Uniform 8-polytope

Graphs of three regular and related uniform polytopes.

8-simplex
Rectified 8-simplex
Truncated 8-simplex

Cantellated 8-simplex
Runcinated 8-simplex
Stericated 8-simplex

Pentellated 8-simplex
Hexicated 8-simplex
Heptellated 8-simplex

8-orthoplex
Rectified 8-orthoplex
Truncated 8-orthoplex

Cantellated 8-orthoplex
Runcinated 8-orthoplex

Hexicated 8-orthoplex
Cantellated 8-cube

Runcinated 8-cube
Stericated 8-cube
Pentellated 8-cube

Hexicated 8-cube
Heptellated 8-cube

8-cube
Rectified 8-cube
Truncated 8-cube

8-demicube
Truncated 8-demicube
Cantellated 8-demicube

Runcinated 8-demicube
Stericated 8-demicube

Pentellated 8-demicube
Hexicated 8-demicube

421
142
241

In eight-dimensional geometry, a polyzetton[1] (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 polyzetton is one which is vertex-transitive, and constructed from uniform (7-polytope) facets.

Contents

Regular 8-polytopes

Regular polyzetta 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.

Euler characteristic

The Euler characteristic for 8-polytopes that are topological 7-spheres (including all convex 8-polytopes) is zero. χ=V-E+F-C+f4-f5+f6-f7=0.

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 Coxeter-Dynkin diagram 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. {31,5,1} - 8-demicube or demiocteract, 151 - ; also as h{4,36} .
      2. {35,1,1} - 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. {34,2,1} - Thorold Gosset's semiregular 421,
      2. {31,4,2} - the uniform 142, ,
      3. {32,4,1} - the uniform 241,

Uniform prismatic forms

There are many uniform prismatic families, including:

