430080
t1,3,4,6{4,36}	Bipentiruncicantellated 8-cubeBiteriprismatorhombi-octeractidiacosipentacontahexazetton							3870720	645120
t1,3,4,5{4,36}	Bisteriruncicantellated 8-cubeBicelliprismatorhombated octeract							2150400	430080
t0,1,5,6{36,4}	Hexipentitruncated 8-orthoplexPetiteritruncated diacosipentacontahexazetton							1182720	215040
t0,2,5,6{36,4}	Hexipenticantellated 8-orthoplexPetiterirhombated diacosipentacontahexazetton							2795520	430080
t1,2,5,6{4,36}	Bipentisteritruncated 8-cubeBitericellitrunki-octeractidiacosipentacontahexazetton							2150400	430080
t0,3,5,6{36,4}	Hexipentiruncinated 8-orthoplexPetiteriprismated diacosipentacontahexazetton							2795520	430080
t1,2,4,6{4,36}	Bipentiruncitruncated 8-cubeBiteriprismatotruncated octeract							3870720	645120
t1,2,4,5{4,36}	Bisteriruncitruncated 8-cubeBicelliprismatotruncated octeract							1935360	430080
t0,4,5,6{36,4}	Hexipentistericated 8-orthoplexPetitericellated diacosipentacontahexazetton							1182720	215040
t1,2,3,6{4,36}	Bipenticantitruncated 8-cubeBiterigreatorhombated octeract							2580480	430080
t1,2,3,5{4,36}	Bistericantitruncated 8-cubeBicelligreatorhombated octeract							2365440	430080
t1,2,3,4{4,36}	Biruncicantitruncated 8-cubeGreat biprismated octeract							860160	215040
t0,1,2,7{36,4}	Hepticantitruncated 8-orthoplexExigreatorhombated diacosipentacontahexazetton							516096	86016
t0,1,3,7{36,4}	Heptiruncitruncated 8-orthoplexExiprismatotruncated diacosipentacontahexazetton							1612800	215040
t0,2,3,7{36,4}	Heptiruncicantellated 8-orthoplexExiprismatorhombated diacosipentacontahexazetton							1290240	215040
t0,4,5,6{4,36}	Hexipentistericated 8-cubePetitericellated octeract							1182720	215040
t0,1,4,7{36,4}	Heptisteritruncated 8-orthoplexExicellitruncated diacosipentacontahexazetton							2293760	286720
t0,2,4,7{36,4}	Heptistericantellated 8-orthoplexExicellirhombated diacosipentacontahexazetton							3225600	430080
t0,3,5,6{4,36}	Hexipentiruncinated 8-cubePetiteriprismated octeract							2795520	430080
t0,3,4,7{4,36}	Heptisteriruncinated 8-cubeExicelliprismato-octeractidiacosipentacontahexazetton							1720320	286720
t0,3,4,6{4,36}	Hexisteriruncinated 8-cubePeticelliprismated octeract							2580480	430080
t0,3,4,5{4,36}	Pentisteriruncinated 8-cubeTericelliprismated octeract							1433600	286720
t0,1,5,7{36,4}	Heptipentitruncated 8-orthoplexExiteritruncated diacosipentacontahexazetton							1612800	215040
t0,2,5,7{4,36}	Heptipenticantellated 8-cubeExiterirhombi-octeractidiacosipentacontahexazetton							3440640	430080
t0,2,5,6{4,36}	Hexipenticantellated 8-cubePetiterirhombated octeract							2795520	430080
t0,2,4,7{4,36}	Heptistericantellated 8-cubeExicellirhombated octeract							3225600	430080
t0,2,4,6{4,36}	Hexistericantellated 8-cubePeticellirhombated octeract							4515840	645120
t0,2,4,5{4,36}	Pentistericantellated 8-cubeTericellirhombated octeract							2365440	430080
t0,2,3,7{4,36}	Heptiruncicantellated 8-cubeExiprismatorhombated octeract							1290240	215040
t0,2,3,6{4,36}	Hexiruncicantellated 8-cubePetiprismatorhombated octeract							2795520	430080
