# Uniform 8-polytope

(Redirected from 8-polytope)

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:

1. {3,3,3,3,3,3,3} - 8-simplex
2. {4,3,3,3,3,3,3} - 8-cube
3. {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.[1]

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.[1]

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.[1]

## 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] 135
2 BC8 [4,36] 255
3 D8 [35,1,1] 191 (64 unique)
4 E8 [34,2,1] 255

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

1. 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 -
2. 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 -
3. 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 -
4. 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:

# Coxeter group Coxeter-Dynkin diagram
7+1
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]×[ ]
6+2
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]
6+1+1
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[ ]
5+3
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]
5+2+1
1 A5I2(p)A1 [3,3,3]×[p]×[ ]
2 B5I2(p)A1 [4,3,3]×[p]×[ ]
3 D5I2(p)A1 [32,1,1]×[p]×[ ]
5+1+1+1
1 A5A1A1A1 [3,3,3]×[ ]×[ ]×[ ]
2 B5A1A1A1 [4,3,3]×[ ]×[ ]×[ ]
3 D5A1A1A1 [32,1,1]×[ ]×[ ]×[ ]
4+4
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]
4+3+1
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]×[ ]
4+2+2
...
4+2+1+1
...
4+1+1+1+1
...
3+3+2
1 A3A3I2(p) [3,3]×[3,3]×[p]
2 B3A3I2(p) [4,3]×[3,3]×[p]
3 H3A3I2(p) [5,3]×[3,3]×[p]
4 B3B3I2(p) [4,3]×[4,3]×[p]
5 H3B3I2(p) [5,3]×[4,3]×[p]
6 H3H3I2(p) [5,3]×[5,3]×[p]
3+3+1+1
1 A32A12 [3,3]×[3,3]×[ ]×[ ]
2 B3A3A12 [4,3]×[3,3]×[ ]×[ ]
3 H3A3A12 [5,3]×[3,3]×[ ]×[ ]
4 B3B3A12 [4,3]×[4,3]×[ ]×[ ]
5 H3B3A12 [5,3]×[4,3]×[ ]×[ ]
6 H3H3A12 [5,3]×[5,3]×[ ]×[ ]
3+2+2+1
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]×[ ]
3+2+1+1+1
1 A3I2(p)A13 [3,3]×[p]×[ ]x[ ]×[ ]
2 B3I2(p)A13 [4,3]×[p]×[ ]x[ ]×[ ]
3 H3I2(p)A13 [5,3]×[p]×[ ]x[ ]×[ ]
3+1+1+1+1+1
1 A3A15 [3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2 B3A15 [4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
3 H3A15 [5,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2+2+2+2
1 I2(p)I2(q)I2(r)I2(s) [p]×[q]×[r]×[s]
2+2+2+1+1
1 I2(p)I2(q)I2(r)A12 [p]×[q]×[r]×[ ]×[ ]
2+2+1+1+1+1
2 I2(p)I2(q)A14 [p]×[q]×[ ]×[ ]×[ ]×[ ]
2+1+1+1+1+1+1
1 I2(p)A16 [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1+1
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 Truncation
indices
Johnson name Basepoint Element counts
7 6 5 4 3 2 1 0
1

t0 8-simplex (ene) (0,0,0,0,0,0,0,0,1) 9 36 84 126 126 84 36 9
2

t1 Rectified 8-simplex (rene) (0,0,0,0,0,0,0,1,1) 18 108 336 630 576 588 252 36
3

t2 Birectified 8-simplex (bene) (0,0,0,0,0,0,1,1,1) 18 144 588 1386 2016 1764 756 84
4