# Coxeter group Coxeter-Dynkin diagram
7+1
1 A7×A1 [3,3,3,3,3,3]×[ ]
2 BC7×A1 [4,3,3,3,3,3]×[ ]
3 D7×A1 [34,1,1]×[ ]
4 E7×A1 [33,2,1]×[ ]
6+2
1 A6×I2(p) [3,3,3,3,3]×[p]
2 BC6×I2(p) [4,3,3,3,3]×[p]
3 D6×I2(p) [33,1,1]×[p]
4 E6×I2(p) [3,3,3,3,3]×[p]
6+1+1
1 A6×A1×A1 [3,3,3,3,3]×[ ]x[ ]
2 BC6×A1×A1 [4,3,3,3,3]×[ ]x[ ]
3 D6×A1×A1 [33,1,1]×[ ]x[ ]
4 E6×A1×A1 [3,3,3,3,3]×[ ]x[ ]
5+3
1 A5×A3 [34]×[3,3]
2 BC5×A3 [4,33]×[3,3]
3 D5×A3 [32,1,1]×[3,3]
4 A5×BC3 [34]×[4,3]
5 BC5×BC3 [4,33]×[4,3]
6 D5×BC3 [32,1,1]×[4,3]
7 A5×H3 [34]×[5,3]
8 BC5×H3 [4,33]×[5,3]
9 D5×H3 [32,1,1]×[5,3]
5+2+1
1 A5×I2(p)×A1 [3,3,3]×[p]×[ ]
2 BC5×I2(p)×A1 [4,3,3]×[p]×[ ]
3 D5×I2(p)×A1 [32,1,1]×[p]×[ ]
5+1+1+1
1 A5×A1×A1×A1 [3,3,3]×[ ]×[ ]×[ ]
2 BC5×A1×A1×A1 [4,3,3]×[ ]×[ ]×[ ]
3 D5×A1×A1×A1 [32,1,1]×[ ]×[ ]×[ ]
4+4
1 A4×A4 [3,3,3]×[3,3,3]
2 BC4×A4 [4,3,3]×[3,3,3]
3 D4×A4 [31,1,1]×[3,3,3]
4 F4×A4 [3,4,3]×[3,3,3]
5 H4×A4 [5,3,3]×[3,3,3]
6 BC4×BC4 [4,3,3]×[4,3,3]
7 D4×BC4 [31,1,1]×[4,3,3]
8 F4×BC4 [3,4,3]×[4,3,3]
9 H4×BC4 [5,3,3]×[4,3,3]
10 D4×D4 [31,1,1]×[31,1,1]
11 F4×D4 [3,4,3]×[31,1,1]
12 H4×D4 [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 H4×H4 [5,3,3]×[5,3,3]
4+3+1
1 A4×A3×A1 [3,3,3]×[3,3]×[ ]
2 A4×BC3×A1 [3,3,3]×[4,3]×[ ]
3 A4×H3×A1 [3,3,3]×[5,3]×[ ]
4 BC4×A3×A1 [4,3,3]×[3,3]×[ ]
5 BC4×BC3×A1 [4,3,3]×[4,3]×[ ]
6 BC4×H3×A1 [4,3,3]×[5,3]×[ ]
7 H4×A3×A1 [5,3,3]×[3,3]×[ ]
8 H4×BC3×A1 [5,3,3]×[4,3]×[ ]
9 H4×H3×A1 [5,3,3]×[5,3]×[ ]
10 F4×A3×A1 [3,4,3]×[3,3]×[ ]
11 F4×BC3×A1 [3,4,3]×[4,3]×[ ]
12 F4×H3×A1 [3,4,3]×[5,3]×[ ]
13 D4×A3×A1 [31,1,1]×[3,3]×[ ]
14 D4×BC3×A1 [31,1,1]×[4,3]×[ ]
15 D4×H3×A1 [31,1,1]×[5,3]×[ ]
4+2+2
...
4+2+1+1
...
4+1+1+1+1
...
3+3+2
1 A3×A3×I2(p) [3,3]×[3,3]×[p]
2 BC3×A3×I2(p) [4,3]×[3,3]×[p]
3 H3×A3×I2(p) [5,3]×[3,3]×[p]
4 BC3×BC3×I2(p) [4,3]×[4,3]×[p]
5 H3×BC3×I2(p) [5,3]×[4,3]×[p]
6 H3×H3×I2(p) [5,3]×[5,3]×[p]
3+3+1+1
1 A3×A3×A1×A1 [3,3]×[3,3]×[ ]×[ ]
2 BC3×A3×A1×A1 [4,3]×[3,3]×[ ]×[ ]
3 H3×A3×A1×A1 [5,3]×[3,3]×[ ]×[ ]
4 BC3×BC3×A1×A1 [4,3]×[4,3]×[ ]×[ ]
5 H3×BC3×A1×A1 [5,3]×[4,3]×[ ]×[ ]
6 H3×H3×A1×A1 [5,3]×[5,3]×[ ]×[ ]
3+2+2+1
1 A3×I2(p)×I2(q)×A1 [3,3]×[p]×[q]×[ ]
2 BC3×I2(p)×I2(q)×A1 [4,3]×[p]×[q]×[ ]
3 H3×I2(p)×I2(q)×A1 [5,3]×[p]×[q]×[ ]
3+2+1+1+1
1 A3×I2(p)×A1×A1×A1 [3,3]×[p]×[ ]x[ ]×[ ]
2 BC3×I2(p)×A1×A1×A1 [4,3]×[p]×[ ]x[ ]×[ ]
3 H3×I2(p)×A1×A1×A1 [5,3]×[p]×[ ]x[ ]×[ ]
3+1+1+1+1+1
1 A3×A1×A1×A1×A1×A1 [3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2 BC3×A1×A1×A1×A1×A1 [4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
3 H3×A1×A1×A1×A1×A1 [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)×A1×A1 [p]×[q]×[r]×[ ]×[ ]
2+2+1+1+1+1
2 I2(p)×I2(q)×A1×A1×A1×A1 [p]×[q]×[ ]×[ ]×[ ]×[ ]
2+1+1+1+1+1+1
1 I2(p)×A1×A1×A1×A1×A1×A1 [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1+1
1 A1×A1×A1×A1×A1×A1×A1×A1 [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