t0,2,3,5{4,36}	Pentiruncicantellated 8-cubeTeriprismatorhombated octeract							2580480	430080
t0,2,3,4{4,36}	Steriruncicantellated 8-cubeCelliprismatorhombated octeract							967680	215040
t0,1,6,7{4,36}	Heptihexitruncated 8-cubeExipetitrunki-octeractidiacosipentacontahexazetton							516096	86016
t0,1,5,7{4,36}	Heptipentitruncated 8-cubeExiteritruncated octeract							1612800	215040
t0,1,5,6{4,36}	Hexipentitruncated 8-cubePetiteritruncated octeract							1182720	215040
t0,1,4,7{4,36}	Heptisteritruncated 8-cubeExicellitruncated octeract							2293760	286720
t0,1,4,6{4,36}	Hexisteritruncated 8-cubePeticellitruncated octeract							3010560	430080
t0,1,4,5{4,36}	Pentisteritruncated 8-cubeTericellitruncated octeract							1433600	286720
t0,1,3,7{4,36}	Heptiruncitruncated 8-cubeExiprismatotruncated octeract							1612800	215040
t0,1,3,6{4,36}	Hexiruncitruncated 8-cubePetiprismatotruncated octeract							3225600	430080
t0,1,3,5{4,36}	Pentiruncitruncated 8-cubeTeriprismatotruncated octeract							2795520	430080
t0,1,3,4{4,36}	Steriruncitruncated 8-cubeCelliprismatotruncated octeract							967680	215040
t0,1,2,7{4,36}	Hepticantitruncated 8-cubeExigreatorhombated octeract							516096	86016
t0,1,2,6{4,36}	Hexicantitruncated 8-cubePetigreatorhombated octeract							1505280	215040
t0,1,2,5{4,36}	Penticantitruncated 8-cubeTerigreatorhombated octeract							2007040	286720
t0,1,2,4{4,36}	Stericantitruncated 8-cubeCelligreatorhombated octeract							1290240	215040
t0,1,2,3{4,36}	Runcicantitruncated 8-cubeGreat prismated octeract							344064	86016
t0,1,2,3,4{36,4}	Steriruncicantitruncated 8-orthoplexGreat cellated diacosipentacontahexazetton							1075200	215040
t0,1,2,3,5{36,4}	Pentiruncicantitruncated 8-orthoplexTerigreatoprismated diacosipentacontahexazetton							4193280	645120
t0,1,2,4,5{36,4}	Pentistericantitruncated 8-orthoplexTericelligreatorhombated diacosipentacontahexazetton							3225600	645120
t0,1,3,4,5{36,4}	Pentisteriruncitruncated 8-orthoplexTericelliprismatotruncated diacosipentacontahexazetton							3225600	645120
t0,2,3,4,5{36,4}	Pentisteriruncicantellated 8-orthoplexTericelliprismatorhombated diacosipentacontahexazetton							3225600	645120
t1,2,3,4,5{36,4}	Bisteriruncicantitruncated 8-orthoplexGreat bicellated diacosipentacontahexazetton							2903040	645120
t0,1,2,3,6{36,4}	Hexiruncicantitruncated 8-orthoplexPetigreatoprismated diacosipentacontahexazetton							5160960	860160
t0,1,2,4,6{36,4}	Hexistericantitruncated 8-orthoplexPeticelligreatorhombated diacosipentacontahexazetton							7741440	1290240
t0,1,3,4,6{36,4}	Hexisteriruncitruncated 8-orthoplexPeticelliprismatotruncated diacosipentacontahexazetton							7096320	1290240
t0,2,3,4,6{36,4}	Hexisteriruncicantellated 8-orthoplexPeticelliprismatorhombated diacosipentacontahexazetton							7096320	1290240
t1,2,3,4,6{36,4}	Bipentiruncicantitruncated 8-orthoplexBiterigreatoprismated diacosipentacontahexazetton							6451200	1290240
t0,1,2,5,6{36,4}	Hexipenticantitruncated 8-orthoplexPetiterigreatorhombated diacosipentacontahexazetton							4300800	860160
t0,1,3,5,6{36,4}	Hexipentiruncitruncated 8-orthoplexPetiteriprismatotruncated diacosipentacontahexazetton							7096320	1290240
