Uniform 8-polytope
Polytope contained by 7-polytope facets
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 .[ 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 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:
Simplex family: A8 [37 ] -
135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
{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:
{4,36 } - 8-cube or octeract -
{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:
{3,35,1 } - 8-demicube or demiocteract , 151 - ; also as h{4,36 } .
{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:
{3,3,3,3,32,1 } - Thorold Gosset 's semiregular 421 ,
{3,34,2 } - the uniform 142 , ,
{3,3,34,1 } - the uniform 241 ,
There are many uniform prismatic families, including:
Uniform 8-polytope prism families
#
Coxeter group
Coxeter-Dynkin diagram
7+1
1
A7 A1
[3,3,3,3,3,3]×[ ]
2
B7 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
B6 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
B6 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
B5 A3
[4,33 ]×[3,3]
3
D5 A3
[32,1,1 ]×[3,3]
4
A5 B3
[34 ]×[4,3]
5
B5 B3
[4,33 ]×[4,3]
6
D5 B3
[32,1,1 ]×[4,3]
7
A5 H3
[34 ]×[5,3]
8
B5 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
B5 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
B5 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
B4 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
B4 B4
[4,3,3]×[4,3,3]
7
D4 B4
[31,1,1 ]×[4,3,3]
8
F4 B4
[3,4,3]×[4,3,3]
9
H4 B4
[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 B3 A1
[3,3,3]×[4,3]×[ ]
3
A4 H3 A1
[3,3,3]×[5,3]×[ ]
4
B4 A3 A1
[4,3,3]×[3,3]×[ ]
5
B4 B3 A1
[4,3,3]×[4,3]×[ ]
6
B4 H3 A1
[4,3,3]×[5,3]×[ ]
7
H4 A3 A1
[5,3,3]×[3,3]×[ ]
8
H4 B3 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 B3 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 B3 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
B3 A3 I2 (p)
[4,3]×[3,3]×[p]
3
H3 A3 I2 (p)
[5,3]×[3,3]×[p]
4
B3 B3 I2 (p)
[4,3]×[4,3]×[p]
5
H3 B3 I2 (p)
[5,3]×[4,3]×[p]
6
H3 H3 I2 (p)
[5,3]×[5,3]×[p]
3+3+1+1
1
A3 2 A1 2
[3,3]×[3,3]×[ ]×[ ]
2
B3 A3 A1 2
[4,3]×[3,3]×[ ]×[ ]
3
H3 A3 A1 2
[5,3]×[3,3]×[ ]×[ ]
4
B3 B3 A1 2
[4,3]×[4,3]×[ ]×[ ]
5
H3 B3 A1 2
[5,3]×[4,3]×[ ]×[ ]
6
H3 H3 A1 2
[5,3]×[5,3]×[ ]×[ ]
3+2+2+1
1
A3 I2 (p)I2 (q)A1
[3,3]×[p]×[q]×[ ]
2
B3 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 3
[3,3]×[p]×[ ]x[ ]×[ ]
2
B3 I2 (p)A1 3
[4,3]×[p]×[ ]x[ ]×[ ]
3
H3 I2 (p)A1 3
[5,3]×[p]×[ ]x[ ]×[ ]
3+1+1+1+1+1
1
A3 A1 5
[3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2
B3 A1 5
[4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
3
H3 A1 5
[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 2
[p]×[q]×[r]×[ ]×[ ]
2+2+1+1+1+1
2
I2 (p)I2 (q)A1 4
[p]×[q]×[ ]×[ ]×[ ]×[ ]
2+1+1+1+1+1+1
1
I2 (p)A1 6
[p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1+1
1
A1 8
[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
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.
A8 uniform 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.
B8 uniform 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.