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
18 t3,4{4,36} Quadritruncated 8-cube
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 t0,2,4,7{4,36} Heptistericantellated 8-cube
Exicellirhombated octeract		 3225600 430080
142 t0,2,4,6{4,36} Hexistericantellated 8-cube
Peticellirhombated octeract		 4515840 645120
143 t0,2,4,5{4,36} Pentistericantellated 8-cube
Tericellirhombated octeract		 2365440 430080
144 t0,2,3,7{4,36} Heptiruncicantellated 8-cube
Exiprismatorhombated octeract		 1290240 215040
145 t0,2,3,6{4,36} Hexiruncicantellated 8-cube
Petiprismatorhombated octeract		 2795520 430080
146 t0,2,3,5{4,36} Pentiruncicantellated 8-cube
Teriprismatorhombated octeract		 2580480 430080
147 t0,2,3,4{4,36} Steriruncicantellated 8-cube
Celliprismatorhombated octeract		 967680 215040
148 t0,1,6,7{4,36} Heptihexitruncated 8-cube
Exipetitrunki-octeractidiacosipentacontahexazetton		 516096 86016
149 t0,1,5,7{4,36} Heptipentitruncated 8-cube
Exiteritruncated octeract		 1612800 215040
150 t0,1,5,6{4,36} Hexipentitruncated 8-cube
Petiteritruncated octeract		 1182720 215040
151 t0,1,4,7{4,36} Heptisteritruncated 8-cube
Exicellitruncated octeract		 2293760 286720
152 t0,1,4,6{4,36} Hexisteritruncated 8-cube
Peticellitruncated octeract		 3010560 430080
153 t0,1,4,5{4,36} Pentisteritruncated 8-cube
Tericellitruncated octeract		 1433600 286720
154 t0,1,3,7{4,36} Heptiruncitruncated 8-cube
Exiprismatotruncated octeract		 1612800 215040
155 t0,1,3,6{4,36} Hexiruncitruncated 8-cube
Petiprismatotruncated octeract		 3225600 430080
156 t0,1,3,5{4,36} Pentiruncitruncated 8-cube
Teriprismatotruncated octeract		 2795520 430080
157 t0,1,3,4{4,36} Steriruncitruncated 8-cube
Celliprismatotruncated octeract		 967680 215040
158 t0,1,2,7{4,36} Hepticantitruncated 8-cube
Exigreatorhombated octeract		 516096 86016
159 t0,1,2,6{4,36} Hexicantitruncated 8-cube
Petigreatorhombated octeract		 1505280 215040
160 t0,1,2,5{4,36} Penticantitruncated 8-cube
Terigreatorhombated octeract		 2007040 286720
161 t0,1,2,4{4,36} Stericantitruncated 8-cube
Celligreatorhombated octeract		 1290240 215040
162 t0,1,2,3{4,36} Runcicantitruncated 8-cube
Great prismated octeract		 344064 86016
163 t0,1,2,3,4{36,4} Steriruncicantitruncated 8-orthoplex
Great cellated diacosipentacontahexazetton		 1075200 215040
164 t0,1,2,3,5{36,4} Pentiruncicantitruncated 8-orthoplex
Terigreatoprismated diacosipentacontahexazetton		 4193280 645120
165 t0,1,2,4,5{36,4} Pentistericantitruncated 8-orthoplex
Tericelligreatorhombated diacosipentacontahexazetton		 3225600 645120
166 t0,1,3,4,5{36,4} Pentisteriruncitruncated 8-orthoplex
Tericelliprismatotruncated diacosipentacontahexazetton		 3225600 645120
167 t0,2,3,4,5{36,4} Pentisteriruncicantellated 8-orthoplex
Tericelliprismatorhombated diacosipentacontahexazetton		 3225600 645120