t0,2,3,5,6{36,4}	Hexipentiruncicantellated 8-orthoplexPetiteriprismatorhombated diacosipentacontahexazetton							6451200	1290240
t1,2,3,5,6{36,4}	Bipentistericantitruncated 8-orthoplexBitericelligreatorhombated diacosipentacontahexazetton							5806080	1290240
t0,1,4,5,6{36,4}	Hexipentisteritruncated 8-orthoplexPetitericellitruncated diacosipentacontahexazetton							4300800	860160
t0,2,4,5,6{36,4}	Hexipentistericantellated 8-orthoplexPetitericellirhombated diacosipentacontahexazetton							7096320	1290240
t1,2,3,5,6{4,36}	Bipentistericantitruncated 8-cubeBitericelligreatorhombated octeract							5806080	1290240
t0,3,4,5,6{36,4}	Hexipentisteriruncinated 8-orthoplexPetitericelliprismated diacosipentacontahexazetton							4300800	860160
t1,2,3,4,6{4,36}	Bipentiruncicantitruncated 8-cubeBiterigreatoprismated octeract							6451200	1290240
t1,2,3,4,5{4,36}	Bisteriruncicantitruncated 8-cubeGreat bicellated octeract							3440640	860160
t0,1,2,3,7{36,4}	Heptiruncicantitruncated 8-orthoplexExigreatoprismated diacosipentacontahexazetton							2365440	430080
t0,1,2,4,7{36,4}	Heptistericantitruncated 8-orthoplexExicelligreatorhombated diacosipentacontahexazetton							5591040	860160
t0,1,3,4,7{36,4}	Heptisteriruncitruncated 8-orthoplexExicelliprismatotruncated diacosipentacontahexazetton							4730880	860160
t0,2,3,4,7{36,4}	Heptisteriruncicantellated 8-orthoplexExicelliprismatorhombated diacosipentacontahexazetton							4730880	860160
t0,3,4,5,6{4,36}	Hexipentisteriruncinated 8-cubePetitericelliprismated octeract							4300800	860160
t0,1,2,5,7{36,4}	Heptipenticantitruncated 8-orthoplexExiterigreatorhombated diacosipentacontahexazetton							5591040	860160
t0,1,3,5,7{36,4}	Heptipentiruncitruncated 8-orthoplexExiteriprismatotruncated diacosipentacontahexazetton							8386560	1290240
t0,2,3,5,7{36,4}	Heptipentiruncicantellated 8-orthoplexExiteriprismatorhombated diacosipentacontahexazetton							7741440	1290240
t0,2,4,5,6{4,36}	Hexipentistericantellated 8-cubePetitericellirhombated octeract							7096320	1290240
t0,1,4,5,7{36,4}	Heptipentisteritruncated 8-orthoplexExitericellitruncated diacosipentacontahexazetton							4730880	860160
t0,2,3,5,7{4,36}	Heptipentiruncicantellated 8-cubeExiteriprismatorhombated octeract							7741440	1290240
t0,2,3,5,6{4,36}	Hexipentiruncicantellated 8-cubePetiteriprismatorhombated octeract							6451200	1290240
t0,2,3,4,7{4,36}	Heptisteriruncicantellated 8-cubeExicelliprismatorhombated octeract							4730880	860160
t0,2,3,4,6{4,36}	Hexisteriruncicantellated 8-cubePeticelliprismatorhombated octeract							7096320	1290240
t0,2,3,4,5{4,36}	Pentisteriruncicantellated 8-cubeTericelliprismatorhombated octeract							3870720	860160
t0,1,2,6,7{36,4}	Heptihexicantitruncated 8-orthoplexExipetigreatorhombated diacosipentacontahexazetton							2365440	430080
t0,1,3,6,7{36,4}	Heptihexiruncitruncated 8-orthoplexExipetiprismatotruncated diacosipentacontahexazetton							5591040	860160
t0,1,4,5,7{4,36}	Heptipentisteritruncated 8-cubeExitericellitruncated octeract							4730880	860160
t0,1,4,5,6{4,36}	Hexipentisteritruncated 8-cubePetitericellitruncated octeract							4300800	860160