t3 Trirectified 8-simplex (trene) (0,0,0,0,0,1,1,1,1) 1260 126
5

t0,1 Truncated 8-simplex (tene) (0,0,0,0,0,0,0,1,2) 288 72
6

t0,2 Cantellated 8-simplex (0,0,0,0,0,0,1,1,2) 1764 252
7

t1,2 Bitruncated 8-simplex (0,0,0,0,0,0,1,2,2) 1008 252
8

t0,3 Runcinated 8-simplex (0,0,0,0,0,1,1,1,2) 4536 504
9

t1,3 Bicantellated 8-simplex (0,0,0,0,0,1,1,2,2) 5292 756
10

t2,3 Tritruncated 8-simplex (0,0,0,0,0,1,2,2,2) 2016 504
11

t0,4 Stericated 8-simplex (0,0,0,0,1,1,1,1,2) 6300 630
12

t1,4 Biruncinated 8-simplex (0,0,0,0,1,1,1,2,2) 11340 1260
13

t2,4 Tricantellated 8-simplex (0,0,0,0,1,1,2,2,2) 8820 1260
14

t3,4 Quadritruncated 8-simplex (0,0,0,0,1,2,2,2,2) 2520 630
15

t0,5 Pentellated 8-simplex (0,0,0,1,1,1,1,1,2) 5040 504
16

t1,5 Bistericated 8-simplex (0,0,0,1,1,1,1,2,2) 12600 1260
17

t2,5 Triruncinated 8-simplex (0,0,0,1,1,1,2,2,2) 15120 1680
18

t0,6 Hexicated 8-simplex (0,0,1,1,1,1,1,1,2) 2268 252
19

t1,6 Bipentellated 8-simplex (0,0,1,1,1,1,1,2,2) 7560 756
20

t0,7 Heptellated 8-simplex (0,1,1,1,1,1,1,1,2) 504 72
21

t0,1,2 Cantitruncated 8-simplex (0,0,0,0,0,0,1,2,3) 2016 504
22

t0,1,3 Runcitruncated 8-simplex (0,0,0,0,0,1,1,2,3) 9828 1512
23

t0,2,3 Runcicantellated 8-simplex (0,0,0,0,0,1,2,2,3) 6804 1512
24

t1,2,3 Bicantitruncated 8-simplex (0,0,0,0,0,1,2,3,3) 6048 1512
25

t0,1,4 Steritruncated 8-simplex (0,0,0,0,1,1,1,2,3) 20160 2520
26

t0,2,4 Stericantellated 8-simplex (0,0,0,0,1,1,2,2,3) 26460 3780
27

t1,2,4 Biruncitruncated 8-simplex (0,0,0,0,1,1,2,3,3) 22680 3780
28

t0,3,4 Steriruncinated 8-simplex (0,0,0,0,1,2,2,2,3) 12600 2520
29

t1,3,4 Biruncicantellated 8-simplex (0,0,0,0,1,2,2,3,3) 18900 3780
30

t2,3,4 Tricantitruncated 8-simplex (0,0,0,0,1,2,3,3,3) 10080 2520
31

t0,1,5 Pentitruncated 8-simplex (0,0,0,1,1,1,1,2,3) 21420 2520
32

t0,2,5 Penticantellated 8-simplex (0,0,0,1,1,1,2,2,3) 42840 5040
33

t1,2,5 Bisteritruncated 8-simplex (0,0,0,1,1,1,2,3,3) 35280 5040
34

t0,3,5 Pentiruncinated 8-simplex (0,0,0,1,1,2,2,2,3) 37800 5040
35

t1,3,5 Bistericantellated 8-simplex (0,0,0,1,1,2,2,3,3) 52920 7560
36

t2,3,5 Triruncitruncated 8-simplex (0,0,0,1,1,2,3,3,3) 27720 5040
37

t0,4,5 Pentistericated 8-simplex (0,0,0,1,2,2,2,2,3) 13860 2520
38

t1,4,5 Bisteriruncinated 8-simplex (0,0,0,1,2,2,2,3,3) 30240 5040
39

t0,1,6 Hexitruncated 8-simplex (0,0,1,1,1,1,1,2,3) 12096 1512
40

t0,2,6 Hexicantellated 8-simplex (0,0,1,1,1,1,2,2,3) 34020 3780
41

t1,2,6 Bipentitruncated 8-simplex (0,0,1,1,1,1,2,3,3) 26460 3780
42

t0,3,6 Hexiruncinated 8-simplex (0,0,1,1,1,2,2,2,3) 45360 5040
43

t1,3,6 Bipenticantellated 8-simplex (0,0,1,1,1,2,2,3,3) 60480 7560
44

t0,4,6 Hexistericated 8-simplex (0,0,1,1,2,2,2,2,3) 30240 3780
45

t0,5,6 Hexipentellated 8-simplex (0,0,1,2,2,2,2,2,3) 9072 1512
46

t0,1,7 Heptitruncated 8-simplex (0,1,1,1,1,1,1,2,3) 3276 504
47

t0,2,7 Hepticantellated 8-simplex (0,1,1,1,1,1,2,2,3) 12852 1512
48

t0,3,7 Heptiruncinated 8-simplex (0,1,1,1,1,2,2,2,3) 23940 2520
49

t0,1,2,3 Runcicantitruncated 8-simplex (0,0,0,0,0,1,2,3,4) 12096 3024
50

t0,1,2,4 Stericantitruncated 8-simplex (0,0,0,0,1,1,2,3,4) 45360 7560
51

t0,1,3,4 Steriruncitruncated 8-simplex (0,0,0,0,1,2,2,3,4) 34020 7560