168 t1,2,3,4,5{36,4} Bisteriruncicantitruncated 8-orthoplex
Great bicellated diacosipentacontahexazetton		 2903040 645120
169 t0,1,2,3,6{36,4} Hexiruncicantitruncated 8-orthoplex
Petigreatoprismated diacosipentacontahexazetton		 5160960 860160
170 t0,1,2,4,6{36,4} Hexistericantitruncated 8-orthoplex
Peticelligreatorhombated diacosipentacontahexazetton		 7741440 1290240
171 t0,1,3,4,6{36,4} Hexisteriruncitruncated 8-orthoplex
Peticelliprismatotruncated diacosipentacontahexazetton		 7096320 1290240
172 t0,2,3,4,6{36,4} Hexisteriruncicantellated 8-orthoplex
Peticelliprismatorhombated diacosipentacontahexazetton		 7096320 1290240
173 t1,2,3,4,6{36,4} Bipentiruncicantitruncated 8-orthoplex
Biterigreatoprismated diacosipentacontahexazetton		 6451200 1290240
174 t0,1,2,5,6{36,4} Hexipenticantitruncated 8-orthoplex
Petiterigreatorhombated diacosipentacontahexazetton		 4300800 860160
175 t0,1,3,5,6{36,4} Hexipentiruncitruncated 8-orthoplex
Petiteriprismatotruncated diacosipentacontahexazetton		 7096320 1290240
176 t0,2,3,5,6{36,4} Hexipentiruncicantellated 8-orthoplex
Petiteriprismatorhombated diacosipentacontahexazetton		 6451200 1290240
177 t1,2,3,5,6{36,4} Bipentistericantitruncated 8-orthoplex
Bitericelligreatorhombated diacosipentacontahexazetton		 5806080 1290240
178 t0,1,4,5,6{36,4} Hexipentisteritruncated 8-orthoplex
Petitericellitruncated diacosipentacontahexazetton		 4300800 860160
179 t0,2,4,5,6{36,4} Hexipentistericantellated 8-orthoplex
Petitericellirhombated diacosipentacontahexazetton		 7096320 1290240
180 t1,2,3,5,6{4,36} Bipentistericantitruncated 8-cube
Bitericelligreatorhombated octeract		 5806080 1290240
181 t0,3,4,5,6{36,4} Hexipentisteriruncinated 8-orthoplex
Petitericelliprismated diacosipentacontahexazetton		 4300800 860160
182 t1,2,3,4,6{4,36} Bipentiruncicantitruncated 8-cube
Biterigreatoprismated octeract		 6451200 1290240
183 t1,2,3,4,5{4,36} Bisteriruncicantitruncated 8-cube
Great bicellated octeract		 3440640 860160
184 t0,1,2,3,7{36,4} Heptiruncicantitruncated 8-orthoplex
Exigreatoprismated diacosipentacontahexazetton		 2365440 430080
185 t0,1,2,4,7{36,4} Heptistericantitruncated 8-orthoplex
Exicelligreatorhombated diacosipentacontahexazetton		 5591040 860160
186 t0,1,3,4,7{36,4} Heptisteriruncitruncated 8-orthoplex
Exicelliprismatotruncated diacosipentacontahexazetton		 4730880 860160
187 t0,2,3,4,7{36,4} Heptisteriruncicantellated 8-orthoplex
Exicelliprismatorhombated diacosipentacontahexazetton		 4730880 860160
188 t0,3,4,5,6{4,36} Hexipentisteriruncinated 8-cube
Petitericelliprismated octeract		 4300800 860160
189 t0,1,2,5,7{36,4} Heptipenticantitruncated 8-orthoplex
Exiterigreatorhombated diacosipentacontahexazetton		 5591040 860160
190 t0,1,3,5,7{36,4} Heptipentiruncitruncated 8-orthoplex