D8 uniform polytopes
#
Coxeter-Dynkin diagram
Name
Base point (Alternately signed)
Element counts
Circumrad
7
6
5
4
3
2
1
0
1
=
8-demicube h{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
2
=
cantic 8-cube h2 {4,3,3,3,3,3,3}
(1,1,3,3,3,3,3,3)
23296
3584
2.6457512
3
=
runcic 8-cube h3 {4,3,3,3,3,3,3}
(1,1,1,3,3,3,3,3)
64512
7168
2.4494896
4
=
steric 8-cube h4 {4,3,3,3,3,3,3}
(1,1,1,1,3,3,3,3)
98560
8960
2.2360678
5
=
pentic 8-cube h5 {4,3,3,3,3,3,3}
(1,1,1,1,1,3,3,3)
89600
7168
1.9999999
6
=
hexic 8-cube h6 {4,3,3,3,3,3,3}
(1,1,1,1,1,1,3,3)
48384
3584
1.7320508
7
=
heptic 8-cube h7 {4,3,3,3,3,3,3}
(1,1,1,1,1,1,1,3)
14336
1024
1.4142135
8
=
runcicantic 8-cube h2,3 {4,3,3,3,3,3,3}
(1,1,3,5,5,5,5,5)
86016
21504
4.1231055
9
=
stericantic 8-cube h2,4 {4,3,3,3,3,3,3}
(1,1,3,3,5,5,5,5)
349440
53760
3.8729835
10
=
steriruncic 8-cube h3,4 {4,3,3,3,3,3,3}
(1,1,1,3,5,5,5,5)
179200
35840
3.7416575
11
=
penticantic 8-cube h2,5 {4,3,3,3,3,3,3}
(1,1,3,3,3,5,5,5)
573440
71680
3.6055512
12
=
pentiruncic 8-cube h3,5 {4,3,3,3,3,3,3}
(1,1,1,3,3,5,5,5)
537600
71680
3.4641016
13
=
pentisteric 8-cube h4,5 {4,3,3,3,3,3,3}
(1,1,1,1,3,5,5,5)
232960
35840
3.3166249
14
=
hexicantic 8-cube h2,6 {4,3,3,3,3,3,3}
(1,1,3,3,3,3,5,5)
456960
53760
3.3166249
15
=
hexicruncic 8-cube h3,6 {4,3,3,3,3,3,3}
(1,1,1,3,3,3,5,5)
645120
71680
3.1622777
16
=
hexisteric 8-cube h4,6 {4,3,3,3,3,3,3}
(1,1,1,1,3,3,5,5)
483840
53760
3
17
=
hexipentic 8-cube h5,6 {4,3,3,3,3,3,3}
(1,1,1,1,1,3,5,5)
182784
21504
2.8284271
18
=
hepticantic 8-cube h2,7 {4,3,3,3,3,3,3}
(1,1,3,3,3,3,3,5)
172032
21504
3
19
=
heptiruncic 8-cube h3,7 {4,3,3,3,3,3,3}
(1,1,1,3,3,3,3,5)
340480
35840
2.8284271
20
=
heptsteric 8-cube h4,7 {4,3,3,3,3,3,3}
(1,1,1,1,3,3,3,5)
376320
35840
2.6457512
21
=
heptipentic 8-cube h5,7 {4,3,3,3,3,3,3}
(1,1,1,1,1,3,3,5)
236544
21504
2.4494898
22
=
heptihexic 8-cube h6,7 {4,3,3,3,3,3,3}
(1,1,1,1,1,1,3,5)
78848
7168
2.236068
23
=
steriruncicantic 8-cube h2,3,4 {4,36 }
(1,1,3,5,7,7,7,7)
430080
107520
5.3851647
24
=
pentiruncicantic 8-cube h2,3,5 {4,36 }
(1,1,3,5,5,7,7,7)
1182720
215040
5.0990195
25
=
pentistericantic 8-cube h2,4,5 {4,36 }
(1,1,3,3,5,7,7,7)
1075200
215040
4.8989797
26
=
pentisterirunic 8-cube h3,4,5 {4,36 }
(1,1,1,3,5,7,7,7)
716800
143360
4.7958317
27
=
hexiruncicantic 8-cube h2,3,6 {4,36 }
(1,1,3,5,5,5,7,7)
1290240
215040
4.7958317
28
=
hexistericantic 8-cube h2,4,6 {4,36 }
(1,1,3,3,5,5,7,7)
2096640
322560
4.5825758
29
=
hexisterirunic 8-cube h3,4,6 {4,36 }
(1,1,1,3,5,5,7,7)
1290240
215040
4.472136
30
=