52

t0,2,3,4 Steriruncicantellated 8-simplex (0,0,0,0,1,2,3,3,4) 34020 7560
53

t1,2,3,4 Biruncicantitruncated 8-simplex (0,0,0,0,1,2,3,4,4) 30240 7560
54

t0,1,2,5 Penticantitruncated 8-simplex (0,0,0,1,1,1,2,3,4) 70560 10080
55

t0,1,3,5 Pentiruncitruncated 8-simplex (0,0,0,1,1,2,2,3,4) 98280 15120
56

t0,2,3,5 Pentiruncicantellated 8-simplex (0,0,0,1,1,2,3,3,4) 90720 15120
57

t1,2,3,5 Bistericantitruncated 8-simplex (0,0,0,1,1,2,3,4,4) 83160 15120
58

t0,1,4,5 Pentisteritruncated 8-simplex (0,0,0,1,2,2,2,3,4) 50400 10080
59

t0,2,4,5 Pentistericantellated 8-simplex (0,0,0,1,2,2,3,3,4) 83160 15120
60

t1,2,4,5 Bisteriruncitruncated 8-simplex (0,0,0,1,2,2,3,4,4) 68040 15120
61

t0,3,4,5 Pentisteriruncinated 8-simplex (0,0,0,1,2,3,3,3,4) 50400 10080
62

t1,3,4,5 Bisteriruncicantellated 8-simplex (0,0,0,1,2,3,3,4,4) 75600 15120
63

t2,3,4,5 Triruncicantitruncated 8-simplex (0,0,0,1,2,3,4,4,4) 40320 10080
64

t0,1,2,6 Hexicantitruncated 8-simplex (0,0,1,1,1,1,2,3,4) 52920 7560
65

t0,1,3,6 Hexiruncitruncated 8-simplex (0,0,1,1,1,2,2,3,4) 113400 15120
66

t0,2,3,6 Hexiruncicantellated 8-simplex (0,0,1,1,1,2,3,3,4) 98280 15120
67

t1,2,3,6 Bipenticantitruncated 8-simplex (0,0,1,1,1,2,3,4,4) 90720 15120
68

t0,1,4,6 Hexisteritruncated 8-simplex (0,0,1,1,2,2,2,3,4) 105840 15120
69

t0,2,4,6 Hexistericantellated 8-simplex (0,0,1,1,2,2,3,3,4) 158760 22680
70

t1,2,4,6 Bipentiruncitruncated 8-simplex (0,0,1,1,2,2,3,4,4) 136080 22680
71

t0,3,4,6 Hexisteriruncinated 8-simplex (0,0,1,1,2,3,3,3,4) 90720 15120
72

t1,3,4,6 Bipentiruncicantellated 8-simplex (0,0,1,1,2,3,3,4,4) 136080 22680
73

t0,1,5,6 Hexipentitruncated 8-simplex (0,0,1,2,2,2,2,3,4) 41580 7560
74

t0,2,5,6 Hexipenticantellated 8-simplex (0,0,1,2,2,2,3,3,4) 98280 15120
75

t1,2,5,6 Bipentisteritruncated 8-simplex (0,0,1,2,2,2,3,4,4) 75600 15120
76

t0,3,5,6 Hexipentiruncinated 8-simplex (0,0,1,2,2,3,3,3,4) 98280 15120
77

t0,4,5,6 Hexipentistericated 8-simplex (0,0,1,2,3,3,3,3,4) 41580 7560
78

t0,1,2,7 Hepticantitruncated 8-simplex (0,1,1,1,1,1,2,3,4) 18144 3024
79

t0,1,3,7 Heptiruncitruncated 8-simplex (0,1,1,1,1,2,2,3,4) 56700 7560
80

t0,2,3,7 Heptiruncicantellated 8-simplex (0,1,1,1,1,2,3,3,4) 45360 7560
81

t0,1,4,7 Heptisteritruncated 8-simplex (0,1,1,1,2,2,2,3,4) 80640 10080
82

t0,2,4,7 Heptistericantellated 8-simplex (0,1,1,1,2,2,3,3,4) 113400 15120
83

t0,3,4,7 Heptisteriruncinated 8-simplex (0,1,1,1,2,3,3,3,4) 60480 10080
84

t0,1,5,7 Heptipentitruncated 8-simplex (0,1,1,2,2,2,2,3,4) 56700 7560
85

t0,2,5,7 Heptipenticantellated 8-simplex (0,1,1,2,2,2,3,3,4) 120960 15120
86

t0,1,6,7 Heptihexitruncated 8-simplex (0,1,2,2,2,2,2,3,4) 18144 3024
87

t0,1,2,3,4 Steriruncicantitruncated 8-simplex (0,0,0,0,1,2,3,4,5) 60480 15120
88

t0,1,2,3,5 Pentiruncicantitruncated 8-simplex (0,0,0,1,1,2,3,4,5) 166320 30240
89

t0,1,2,4,5 Pentistericantitruncated 8-simplex (0,0,0,1,2,2,3,4,5) 136080 30240
90

t0,1,3,4,5 Pentisteriruncitruncated 8-simplex (0,0,0,1,2,3,3,4,5) 136080 30240
91

t0,2,3,4,5 Pentisteriruncicantellated 8-simplex (0,0,0,1,2,3,4,4,5) 136080 30240
92

t1,2,3,4,5 Bisteriruncicantitruncated 8-simplex (0,0,0,1,2,3,4,5,5) 120960 30240
93