Exiteriprismatotruncated diacosipentacontahexazetton		 8386560 1290240
191 t0,2,3,5,7{36,4} Heptipentiruncicantellated 8-orthoplex
Exiteriprismatorhombated diacosipentacontahexazetton		 7741440 1290240
192 t0,2,4,5,6{4,36} Hexipentistericantellated 8-cube
Petitericellirhombated octeract		 7096320 1290240
193 t0,1,4,5,7{36,4} Heptipentisteritruncated 8-orthoplex
Exitericellitruncated diacosipentacontahexazetton		 4730880 860160
194 t0,2,3,5,7{4,36} Heptipentiruncicantellated 8-cube
Exiteriprismatorhombated octeract		 7741440 1290240
195 t0,2,3,5,6{4,36} Hexipentiruncicantellated 8-cube
Petiteriprismatorhombated octeract		 6451200 1290240
196 t0,2,3,4,7{4,36} Heptisteriruncicantellated 8-cube
Exicelliprismatorhombated octeract		 4730880 860160
197 t0,2,3,4,6{4,36} Hexisteriruncicantellated 8-cube
Peticelliprismatorhombated octeract		 7096320 1290240
198 t0,2,3,4,5{4,36} Pentisteriruncicantellated 8-cube
Tericelliprismatorhombated octeract		 3870720 860160
199 t0,1,2,6,7{36,4} Heptihexicantitruncated 8-orthoplex
Exipetigreatorhombated diacosipentacontahexazetton		 2365440 430080
200 t0,1,3,6,7{36,4} Heptihexiruncitruncated 8-orthoplex
Exipetiprismatotruncated diacosipentacontahexazetton		 5591040 860160
201 t0,1,4,5,7{4,36} Heptipentisteritruncated 8-cube
Exitericellitruncated octeract		 4730880 860160
202 t0,1,4,5,6{4,36} Hexipentisteritruncated 8-cube
Petitericellitruncated octeract		 4300800 860160
203 t0,1,3,6,7{4,36} Heptihexiruncitruncated 8-cube
Exipetiprismatotruncated octeract		 5591040 860160
204 t0,1,3,5,7{4,36} Heptipentiruncitruncated 8-cube
Exiteriprismatotruncated octeract		 8386560 1290240
205 t0,1,3,5,6{4,36} Hexipentiruncitruncated 8-cube
Petiteriprismatotruncated octeract		 7096320 1290240
206 t0,1,3,4,7{4,36} Heptisteriruncitruncated 8-cube
Exicelliprismatotruncated octeract		 4730880 860160
207 t0,1,3,4,6{4,36} Hexisteriruncitruncated 8-cube
Peticelliprismatotruncated octeract		 7096320 1290240
208 t0,1,3,4,5{4,36} Pentisteriruncitruncated 8-cube
Tericelliprismatotruncated octeract		 3870720 860160
209 t0,1,2,6,7{4,36} Heptihexicantitruncated 8-cube
Exipetigreatorhombated octeract		 2365440 430080
210 t0,1,2,5,7{4,36} Heptipenticantitruncated 8-cube
Exiterigreatorhombated octeract		 5591040 860160
211 t0,1,2,5,6{4,36} Hexipenticantitruncated 8-cube
Petiterigreatorhombated octeract		 4300800 860160
212 t0,1,2,4,7{4,36} Heptistericantitruncated 8-cube
Exicelligreatorhombated octeract		 5591040 860160
213 t0,1,2,4,6{4,36} Hexistericantitruncated 8-cube
Peticelligreatorhombated octeract		 7741440 1290240
214 t0,1,2,4,5{4,36} Pentistericantitruncated 8-cube
Tericelligreatorhombated octeract		 3870720 860160
215 t0,1,2,3,7{4,36} Heptiruncicantitruncated 8-cube
Exigreatoprismated octeract		 2365440 430080
216 t0,1,2,3,6{4,36} Hexiruncicantitruncated 8-cube