t0,1,3,6,7{4,36}	Heptihexiruncitruncated 8-cubeExipetiprismatotruncated octeract							5591040	860160
t0,1,3,5,7{4,36}	Heptipentiruncitruncated 8-cubeExiteriprismatotruncated octeract							8386560	1290240
t0,1,3,5,6{4,36}	Hexipentiruncitruncated 8-cubePetiteriprismatotruncated octeract							7096320	1290240
t0,1,3,4,7{4,36}	Heptisteriruncitruncated 8-cubeExicelliprismatotruncated octeract							4730880	860160
t0,1,3,4,6{4,36}	Hexisteriruncitruncated 8-cubePeticelliprismatotruncated octeract							7096320	1290240
t0,1,3,4,5{4,36}	Pentisteriruncitruncated 8-cubeTericelliprismatotruncated octeract							3870720	860160
t0,1,2,6,7{4,36}	Heptihexicantitruncated 8-cubeExipetigreatorhombated octeract							2365440	430080
t0,1,2,5,7{4,36}	Heptipenticantitruncated 8-cubeExiterigreatorhombated octeract							5591040	860160
t0,1,2,5,6{4,36}	Hexipenticantitruncated 8-cubePetiterigreatorhombated octeract							4300800	860160
t0,1,2,4,7{4,36}	Heptistericantitruncated 8-cubeExicelligreatorhombated octeract							5591040	860160
t0,1,2,4,6{4,36}	Hexistericantitruncated 8-cubePeticelligreatorhombated octeract							7741440	1290240
t0,1,2,4,5{4,36}	Pentistericantitruncated 8-cubeTericelligreatorhombated octeract							3870720	860160
t0,1,2,3,7{4,36}	Heptiruncicantitruncated 8-cubeExigreatoprismated octeract							2365440	430080
t0,1,2,3,6{4,36}	Hexiruncicantitruncated 8-cubePetigreatoprismated octeract							5160960	860160
t0,1,2,3,5{4,36}	Pentiruncicantitruncated 8-cubeTerigreatoprismated octeract							4730880	860160
t0,1,2,3,4{4,36}	Steriruncicantitruncated 8-cubeGreat cellated octeract							1720320	430080
t0,1,2,3,4,5{36,4}	Pentisteriruncicantitruncated 8-orthoplexGreat terated diacosipentacontahexazetton							5806080	1290240
t0,1,2,3,4,6{36,4}	Hexisteriruncicantitruncated 8-orthoplexPetigreatocellated diacosipentacontahexazetton							12902400	2580480
t0,1,2,3,5,6{36,4}	Hexipentiruncicantitruncated 8-orthoplexPetiterigreatoprismated diacosipentacontahexazetton							11612160	2580480
t0,1,2,4,5,6{36,4}	Hexipentistericantitruncated 8-orthoplexPetitericelligreatorhombated diacosipentacontahexazetton							11612160	2580480
t0,1,3,4,5,6{36,4}	Hexipentisteriruncitruncated 8-orthoplexPetitericelliprismatotruncated diacosipentacontahexazetton							11612160	2580480
t0,2,3,4,5,6{36,4}	Hexipentisteriruncicantellated 8-orthoplexPetitericelliprismatorhombated diacosipentacontahexazetton							11612160	2580480
t1,2,3,4,5,6{4,36}	Bipentisteriruncicantitruncated 8-cubeGreat biteri-octeractidiacosipentacontahexazetton							10321920	2580480
t0,1,2,3,4,7{36,4}	Heptisteriruncicantitruncated 8-orthoplexExigreatocellated diacosipentacontahexazetton							8601600	1720320
t0,1,2,3,5,7{36,4}	Heptipentiruncicantitruncated 8-orthoplexExiterigreatoprismated diacosipentacontahexazetton							14192640	2580480
t0,1,2,4,5,7{36,4}	Heptipentistericantitruncated 8-orthoplexExitericelligreatorhombated diacosipentacontahexazetton							12902400	2580480
t0,1,3,4,5,7{36,4}	Heptipentisteriruncitruncated 8-orthoplexExitericelliprismatotruncated diacosipentacontahexazetton							12902400	2580480