t0,1,2,3,6 Hexiruncicantitruncated 8-simplex (0,0,1,1,1,2,3,4,5) 181440 30240
94

t0,1,2,4,6 Hexistericantitruncated 8-simplex (0,0,1,1,2,2,3,4,5) 272160 45360
95

t0,1,3,4,6 Hexisteriruncitruncated 8-simplex (0,0,1,1,2,3,3,4,5) 249480 45360
96

t0,2,3,4,6 Hexisteriruncicantellated 8-simplex (0,0,1,1,2,3,4,4,5) 249480 45360
97

t1,2,3,4,6 Bipentiruncicantitruncated 8-simplex (0,0,1,1,2,3,4,5,5) 226800 45360
98

t0,1,2,5,6 Hexipenticantitruncated 8-simplex (0,0,1,2,2,2,3,4,5) 151200 30240
99

t0,1,3,5,6 Hexipentiruncitruncated 8-simplex (0,0,1,2,2,3,3,4,5) 249480 45360
100

t0,2,3,5,6 Hexipentiruncicantellated 8-simplex (0,0,1,2,2,3,4,4,5) 226800 45360
101

t1,2,3,5,6 Bipentistericantitruncated 8-simplex (0,0,1,2,2,3,4,5,5) 204120 45360
102

t0,1,4,5,6 Hexipentisteritruncated 8-simplex (0,0,1,2,3,3,3,4,5) 151200 30240
103

t0,2,4,5,6 Hexipentistericantellated 8-simplex (0,0,1,2,3,3,4,4,5) 249480 45360
104

t0,3,4,5,6 Hexipentisteriruncinated 8-simplex (0,0,1,2,3,4,4,4,5) 151200 30240
105

t0,1,2,3,7 Heptiruncicantitruncated 8-simplex (0,1,1,1,1,2,3,4,5) 83160 15120
106

t0,1,2,4,7 Heptistericantitruncated 8-simplex (0,1,1,1,2,2,3,4,5) 196560 30240
107

t0,1,3,4,7 Heptisteriruncitruncated 8-simplex (0,1,1,1,2,3,3,4,5) 166320 30240
108

t0,2,3,4,7 Heptisteriruncicantellated 8-simplex (0,1,1,1,2,3,4,4,5) 166320 30240
109

t0,1,2,5,7 Heptipenticantitruncated 8-simplex (0,1,1,2,2,2,3,4,5) 196560 30240
110

t0,1,3,5,7 Heptipentiruncitruncated 8-simplex (0,1,1,2,2,3,3,4,5) 294840 45360
111

t0,2,3,5,7 Heptipentiruncicantellated 8-simplex (0,1,1,2,2,3,4,4,5) 272160 45360
112

t0,1,4,5,7 Heptipentisteritruncated 8-simplex (0,1,1,2,3,3,3,4,5) 166320 30240
113

t0,1,2,6,7 Heptihexicantitruncated 8-simplex (0,1,2,2,2,2,3,4,5) 83160 15120
114

t0,1,3,6,7 Heptihexiruncitruncated 8-simplex (0,1,2,2,2,3,3,4,5) 196560 30240
115

t0,1,2,3,4,5 Pentisteriruncicantitruncated 8-simplex (0,0,0,1,2,3,4,5,6) 241920 60480
116

t0,1,2,3,4,6 Hexisteriruncicantitruncated 8-simplex (0,0,1,1,2,3,4,5,6) 453600 90720
117

t0,1,2,3,5,6 Hexipentiruncicantitruncated 8-simplex (0,0,1,2,2,3,4,5,6) 408240 90720
118

t0,1,2,4,5,6 Hexipentistericantitruncated 8-simplex (0,0,1,2,3,3,4,5,6) 408240 90720
119

t0,1,3,4,5,6 Hexipentisteriruncitruncated 8-simplex (0,0,1,2,3,4,4,5,6) 408240 90720
120

t0,2,3,4,5,6 Hexipentisteriruncicantellated 8-simplex (0,0,1,2,3,4,5,5,6) 408240 90720
121

t1,2,3,4,5,6 Bipentisteriruncicantitruncated 8-simplex (0,0,1,2,3,4,5,6,6) 362880 90720
122

t0,1,2,3,4,7 Heptisteriruncicantitruncated 8-simplex (0,1,1,1,2,3,4,5,6) 302400 60480
123

t0,1,2,3,5,7 Heptipentiruncicantitruncated 8-simplex (0,1,1,2,2,3,4,5,6) 498960 90720
124

t0,1,2,4,5,7 Heptipentistericantitruncated 8-simplex (0,1,1,2,3,3,4,5,6) 453600 90720
125

t0,1,3,4,5,7 Heptipentisteriruncitruncated 8-simplex (0,1,1,2,3,4,4,5,6) 453600 90720
126

t0,2,3,4,5,7 Heptipentisteriruncicantellated 8-simplex (0,1,1,2,3,4,5,5,6) 453600 90720
127

t0,1,2,3,6,7 Heptihexiruncicantitruncated 8-simplex (0,1,2,2,2,3,4,5,6) 302400 60480
128