Petigreatoprismated octeract		 5160960 860160
217 t0,1,2,3,5{4,36} Pentiruncicantitruncated 8-cube
Terigreatoprismated octeract		 4730880 860160
218 t0,1,2,3,4{4,36} Steriruncicantitruncated 8-cube
Great cellated octeract		 1720320 430080
219 t0,1,2,3,4,5{36,4} Pentisteriruncicantitruncated 8-orthoplex
Great terated diacosipentacontahexazetton		 5806080 1290240
220 t0,1,2,3,4,6{36,4} Hexisteriruncicantitruncated 8-orthoplex
Petigreatocellated diacosipentacontahexazetton		 12902400 2580480
221 t0,1,2,3,5,6{36,4} Hexipentiruncicantitruncated 8-orthoplex
Petiterigreatoprismated diacosipentacontahexazetton		 11612160 2580480
222 t0,1,2,4,5,6{36,4} Hexipentistericantitruncated 8-orthoplex
Petitericelligreatorhombated diacosipentacontahexazetton		 11612160 2580480
223 t0,1,3,4,5,6{36,4} Hexipentisteriruncitruncated 8-orthoplex
Petitericelliprismatotruncated diacosipentacontahexazetton		 11612160 2580480
224 t0,2,3,4,5,6{36,4} Hexipentisteriruncicantellated 8-orthoplex
Petitericelliprismatorhombated diacosipentacontahexazetton		 11612160 2580480
225 t1,2,3,4,5,6{4,36} Bipentisteriruncicantitruncated 8-cube
Great biteri-octeractidiacosipentacontahexazetton		 10321920 2580480
226 t0,1,2,3,4,7{36,4} Heptisteriruncicantitruncated 8-orthoplex
Exigreatocellated diacosipentacontahexazetton		 8601600 1720320
227 t0,1,2,3,5,7{36,4} Heptipentiruncicantitruncated 8-orthoplex
Exiterigreatoprismated diacosipentacontahexazetton		 14192640 2580480
228 t0,1,2,4,5,7{36,4} Heptipentistericantitruncated 8-orthoplex
Exitericelligreatorhombated diacosipentacontahexazetton		 12902400 2580480
229 t0,1,3,4,5,7{36,4} Heptipentisteriruncitruncated 8-orthoplex
Exitericelliprismatotruncated diacosipentacontahexazetton		 12902400 2580480
230 t0,2,3,4,5,7{4,36} Heptipentisteriruncicantellated 8-cube
Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton		 12902400 2580480
231 t0,2,3,4,5,6{4,36} Hexipentisteriruncicantellated 8-cube
Petitericelliprismatorhombated octeract		 11612160 2580480
232 t0,1,2,3,6,7{36,4} Heptihexiruncicantitruncated 8-orthoplex
Exipetigreatoprismated diacosipentacontahexazetton		 8601600 1720320
233 t0,1,2,4,6,7{36,4} Heptihexistericantitruncated 8-orthoplex
Exipeticelligreatorhombated diacosipentacontahexazetton		 14192640 2580480
234 t0,1,3,4,6,7{4,36} Heptihexisteriruncitruncated 8-cube
Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton		 12902400 2580480
235 t0,1,3,4,5,7{4,36} Heptipentisteriruncitruncated 8-cube
Exitericelliprismatotruncated octeract		 12902400 2580480
236 t0,1,3,4,5,6{4,36} Hexipentisteriruncitruncated 8-cube
Petitericelliprismatotruncated octeract		 11612160 2580480
237 t0,1,2,5,6,7{4,36} Heptihexipenticantitruncated 8-cube
Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton		 8601600 1720320