t0,2,3,4,5,7{4,36}	Heptipentisteriruncicantellated 8-cubeExitericelliprismatorhombi-octeractidiacosipentacontahexazetton							12902400	2580480
t0,2,3,4,5,6{4,36}	Hexipentisteriruncicantellated 8-cubePetitericelliprismatorhombated octeract							11612160	2580480
t0,1,2,3,6,7{36,4}	Heptihexiruncicantitruncated 8-orthoplexExipetigreatoprismated diacosipentacontahexazetton							8601600	1720320
t0,1,2,4,6,7{36,4}	Heptihexistericantitruncated 8-orthoplexExipeticelligreatorhombated diacosipentacontahexazetton							14192640	2580480
t0,1,3,4,6,7{4,36}	Heptihexisteriruncitruncated 8-cubeExipeticelliprismatotrunki-octeractidiacosipentacontahexazetton							12902400	2580480
t0,1,3,4,5,7{4,36}	Heptipentisteriruncitruncated 8-cubeExitericelliprismatotruncated octeract							12902400	2580480
t0,1,3,4,5,6{4,36}	Hexipentisteriruncitruncated 8-cubePetitericelliprismatotruncated octeract							11612160	2580480
t0,1,2,5,6,7{4,36}	Heptihexipenticantitruncated 8-cubeExipetiterigreatorhombi-octeractidiacosipentacontahexazetton							8601600	1720320
t0,1,2,4,6,7{4,36}	Heptihexistericantitruncated 8-cubeExipeticelligreatorhombated octeract							14192640	2580480
t0,1,2,4,5,7{4,36}	Heptipentistericantitruncated 8-cubeExitericelligreatorhombated octeract							12902400	2580480
t0,1,2,4,5,6{4,36}	Hexipentistericantitruncated 8-cubePetitericelligreatorhombated octeract							11612160	2580480
t0,1,2,3,6,7{4,36}	Heptihexiruncicantitruncated 8-cubeExipetigreatoprismated octeract							8601600	1720320
t0,1,2,3,5,7{4,36}	Heptipentiruncicantitruncated 8-cubeExiterigreatoprismated octeract							14192640	2580480
t0,1,2,3,5,6{4,36}	Hexipentiruncicantitruncated 8-cubePetiterigreatoprismated octeract							11612160	2580480
t0,1,2,3,4,7{4,36}	Heptisteriruncicantitruncated 8-cubeExigreatocellated octeract							8601600	1720320
t0,1,2,3,4,6{4,36}	Hexisteriruncicantitruncated 8-cubePetigreatocellated octeract							12902400	2580480
t0,1,2,3,4,5{4,36}	Pentisteriruncicantitruncated 8-cubeGreat terated octeract							6881280	1720320
t0,1,2,3,4,5,6{36,4}	Hexipentisteriruncicantitruncated 8-orthoplexGreat petated diacosipentacontahexazetton							20643840	5160960
t0,1,2,3,4,5,7{36,4}	Heptipentisteriruncicantitruncated 8-orthoplexExigreatoterated diacosipentacontahexazetton							23224320	5160960
t0,1,2,3,4,6,7{36,4}	Heptihexisteriruncicantitruncated 8-orthoplexExipetigreatocellated diacosipentacontahexazetton							23224320	5160960
t0,1,2,3,5,6,7{36,4}	Heptihexipentiruncicantitruncated 8-orthoplexExipetiterigreatoprismated diacosipentacontahexazetton							23224320	5160960
t0,1,2,3,5,6,7{4,36}	Heptihexipentiruncicantitruncated 8-cubeExipetiterigreatoprismated octeract							23224320	5160960
t0,1,2,3,4,6,7{4,36}	Heptihexisteriruncicantitruncated 8-cubeExipetigreatocellated octeract							23224320	5160960
t0,1,2,3,4,5,7{4,36}	Heptipentisteriruncicantitruncated 8-cubeExigreatoterated octeract							23224320	5160960
t0,1,2,3,4,5,6{4,36}	Hexipentisteriruncicantitruncated 8-cubeGreat petated octeract							20643840	5160960
t0,1,2,3,4,5,6,7{4,36}	Omnitruncated 8-cubeGreat exi-octeractidiacosipentacontahexazetton							41287680	10321920