t0,1,2,4,6,7 Heptihexistericantitruncated 8-simplex (0,1,2,2,3,3,4,5,6) 498960 90720
129

t0,1,3,4,6,7 Heptihexisteriruncitruncated 8-simplex (0,1,2,2,3,4,4,5,6) 453600 90720
130

t0,1,2,5,6,7 Heptihexipenticantitruncated 8-simplex (0,1,2,3,3,3,4,5,6) 302400 60480
131

t0,1,2,3,4,5,6 Hexipentisteriruncicantitruncated 8-simplex (0,0,1,2,3,4,5,6,7) 725760 181440
132

t0,1,2,3,4,5,7 Heptipentisteriruncicantitruncated 8-simplex (0,1,1,2,3,4,5,6,7) 816480 181440
133

t0,1,2,3,4,6,7 Heptihexisteriruncicantitruncated 8-simplex (0,1,2,2,3,4,5,6,7) 816480 181440
134

t0,1,2,3,5,6,7 Heptihexipentiruncicantitruncated 8-simplex (0,1,2,3,3,4,5,6,7) 816480 181440
135

t0,1,2,3,4,5,6,7 Omnitruncated 8-simplex (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 Schläfli
symbol
Name Element counts
7 6 5 4 3 2 1 0
1 t0{36,4} 8-orthoplex
Diacosipentacontahexazetton (ek)
256 1024 1792 1792 1120 448 112 16
2 t1{36,4} Rectified 8-orthoplex
Rectified diacosipentacontahexazetton (rek)
272 3072 8960 12544 10080 4928 1344 112
3 t2{36,4} Birectified 8-orthoplex
Birectified diacosipentacontahexazetton (bark)
272 3184 16128 34048 36960 22400 6720 448
4 t3{36,4} Trirectified 8-orthoplex
Trirectified diacosipentacontahexazetton (tark)
272 3184 16576 48384 71680 53760 17920 1120
5 t3{4,36} Trirectified 8-cube
Trirectified octeract (tro)
272 3184 16576 47712 80640 71680 26880 1792
6 t2{4,36} Birectified 8-cube
Birectified octeract (bro)
272 3184 14784 36960 55552 50176 21504 1792
7 t1{4,36} Rectified 8-cube
Rectified octeract (recto)
272 2160 7616 15456 19712 16128 7168 1024
8 t0{4,36} 8-cube
Octeract (octo)
16 112 448 1120 1792 1792 1024 256
9 t0,1{36,4} Truncated 8-orthoplex
Truncated diacosipentacontahexazetton (tek)
1456 224
10 t0,2{36,4} Cantellated 8-orthoplex
Small rhombated diacosipentacontahexazetton (srek)
14784 1344
11 t1,2{36,4} Bitruncated 8-orthoplex
Bitruncated diacosipentacontahexazetton (batek)
8064 1344
12 t0,3{36,4} Runcinated 8-orthoplex
Small prismated diacosipentacontahexazetton (spek)
60480 4480
13 t1,3{36,4} Bicantellated 8-orthoplex
Small birhombated diacosipentacontahexazetton (sabork)
67200 6720
14 t2,3{36,4} Tritruncated 8-orthoplex
Tritruncated diacosipentacontahexazetton (tatek)
24640 4480
15 t0,4{36,4} Stericated 8-orthoplex
Small cellated diacosipentacontahexazetton (scak)
125440 8960
16 t1,4{36,4} Biruncinated 8-orthoplex
Small biprismated diacosipentacontahexazetton (sabpek)
215040 17920
17 t2,4{36,4} Tricantellated 8-orthoplex
Small trirhombated diacosipentacontahexazetton (satrek)
161280 17920
Octeractidiacosipentacontahexazetton (oke)
44800 8960
19 t0,5{36,4} Pentellated 8-orthoplex
Small terated diacosipentacontahexazetton (setek)
134400 10752
20 t1,5{36,4} Bistericated 8-orthoplex
Small bicellated diacosipentacontahexazetton (sibcak)
322560 26880
21 t2,5{4,36} Triruncinated 8-cube
Small triprismato-octeractidiacosipentacontahexazetton (sitpoke)
376320 35840
22 t2,4{4,36} Tricantellated 8-cube
Small trirhombated octeract (satro)
215040 26880
23 t2,3{4,36} Tritruncated 8-cube
Tritruncated octeract (tato)
48384 10752
24 t0,6{36,4} Hexicated 8-orthoplex
Small petated diacosipentacontahexazetton (supek)
64512 7168
25 t1,6{4,36} Bipentellated 8-cube
Small biteri-octeractidiacosipentacontahexazetton (sabtoke)
215040 21504
26 t1,5{4,36} Bistericated 8-cube
Small bicellated octeract (sobco)
358400 35840
27 t1,4{4,36} Biruncinated 8-cube
Small biprismated octeract (sabepo)
322560 35840
28 t1,3{4,36} Bicantellated 8-cube
Small birhombated octeract (subro)
150528 21504
29 t1,2{4,36} Bitruncated 8-cube
Bitruncated octeract (bato)
28672 7168