hexipenticantic 8-cube h2,5,6 {4,36 }
(1,1,3,3,3,5,7,7)
1290240
215040
4.3588991
31
=
hexipentirunic 8-cube h3,5,6 {4,36 }
(1,1,1,3,3,5,7,7)
1397760
215040
4.2426405
32
=
hexipentisteric 8-cube h4,5,6 {4,36 }
(1,1,1,1,3,5,7,7)
698880
107520
4.1231055
33
=
heptiruncicantic 8-cube h2,3,7 {4,36 }
(1,1,3,5,5,5,5,7)
591360
107520
4.472136
34
=
heptistericantic 8-cube h2,4,7 {4,36 }
(1,1,3,3,5,5,5,7)
1505280
215040
4.2426405
35
=
heptisterruncic 8-cube h3,4,7 {4,36 }
(1,1,1,3,5,5,5,7)
860160
143360
4.1231055
36
=
heptipenticantic 8-cube h2,5,7 {4,36 }
(1,1,3,3,3,5,5,7)
1612800
215040
4
37
=
heptipentiruncic 8-cube h3,5,7 {4,36 }
(1,1,1,3,3,5,5,7)
1612800
215040
3.8729835
38
=
heptipentisteric 8-cube h4,5,7 {4,36 }
(1,1,1,1,3,5,5,7)
752640
107520
3.7416575
39
=
heptihexicantic 8-cube h2,6,7 {4,36 }
(1,1,3,3,3,3,5,7)
752640
107520
3.7416575
40
=
heptihexiruncic 8-cube h3,6,7 {4,36 }
(1,1,1,3,3,3,5,7)
1146880
143360
3.6055512
41
=
heptihexisteric 8-cube h4,6,7 {4,36 }
(1,1,1,1,3,3,5,7)
913920
107520
3.4641016
42
=
heptihexipentic 8-cube h5,6,7 {4,36 }
(1,1,1,1,1,3,5,7)
365568
43008
3.3166249
43
=
pentisteriruncicantic 8-cube h2,3,4,5 {4,36 }
(1,1,3,5,7,9,9,9)
1720320
430080
6.4031243
44
=
hexisteriruncicantic 8-cube h2,3,4,6 {4,36 }
(1,1,3,5,7,7,9,9)
3225600
645120
6.0827627
45
=
hexipentiruncicantic 8-cube h2,3,5,6 {4,36 }
(1,1,3,5,5,7,9,9)
2903040
645120
5.8309517
46
=
hexipentistericantic 8-cube h2,4,5,6 {4,36 }
(1,1,3,3,5,7,9,9)
3225600
645120
5.6568542
47
=
hexipentisteriruncic 8-cube h3,4,5,6 {4,36 }
(1,1,1,3,5,7,9,9)
2150400
430080
5.5677648
48
=
heptsteriruncicantic 8-cube h2,3,4,7 {4,36 }
(1,1,3,5,7,7,7,9)
2150400
430080
5.7445626
49
=
heptipentiruncicantic 8-cube h2,3,5,7 {4,36 }
(1,1,3,5,5,7,7,9)
3548160
645120
5.4772258
50
=
heptipentistericantic 8-cube h2,4,5,7 {4,36 }
(1,1,3,3,5,7,7,9)
3548160
645120
5.291503
51
=
heptipentisteriruncic 8-cube h3,4,5,7 {4,36 }
(1,1,1,3,5,7,7,9)
2365440
430080
5.1961527
52
=
heptihexiruncicantic 8-cube h2,3,6,7 {4,36 }
(1,1,3,5,5,5,7,9)
2150400
430080
5.1961527
53
=
heptihexistericantic 8-cube h2,4,6,7 {4,36 }
(1,1,3,3,5,5,7,9)
3870720
645120
5
54
=
heptihexisteriruncic 8-cube h3,4,6,7 {4,36 }
(1,1,1,3,5,5,7,9)
2365440
430080
4.8989797
55
=
heptihexipenticantic 8-cube h2,5,6,7 {4,36 }
(1,1,3,3,3,5,7,9)
2580480
430080
4.7958317
56
=
heptihexipentiruncic 8-cube h3,5,6,7 {4,36 }
(1,1,1,3,3,5,7,9)
2795520
430080
4.6904159
57
=
heptihexipentisteric 8-cube h4,5,6,7 {4,36 }
(1,1,1,1,3,5,7,9)
1397760
215040
4.5825758
58
=
hexipentisteriruncicantic 8-cube h2,3,4,5,6 {4,36 }