The D₈ family

The D8 family has symmetry of order 5,160,960 (8 factorial x 27).

This family has 191 Wythoffian uniform polytopes, from 3x64-1 permutations of the D8 Coxeter-Dynkin diagram with one or more rings. 127 (2x64-1) are repeated from the B8 family and 64 are unique to this family, all listed below.

See list of D8 polytopes for Coxeter plane graphs of these polytopes.

#	Coxeter-Dynkin diagram	Name	Base point(Alternately signed)	Element counts	Circumrad	7		6		5
=	8-demicubeh{4,3,3,3,3,3,3}	(1,1,1,1,1,1,1,1)	144	1136	4032	8288	10752	7168	1792	128	1.0000000
=	cantic 8-cubeh2{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,3)							23296	3584	2.6457512
=	runcic 8-cubeh3{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,3)							64512	7168	2.4494896
=	steric 8-cubeh4{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,3)							98560	8960	2.2360678
=	pentic 8-cubeh5{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,3)							89600	7168	1.9999999
=	hexic 8-cubeh6{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,3)							48384	3584	1.7320508
=	heptic 8-cubeh7{4,3,3,3,3,3,3}	(1,1,1,1,1,1,1,3)							14336	1024	1.4142135
=	runcicantic 8-cubeh2,3{4,3,3,3,3,3,3}	(1,1,3,5,5,5,5,5)							86016	21504	4.1231055
=	stericantic 8-cubeh2,4{4,3,3,3,3,3,3}	(1,1,3,3,5,5,5,5)							349440	53760	3.8729835
=	steriruncic 8-cubeh3,4{4,3,3,3,3,3,3}	(1,1,1,3,5,5,5,5)							179200	35840	3.7416575
=	penticantic 8-cubeh2,5{4,3,3,3,3,3,3}	(1,1,3,3,3,5,5,5)							573440	71680	3.6055512
=	pentiruncic 8-cubeh3,5{4,3,3,3,3,3,3}	(1,1,1,3,3,5,5,5)							537600	71680	3.4641016
=	pentisteric 8-cubeh4,5{4,3,3,3,3,3,3}	(1,1,1,1,3,5,5,5)							232960	35840	3.3166249
=	hexicantic 8-cubeh2,6{4,3,3,3,3,3,3}	(1,1,3,3,3,3,5,5)							456960	53760	3.3166249
=	hexicruncic 8-cubeh3,6{4,3,3,3,3,3,3}	(1,1,1,3,3,3,5,5)							645120	71680	3.1622777
=	hexisteric 8-cubeh4,6{4,3,3,3,3,3,3}	(1,1,1,1,3,3,5,5)							483840	53760	3
=	hexipentic 8-cubeh5,6{4,3,3,3,3,3,3}	(1,1,1,1,1,3,5,5)							182784	21504	2.8284271
=	hepticantic 8-cubeh2,7{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,5)							172032	21504	3
=	heptiruncic 8-cubeh3,7{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,5)							340480	35840	2.8284271
=	heptsteric 8-cubeh4,7{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,5)							376320	35840	2.6457512
=	heptipentic 8-cubeh5,7{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,5)							236544	21504	2.4494898
=	heptihexic 8-cubeh6,7{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,5)							78848	7168	2.236068
=	steriruncicantic 8-cubeh2,3,4{4,36}	(1,1,3,5,7,7,7,7)							430080	107520	5.3851647
=	pentiruncicantic 8-cubeh2,3,5{4,36}	(1,1,3,5,5,7,7,7)							1182720	215040	5.0990195
=	pentistericantic 8-cubeh2,4,5{4,36}	(1,1,3,3,5,7,7,7)							1075200	215040	4.8989797
=	pentisterirunic 8-cubeh3,4,5{4,36}	(1,1,1,3,5,7,7,7)							716800	143360	4.7958317
=	hexiruncicantic 8-cubeh2,3,6{4,36}	(1,1,3,5,5,5,7,7)							1290240	215040	4.7958317
=	hexistericantic 8-cubeh2,4,6{4,36}	(1,1,3,3,5,5,7,7)							2096640	322560	4.5825758
=	hexisterirunic 8-cubeh3,4,6{4,36}	(1,1,1,3,5,5,7,7)							1290240	215040	4.472136