30 t0,7{4,36} Heptellated 8-cube
Small exi-octeractidiacosipentacontahexazetton (saxoke)
14336 2048
31 t0,6{4,36} Hexicated 8-cube
Small petated octeract (supo)
64512 7168
32 t0,5{4,36} Pentellated 8-cube
Small terated octeract (soto)
143360 14336
33 t0,4{4,36} Stericated 8-cube
Small cellated octeract (soco)
179200 17920
34 t0,3{4,36} Runcinated 8-cube
Small prismated octeract (sopo)
129024 14336
35 t0,2{4,36} Cantellated 8-cube
Small rhombated octeract (soro)
50176 7168
36 t0,1{4,36} Truncated 8-cube
Truncated octeract (tocto)
8192 2048
37 t0,1,2{36,4} Cantitruncated 8-orthoplex
Great rhombated diacosipentacontahexazetton
16128 2688
38 t0,1,3{36,4} Runcitruncated 8-orthoplex
Prismatotruncated diacosipentacontahexazetton
127680 13440
39 t0,2,3{36,4} Runcicantellated 8-orthoplex
Prismatorhombated diacosipentacontahexazetton
80640 13440
40 t1,2,3{36,4} Bicantitruncated 8-orthoplex
Great birhombated diacosipentacontahexazetton
73920 13440
41 t0,1,4{36,4} Steritruncated 8-orthoplex
Cellitruncated diacosipentacontahexazetton
394240 35840
42 t0,2,4{36,4} Stericantellated 8-orthoplex
Cellirhombated diacosipentacontahexazetton
483840 53760
43 t1,2,4{36,4} Biruncitruncated 8-orthoplex
Biprismatotruncated diacosipentacontahexazetton
430080 53760
44 t0,3,4{36,4} Steriruncinated 8-orthoplex
Celliprismated diacosipentacontahexazetton
215040 35840
45 t1,3,4{36,4} Biruncicantellated 8-orthoplex
Biprismatorhombated diacosipentacontahexazetton
322560 53760
46 t2,3,4{36,4} Tricantitruncated 8-orthoplex
Great trirhombated diacosipentacontahexazetton
179200 35840
47 t0,1,5{36,4} Pentitruncated 8-orthoplex
Teritruncated diacosipentacontahexazetton
564480 53760
48 t0,2,5{36,4} Penticantellated 8-orthoplex
Terirhombated diacosipentacontahexazetton
1075200 107520
49 t1,2,5{36,4} Bisteritruncated 8-orthoplex
Bicellitruncated diacosipentacontahexazetton
913920 107520
50 t0,3,5{36,4} Pentiruncinated 8-orthoplex
Teriprismated diacosipentacontahexazetton
913920 107520
51 t1,3,5{36,4} Bistericantellated 8-orthoplex
Bicellirhombated diacosipentacontahexazetton
1290240 161280
52 t2,3,5{36,4} Triruncitruncated 8-orthoplex
Triprismatotruncated diacosipentacontahexazetton
698880 107520
53 t0,4,5{36,4} Pentistericated 8-orthoplex
Tericellated diacosipentacontahexazetton
322560 53760
54 t1,4,5{36,4} Bisteriruncinated 8-orthoplex
Bicelliprismated diacosipentacontahexazetton
698880 107520
55 t2,3,5{4,36} Triruncitruncated 8-cube
Triprismatotruncated octeract
645120 107520
56 t2,3,4{4,36} Tricantitruncated 8-cube
Great trirhombated octeract
241920 53760
57 t0,1,6{36,4} Hexitruncated 8-orthoplex
Petitruncated diacosipentacontahexazetton
344064 43008
58 t0,2,6{36,4} Hexicantellated 8-orthoplex
Petirhombated diacosipentacontahexazetton
967680 107520
59 t1,2,6{36,4} Bipentitruncated 8-orthoplex
Biteritruncated diacosipentacontahexazetton
752640 107520
60 t0,3,6{36,4} Hexiruncinated 8-orthoplex
Petiprismated diacosipentacontahexazetton
1290240 143360
61 t1,3,6{36,4} Bipenticantellated 8-orthoplex
Biterirhombated diacosipentacontahexazetton
1720320 215040
62 t1,4,5{4,36} Bisteriruncinated 8-cube
Bicelliprismated octeract
860160 143360
63 t0,4,6{36,4} Hexistericated 8-orthoplex
Peticellated diacosipentacontahexazetton
860160 107520
64 t1,3,6{4,36} Bipenticantellated 8-cube
Biterirhombated octeract
1720320 215040
65 t1,3,5{4,36} Bistericantellated 8-cube
Bicellirhombated octeract
1505280 215040
66 t1,3,4{4,36} Biruncicantellated 8-cube
Biprismatorhombated octeract
537600 107520
67 t0,5,6{36,4} Hexipentellated 8-orthoplex
Petiterated diacosipentacontahexazetton
258048 43008
68 t1,2,6{4,36} Bipentitruncated 8-cube