238 t0,1,2,4,6,7{4,36} Heptihexistericantitruncated 8-cube
Exipeticelligreatorhombated octeract		 14192640 2580480
239 t0,1,2,4,5,7{4,36} Heptipentistericantitruncated 8-cube
Exitericelligreatorhombated octeract		 12902400 2580480
240 t0,1,2,4,5,6{4,36} Hexipentistericantitruncated 8-cube
Petitericelligreatorhombated octeract		 11612160 2580480
241 t0,1,2,3,6,7{4,36} Heptihexiruncicantitruncated 8-cube
Exipetigreatoprismated octeract		 8601600 1720320
242 t0,1,2,3,5,7{4,36} Heptipentiruncicantitruncated 8-cube
Exiterigreatoprismated octeract		 14192640 2580480
243 t0,1,2,3,5,6{4,36} Hexipentiruncicantitruncated 8-cube
Petiterigreatoprismated octeract		 11612160 2580480
244 t0,1,2,3,4,7{4,36} Heptisteriruncicantitruncated 8-cube
Exigreatocellated octeract		 8601600 1720320
245 t0,1,2,3,4,6{4,36} Hexisteriruncicantitruncated 8-cube
Petigreatocellated octeract		 12902400 2580480
246 t0,1,2,3,4,5{4,36} Pentisteriruncicantitruncated 8-cube
Great terated octeract		 6881280 1720320
247 t0,1,2,3,4,5,6{36,4} Hexipentisteriruncicantitruncated 8-orthoplex
Great petated diacosipentacontahexazetton		 20643840 5160960
248 t0,1,2,3,4,5,7{36,4} Heptipentisteriruncicantitruncated 8-orthoplex
Exigreatoterated diacosipentacontahexazetton		 23224320 5160960
249 t0,1,2,3,4,6,7{36,4} Heptihexisteriruncicantitruncated 8-orthoplex
Exipetigreatocellated diacosipentacontahexazetton		 23224320 5160960
250 t0,1,2,3,5,6,7{36,4} Heptihexipentiruncicantitruncated 8-orthoplex
Exipetiterigreatoprismated diacosipentacontahexazetton		 23224320 5160960
251 t0,1,2,3,5,6,7{4,36} Heptihexipentiruncicantitruncated 8-cube
Exipetiterigreatoprismated octeract		 23224320 5160960
252 t0,1,2,3,4,6,7{4,36} Heptihexisteriruncicantitruncated 8-cube
Exipetigreatocellated octeract		 23224320 5160960
253 t0,1,2,3,4,5,7{4,36} Heptipentisteriruncicantitruncated 8-cube
Exigreatoterated octeract		 23224320 5160960
254 t0,1,2,3,4,5,6{4,36} Hexipentisteriruncicantitruncated 8-cube
Great petated octeract		 20643840 5160960
255 t0,1,2,3,4,5,6,7{4,36} Omnitruncated 8-cube
Great 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 4 3 2 1 0
1
8-demicube		 (1,1,1,1,1,1,1,1) 144 1136 4032 8288 10752 7168 1792 128 1.0000000
2
Truncated 8-demicube		 (1,1,3,3,3,3,3,3) 23296 3584 2.6457512
3 (1,1,1,3,3,3,3,3) 64512 7168 2.4494896
4 (1,1,1,1,3,3,3,3) 98560 8960 2.2360678
5 (1,1,1,1,1,3,3,3) 89600 7168 1.9999999
6 (1,1,1,1,1,1,3,3) 48384 3584 1.7320508
7 (1,1,1,1,1,1,1,3) 14336 1024 1.4142135
8 (1,1,3,5,5,5,5,5) 86016 21504 4.1231055
9 (1,1,3,3,5,5,5,5) 349440 53760 3.8729835
10 (1,1,1,3,5,5,5,5) 179200 35840 3.7416575
11 (1,1,3,3,3,5,5,5) 573440 71680 3.6055512
12 (1,1,1,3,3,5,5,5) 537600 71680 3.4641016
13 (1,1,1,1,3,5,5,5) 232960 35840 3.3166249
14 (1,1,3,3,3,3,5,5) 456960 53760 3.3166249
15 (1,1,1,3,3,3,5,5) 645120 71680 3.1622777