=	hexipenticantic 8-cubeh2,5,6{4,36}	(1,1,3,3,3,5,7,7)							1290240	215040	4.3588991
=	hexipentirunic 8-cubeh3,5,6{4,36}	(1,1,1,3,3,5,7,7)							1397760	215040	4.2426405
=	hexipentisteric 8-cubeh4,5,6{4,36}	(1,1,1,1,3,5,7,7)							698880	107520	4.1231055
=	heptiruncicantic 8-cubeh2,3,7{4,36}	(1,1,3,5,5,5,5,7)							591360	107520	4.472136
=	heptistericantic 8-cubeh2,4,7{4,36}	(1,1,3,3,5,5,5,7)							1505280	215040	4.2426405
=	heptisterruncic 8-cubeh3,4,7{4,36}	(1,1,1,3,5,5,5,7)							860160	143360	4.1231055
=	heptipenticantic 8-cubeh2,5,7{4,36}	(1,1,3,3,3,5,5,7)							1612800	215040	4
=	heptipentiruncic 8-cubeh3,5,7{4,36}	(1,1,1,3,3,5,5,7)							1612800	215040	3.8729835
=	heptipentisteric 8-cubeh4,5,7{4,36}	(1,1,1,1,3,5,5,7)							752640	107520	3.7416575
=	heptihexicantic 8-cubeh2,6,7{4,36}	(1,1,3,3,3,3,5,7)							752640	107520	3.7416575
=	heptihexiruncic 8-cubeh3,6,7{4,36}	(1,1,1,3,3,3,5,7)							1146880	143360	3.6055512
=	heptihexisteric 8-cubeh4,6,7{4,36}	(1,1,1,1,3,3,5,7)							913920	107520	3.4641016
=	heptihexipentic 8-cubeh5,6,7{4,36}	(1,1,1,1,1,3,5,7)							365568	43008	3.3166249
=	pentisteriruncicantic 8-cubeh2,3,4,5{4,36}	(1,1,3,5,7,9,9,9)							1720320	430080	6.4031243
=	hexisteriruncicantic 8-cubeh2,3,4,6{4,36}	(1,1,3,5,7,7,9,9)							3225600	645120	6.0827627
=	hexipentiruncicantic 8-cubeh2,3,5,6{4,36}	(1,1,3,5,5,7,9,9)							2903040	645120	5.8309517
=	hexipentistericantic 8-cubeh2,4,5,6{4,36}	(1,1,3,3,5,7,9,9)							3225600	645120	5.6568542
=	hexipentisteriruncic 8-cubeh3,4,5,6{4,36}	(1,1,1,3,5,7,9,9)							2150400	430080	5.5677648
=	heptsteriruncicantic 8-cubeh2,3,4,7{4,36}	(1,1,3,5,7,7,7,9)							2150400	430080	5.7445626
=	heptipentiruncicantic 8-cubeh2,3,5,7{4,36}	(1,1,3,5,5,7,7,9)							3548160	645120	5.4772258
=	heptipentistericantic 8-cubeh2,4,5,7{4,36}	(1,1,3,3,5,7,7,9)							3548160	645120	5.291503
=	heptipentisteriruncic 8-cubeh3,4,5,7{4,36}	(1,1,1,3,5,7,7,9)							2365440	430080	5.1961527
=	heptihexiruncicantic 8-cubeh2,3,6,7{4,36}	(1,1,3,5,5,5,7,9)							2150400	430080	5.1961527
=	heptihexistericantic 8-cubeh2,4,6,7{4,36}	(1,1,3,3,5,5,7,9)							3870720	645120	5
=	heptihexisteriruncic 8-cubeh3,4,6,7{4,36}	(1,1,1,3,5,5,7,9)							2365440	430080	4.8989797
=	heptihexipenticantic 8-cubeh2,5,6,7{4,36}	(1,1,3,3,3,5,7,9)							2580480	430080	4.7958317
=	heptihexipentiruncic 8-cubeh3,5,6,7{4,36}	(1,1,1,3,3,5,7,9)							2795520	430080	4.6904159
=	heptihexipentisteric 8-cubeh4,5,6,7{4,36}	(1,1,1,1,3,5,7,9)							1397760	215040	4.5825758
=	hexipentisteriruncicantic 8-cubeh2,3,4,5,6{4,36}	(1,1,3,5,7,9,11,11)							5160960	1290240	7.1414285
=	heptipentisteriruncicantic 8-cubeh2,3,4,5,7{4,36}	(1,1,3,5,7,9,9,11)							5806080	1290240	6.78233
=	heptihexisteriruncicantic 8-cubeh2,3,4,6,7{4,36}	(1,1,3,5,7,7,9,11)							5806080	1290240	6.480741
=	heptihexipentiruncicantic 8-cubeh2,3,5,6,7{4,36}	(1,1,3,5,5,7,9,11)							5806080	1290240	6.244998
=	heptihexipentistericantic 8-cubeh2,4,5,6,7{4,36}	(1,1,3,3,5,7,9,11)							6451200	1290240	6.0827627
=	heptihexipentisteriruncic 8-cubeh3,4,5,6,7{4,36}	(1,1,1,3,5,7,9,11)							4300800	860160	6.0000000
=	heptihexipentisteriruncicantic 8-cubeh2,3,4,5,6,7{4,36}	(1,1,3,5,7,9,11,13)							2580480	10321920	7.5498347