Biteritruncated octeract
752640 107520
69 t1,2,5{4,36} Bisteritruncated 8-cube
Bicellitruncated octeract
1003520 143360
70 t1,2,4{4,36} Biruncitruncated 8-cube
Biprismatotruncated octeract
645120 107520
71 t1,2,3{4,36} Bicantitruncated 8-cube
Great birhombated octeract
172032 43008
72 t0,1,7{36,4} Heptitruncated 8-orthoplex
Exitruncated diacosipentacontahexazetton
93184 14336
73 t0,2,7{36,4} Hepticantellated 8-orthoplex
Exirhombated diacosipentacontahexazetton
365568 43008
74 t0,5,6{4,36} Hexipentellated 8-cube
Petiterated octeract
258048 43008
75 t0,3,7{36,4} Heptiruncinated 8-orthoplex
Exiprismated diacosipentacontahexazetton
680960 71680
76 t0,4,6{4,36} Hexistericated 8-cube
Peticellated octeract
860160 107520
77 t0,4,5{4,36} Pentistericated 8-cube
Tericellated octeract
394240 71680
78 t0,3,7{4,36} Heptiruncinated 8-cube
Exiprismated octeract
680960 71680
79 t0,3,6{4,36} Hexiruncinated 8-cube
Petiprismated octeract
1290240 143360
80 t0,3,5{4,36} Pentiruncinated 8-cube
Teriprismated octeract
1075200 143360
81 t0,3,4{4,36} Steriruncinated 8-cube
Celliprismated octeract
358400 71680
82 t0,2,7{4,36} Hepticantellated 8-cube
Exirhombated octeract
365568 43008
83 t0,2,6{4,36} Hexicantellated 8-cube
Petirhombated octeract
967680 107520
84 t0,2,5{4,36} Penticantellated 8-cube
Terirhombated octeract
1218560 143360
85 t0,2,4{4,36} Stericantellated 8-cube
Cellirhombated octeract
752640 107520
86 t0,2,3{4,36} Runcicantellated 8-cube
Prismatorhombated octeract
193536 43008
87 t0,1,7{4,36} Heptitruncated 8-cube
Exitruncated octeract
93184 14336
88 t0,1,6{4,36} Hexitruncated 8-cube
Petitruncated octeract
344064 43008
89 t0,1,5{4,36} Pentitruncated 8-cube
Teritruncated octeract
609280 71680
90 t0,1,4{4,36} Steritruncated 8-cube
Cellitruncated octeract
573440 71680
91 t0,1,3{4,36} Runcitruncated 8-cube
Prismatotruncated octeract
279552 43008
92 t0,1,2{4,36} Cantitruncated 8-cube
Great rhombated octeract
57344 14336
93 t0,1,2,3{36,4} Runcicantitruncated 8-orthoplex
Great prismated diacosipentacontahexazetton
147840 26880
94 t0,1,2,4{36,4} Stericantitruncated 8-orthoplex
Celligreatorhombated diacosipentacontahexazetton
860160 107520
95 t0,1,3,4{36,4} Steriruncitruncated 8-orthoplex
Celliprismatotruncated diacosipentacontahexazetton
591360 107520
96 t0,2,3,4{36,4} Steriruncicantellated 8-orthoplex
Celliprismatorhombated diacosipentacontahexazetton
591360 107520
97 t1,2,3,4{36,4} Biruncicantitruncated 8-orthoplex
Great biprismated diacosipentacontahexazetton
537600 107520
98 t0,1,2,5{36,4} Penticantitruncated 8-orthoplex
Terigreatorhombated diacosipentacontahexazetton
1827840 215040
99 t0,1,3,5{36,4} Pentiruncitruncated 8-orthoplex
Teriprismatotruncated diacosipentacontahexazetton
2419200 322560
100 t0,2,3,5{36,4} Pentiruncicantellated 8-orthoplex
Teriprismatorhombated diacosipentacontahexazetton
2257920 322560
101 t1,2,3,5{36,4} Bistericantitruncated 8-orthoplex
Bicelligreatorhombated diacosipentacontahexazetton
2096640 322560
102 t0,1,4,5{36,4} Pentisteritruncated 8-orthoplex
Tericellitruncated diacosipentacontahexazetton
1182720 215040
103 t0,2,4,5{36,4} Pentistericantellated 8-orthoplex
Tericellirhombated diacosipentacontahexazetton
1935360 322560
104 t1,2,4,5{36,4} Bisteriruncitruncated 8-orthoplex
Bicelliprismatotruncated diacosipentacontahexazetton
1612800 322560
105 t0,3,4,5{36,4} Pentisteriruncinated 8-orthoplex
Tericelliprismated diacosipentacontahexazetton
1182720 215040
106 t1,3,4,5{36,4} Bisteriruncicantellated 8-orthoplex
Bicelliprismatorhombated diacosipentacontahexazetton
1774080 322560
107 t2,3,4,5{4,36} Triruncicantitruncated 8-cube