16 (1,1,1,1,3,3,5,5) 483840 53760 3
17 (1,1,1,1,1,3,5,5) 182784 21504 2.8284271
18 (1,1,3,3,3,3,3,5) 172032 21504 3
19 (1,1,1,3,3,3,3,5) 340480 35840 2.8284271
20 (1,1,1,1,3,3,3,5) 376320 35840 2.6457512
21 (1,1,1,1,1,3,3,5) 236544 21504 2.4494898
22 (1,1,1,1,1,1,3,5) 78848 7168 2.236068
23 (1,1,3,5,7,7,7,7) 430080 107520 5.3851647
24 (1,1,3,5,5,7,7,7) 1182720 215040 5.0990195
25 (1,1,3,3,5,7,7,7) 1075200 215040 4.8989797
26 (1,1,1,3,5,7,7,7) 716800 143360 4.7958317
27 (1,1,3,5,5,5,7,7) 1290240 215040 4.7958317
28 (1,1,3,3,5,5,7,7) 2096640 322560 4.5825758
29 (1,1,1,3,5,5,7,7) 1290240 215040 4.472136
30 (1,1,3,3,3,5,7,7) 1290240 215040 4.3588991
31 (1,1,1,3,3,5,7,7) 1397760 215040 4.2426405
32 (1,1,1,1,3,5,7,7) 698880 107520 4.1231055
33 (1,1,3,5,5,5,5,7) 591360 107520 4.472136
34 (1,1,3,3,5,5,5,7) 1505280 215040 4.2426405
35 (1,1,1,3,5,5,5,7) 860160 143360 4.1231055
36 (1,1,3,3,3,5,5,7) 1612800 215040 4
37 (1,1,1,3,3,5,5,7) 1612800 215040 3.8729835
38 (1,1,1,1,3,5,5,7) 752640 107520 3.7416575
39 (1,1,3,3,3,3,5,7) 752640 107520 3.7416575
40 (1,1,1,3,3,3,5,7) 1146880 143360 3.6055512
41 (1,1,1,1,3,3,5,7) 913920 107520 3.4641016
42 (1,1,1,1,1,3,5,7) 365568 43008 3.3166249
43 (1,1,3,5,7,9,9,9) 1720320 430080 6.4031243
44 (1,1,3,5,7,7,9,9) 3225600 645120 6.0827627
45 (1,1,3,5,5,7,9,9) 2903040 645120 5.8309517
46 (1,1,3,3,5,7,9,9) 3225600 645120 5.6568542
47 (1,1,1,3,5,7,9,9) 2150400 430080 5.5677648
48 (1,1,3,5,7,7,7,9) 2150400 430080 5.7445626
49 (1,1,3,5,5,7,7,9) 3548160 645120 5.4772258
50 (1,1,3,3,5,7,7,9) 3548160 645120 5.291503
51 (1,1,1,3,5,7,7,9) 2365440 430080 5.1961527
52 (1,1,3,5,5,5,7,9) 2150400 430080 5.1961527
53 (1,1,3,3,5,5,7,9) 3870720 645120 5
54 (1,1,1,3,5,5,7,9) 2365440 430080 4.8989797
55 (1,1,3,3,3,5,7,9) 2580480 430080 4.7958317
56 (1,1,1,3,3,5,7,9) 2795520 430080 4.6904159
57 (1,1,1,1,3,5,7,9) 1397760 215040 4.5825758
58 (1,1,3,5,7,9,11,11) 5160960 1290240 7.1414285
59 (1,1,3,5,7,9,9,11) 5806080 1290240 6.78233
60 (1,1,3,5,7,7,9,11) 5806080 1290240 6.480741
61 (1,1,3,5,5,7,9,11) 5806080 1290240 6.244998
62 (1,1,3,3,5,7,9,11) 6451200 1290240 6.0827627
63 (1,1,1,3,5,7,9,11) 4300800 860160 6.0000000
64 (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.

# 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

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

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

Regular and uniform tessellations include:

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 noncompact 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

Notes

  1. ^ A proposed name for 8-polytope is polyzetton (plural: polyzetta), created from poly-, and zetta- (following the SI prefix as a variation on hepta, meaning seven) and -on.
  2. ^ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

External links