The E₈ family

The E8 family has symmetry order 696,729,600.

There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Eight forms are shown below, 4 single-ringed, 3 truncations (2 rings), and the final omnitruncation are given below. Bowers-style acronym names are given for cross-referencing.

See also list of E8 polytopes for Coxeter plane graphs of this family.

E8 uniform polytopes	#	Coxeter-Dynkin diagram
Names	Element counts	7-faces	6-faces	5-faces	4-faces	Cells	Faces	Edges	Vertices
1		421 (fy)	19440	207360	483840	483840	241920	60480	6720	240
2		Truncated 421 (tiffy)							188160	13440
3		Rectified 421 (riffy)	19680	375840	1935360	3386880	2661120	1028160	181440	6720
4		Birectified 421 (borfy)	19680	382560	2600640	7741440	9918720	5806080	1451520	60480
5		Trirectified 421 (torfy)	19680	382560	2661120	9313920	16934400	14515200	4838400	241920
6		Rectified 142 (buffy)	19680	382560	2661120	9072000	16934400	16934400	7257600	483840
7		Rectified 241 (robay)	19680	313440	1693440	4717440	7257600	5322240	1451520	69120
8		241 (bay)	17520	144960	544320	1209600	1209600	483840	69120	2160
9		Truncated 241								138240
10		142 (bif)	2400	106080	725760	2298240	3628800	2419200	483840	17280
11		Truncated 142								967680
12		Omnitruncated 421								696729600

Regular and uniform honeycombs

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 7-space:

#	Coxeter group	Coxeter diagram
1	{\tilde{A}}_7	[3[8]]
2	{\tilde{C}}_7	[4,35,4]
3	{\tilde{B}}_7	[4,34,31,1]
4	{\tilde{D}}_7	[31,1,33,31,1]
5	{\tilde{E}}_7	[33,3,1]

Regular and uniform tessellations include:

{\tilde{A}}_7 29 uniquely ringed forms, including:
- 7-simplex honeycomb: {3[8]}
{\tilde{C}}_7 135 uniquely ringed forms, including:
- Regular 7-cube honeycomb: {4,34,4} = {4,34,31,1}, =
{\tilde{B}}_7 191 uniquely ringed forms, 127 shared with {\tilde{C}}_7, and 64 new, including:
- 7-demicube honeycomb: h{4,34,4} = {31,1,34,4}, =
{\tilde{D}}_7, [31,1,33,31,1]: 77 unique ring permutations, and 10 are new, the first Coxeter called a quarter 7-cubic honeycomb.
- , , , , , , , , ,
{\tilde{E}}_7 143 uniquely ringed forms, including:
- 133 honeycomb: {3,33,3},
- 331 honeycomb: {3,3,3,33,1},

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 8, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 4 paracompact hyperbolic Coxeter groups of rank 8, each generating uniform honeycombs in 7-space as permutations of rings of the Coxeter diagrams.

{\bar{P}}_7 = [3,3[7]]:	{\bar{Q}}_7 = [31,1,32,32,1]:	{\bar{S}}_7 = [4,33,32,1]:	{\bar{T}}_7 = [33,2,2]:

References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com,
  - (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
  - (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
  - (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

References

Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topology'', Princeton, 2008.

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