Great triprismato-octeractidiacosipentacontahexazetton
967680 215040
108 t0,1,2,6{36,4} Hexicantitruncated 8-orthoplex
Petigreatorhombated diacosipentacontahexazetton
1505280 215040
109 t0,1,3,6{36,4} Hexiruncitruncated 8-orthoplex
Petiprismatotruncated diacosipentacontahexazetton
3225600 430080
110 t0,2,3,6{36,4} Hexiruncicantellated 8-orthoplex
Petiprismatorhombated diacosipentacontahexazetton
2795520 430080
111 t1,2,3,6{36,4} Bipenticantitruncated 8-orthoplex
Biterigreatorhombated diacosipentacontahexazetton
2580480 430080
112 t0,1,4,6{36,4} Hexisteritruncated 8-orthoplex
Peticellitruncated diacosipentacontahexazetton
3010560 430080
113 t0,2,4,6{36,4} Hexistericantellated 8-orthoplex
Peticellirhombated diacosipentacontahexazetton
4515840 645120
114 t1,2,4,6{36,4} Bipentiruncitruncated 8-orthoplex
Biteriprismatotruncated diacosipentacontahexazetton
3870720 645120
115 t0,3,4,6{36,4} Hexisteriruncinated 8-orthoplex
Peticelliprismated diacosipentacontahexazetton
2580480 430080
116 t1,3,4,6{4,36} Bipentiruncicantellated 8-cube
Biteriprismatorhombi-octeractidiacosipentacontahexazetton
3870720 645120
117 t1,3,4,5{4,36} Bisteriruncicantellated 8-cube
Bicelliprismatorhombated octeract
2150400 430080
118 t0,1,5,6{36,4} Hexipentitruncated 8-orthoplex
Petiteritruncated diacosipentacontahexazetton
1182720 215040
119 t0,2,5,6{36,4} Hexipenticantellated 8-orthoplex
Petiterirhombated diacosipentacontahexazetton
2795520 430080
120 t1,2,5,6{4,36} Bipentisteritruncated 8-cube
Bitericellitrunki-octeractidiacosipentacontahexazetton
2150400 430080
121 t0,3,5,6{36,4} Hexipentiruncinated 8-orthoplex
Petiteriprismated diacosipentacontahexazetton
2795520 430080
122 t1,2,4,6{4,36} Bipentiruncitruncated 8-cube
Biteriprismatotruncated octeract
3870720 645120
123 t1,2,4,5{4,36} Bisteriruncitruncated 8-cube
Bicelliprismatotruncated octeract
1935360 430080
124 t0,4,5,6{36,4} Hexipentistericated 8-orthoplex
Petitericellated diacosipentacontahexazetton
1182720 215040
125 t1,2,3,6{4,36} Bipenticantitruncated 8-cube
Biterigreatorhombated octeract
2580480 430080
126 t1,2,3,5{4,36} Bistericantitruncated 8-cube
Bicelligreatorhombated octeract
2365440 430080
127 t1,2,3,4{4,36} Biruncicantitruncated 8-cube
Great biprismated octeract
860160 215040
128 t0,1,2,7{36,4} Hepticantitruncated 8-orthoplex
Exigreatorhombated diacosipentacontahexazetton
516096 86016
129 t0,1,3,7{36,4} Heptiruncitruncated 8-orthoplex
Exiprismatotruncated diacosipentacontahexazetton
1612800 215040
130 t0,2,3,7{36,4} Heptiruncicantellated 8-orthoplex
Exiprismatorhombated diacosipentacontahexazetton
1290240 215040
131 t0,4,5,6{4,36} Hexipentistericated 8-cube
Petitericellated octeract
1182720 215040
132 t0,1,4,7{36,4} Heptisteritruncated 8-orthoplex
Exicellitruncated diacosipentacontahexazetton
2293760 286720
133 t0,2,4,7{36,4} Heptistericantellated 8-orthoplex
Exicellirhombated diacosipentacontahexazetton
3225600 430080
134 t0,3,5,6{4,36} Hexipentiruncinated 8-cube
Petiteriprismated octeract
2795520 430080
135 t0,3,4,7{4,36} Heptisteriruncinated 8-cube
Exicelliprismato-octeractidiacosipentacontahexazetton
1720320 286720
136 t0,3,4,6{4,36} Hexisteriruncinated 8-cube
Peticelliprismated octeract
2580480 430080
137 t0,3,4,5{4,36} Pentisteriruncinated 8-cube
Tericelliprismated octeract
1433600 286720
138 t0,1,5,7{36,4} Heptipentitruncated 8-orthoplex
Exiteritruncated diacosipentacontahexazetton
1612800 215040
139 t0,2,5,7{4,36} Heptipenticantellated 8-cube
Exiterirhombi-octeractidiacosipentacontahexazetton
3440640 430080
140 t0,2,5,6{4,36} Hexipenticantellated 8-cube
Petiterirhombated octeract
2795520 430080
141