(1,1,3,5,7,9,11,11)
5160960
1290240
7.1414285
59
=
heptipentisteriruncicantic 8-cube h2,3,4,5,7 {4,36 }
(1,1,3,5,7,9,9,11)
5806080
1290240
6.78233
60
=
heptihexisteriruncicantic 8-cube h2,3,4,6,7 {4,36 }
(1,1,3,5,7,7,9,11)
5806080
1290240
6.480741
61
=
heptihexipentiruncicantic 8-cube h2,3,5,6,7 {4,36 }
(1,1,3,5,5,7,9,11)
5806080
1290240
6.244998
62
=
heptihexipentistericantic 8-cube h2,4,5,6,7 {4,36 }
(1,1,3,3,5,7,9,11)
6451200
1290240
6.0827627
63
=
heptihexipentisteriruncic 8-cube h3,4,5,6,7 {4,36 }
(1,1,1,3,5,7,9,11)
4300800
860160
6.0000000
64
=
heptihexipentisteriruncicantic 8-cube h2,3,4,5,6,7 {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
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
Forms
1
A
~ ~ -->
7
{\displaystyle {\tilde {A}}_{7}}
[3[8] ]
29
2
C
~ ~ -->
7
{\displaystyle {\tilde {C}}_{7}}
[4,35 ,4]
135
3
B
~ ~ -->
7
{\displaystyle {\tilde {B}}_{7}}
[4,34 ,31,1 ]
191 (64 new)
4
D
~ ~ -->
7
{\displaystyle {\tilde {D}}_{7}}
[31,1 ,33 ,31,1 ]
77 (10 new)
5
E
~ ~ -->
7
{\displaystyle {\tilde {E}}_{7}}
[33,3,1 ]
143
Regular and uniform tessellations include:
A
~ ~ -->
7
{\displaystyle {\tilde {A}}_{7}}
29 uniquely ringed forms, including:
C
~ ~ -->
7
{\displaystyle {\tilde {C}}_{7}}
135 uniquely ringed forms, including:
B
~ ~ -->
7
{\displaystyle {\tilde {B}}_{7}}
191 uniquely ringed forms, 127 shared with
C
~ ~ -->
7
{\displaystyle {\tilde {C}}_{7}}
, and 64 new, including:
D
~ ~ -->
7
{\displaystyle {\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 .
E
~ ~ -->
7
{\displaystyle {\tilde {E}}_{7}}
143 uniquely ringed forms, including:
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.
P
¯ ¯ -->
7
{\displaystyle {\bar {P}}_{7}}
= [3,3[7] ]:
Q
¯ ¯ -->
7
{\displaystyle {\bar {Q}}_{7}}
= [31,1 ,32 ,32,1 ]:
S
¯ ¯ -->
7
{\displaystyle {\bar {S}}_{7}}
= [4,33 ,32,1 ]:
T
¯ ¯ -->
7
{\displaystyle {\bar {T}}_{7}}
= [33,2,2 ]:
References
^ a b c Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy , Princeton, 2008.
T. Gosset : On the Regular and Semi-Regular Figures in Space of n Dimensions , Messenger of Mathematics , Macmillan, 1900
A. Boole Stott : Geometrical deduction of semiregular from regular polytopes and space fillings , Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
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, ISBN 978-0-471-01003-6 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
(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
Klitzing, Richard. "8D uniform polytopes (polyzetta)" .
External links