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In 8-dimensional geometry, there are 256 uniform polytopes with B8 symmetry. There are two regular forms, the 8-orthoplex, and 8-cube with 16 and 256 vertices respectively. The 8-demicube is added with half the symmetry.

They can be visualized as symmetric orthographic projections in Coxeter planes of the B8 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 256 polytopes can be made in the B8, B7, B6, B5, B4, B3, B2, A7, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.

These 256 polytopes are each shown in these 10 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

Element counts Coxeter-Dynkin diagram
SchlĂ¤fli symbol
Name
B8
[16]
B7
[14]
B6
[12]
B5
[10]
B4
[8]
B3
[6]
B2
[4]
A7
[8]
A5
[6]
A3
[4]
1
t0{3,3,3,3,3,3,4}
8-orthoplex
Diacosipentacontahexazetton (ek)
2
t1{3,3,3,3,3,3,4}
Rectified 8-orthoplex
Rectified diacosipentacontahexazetton (rek)
3
t2{3,3,3,3,3,3,4}
Birectified 8-orthoplex
Birectified diacosipentacontahexazetton (bark)
4
t3{3,3,3,3,3,3,4}
Trirectified 8-orthoplex
Trirectified diacosipentacontahexazetton (tark)
5
t3{4,3,3,3,3,3,3}
Trirectified 8-cube
Trirectified octeract (tro)
6
t2{4,3,3,3,3,3,3}
Birectified 8-cube
Birectified octeract (bro)
7
t1{4,3,3,3,3,3,3}
Rectified 8-cube
Rectified octeract (recto)
8
t0{4,3,3,3,3,3,3}
8-cube
Octeract (octo)
9
t0,1{3,3,3,3,3,3,4}
Truncated 8-orthoplex
Truncated diacosipentacontahexazetton (tek)
10
t0,2{3,3,3,3,3,3,4}
Cantellated 8-orthoplex
Small rhombated diacosipentacontahexazetton (srek)
11
t1,2{3,3,3,3,3,3,4}
Bitruncated 8-orthoplex
Bitruncated diacosipentacontahexazetton (batek)
12
t0,3{3,3,3,3,3,3,4}
Runcinated 8-orthoplex
Small prismated diacosipentacontahexazetton (spek)
13
t1,3{3,3,3,3,3,3,4}
Bicantellated 8-orthoplex
Small birhombated diacosipentacontahexazetton (sabork)
14
t2,3{3,3,3,3,3,3,4}
Tritruncated 8-orthoplex
Tritruncated diacosipentacontahexazetton (tatek)
15
t0,4{3,3,3,3,3,3,4}
Stericated 8-orthoplex
Small cellated diacosipentacontahexazetton (scak)
16
t1,4{3,3,3,3,3,3,4}
Biruncinated 8-orthoplex
Small biprismated diacosipentacontahexazetton (sabpek)
17
t2,4{3,3,3,3,3,3,4}
Tricantellated 8-orthoplex
Small trirhombated diacosipentacontahexazetton (satrek)
18
t3,4{4,3,3,3,3,3,3}
Octeractidiacosipentacontahexazetton (oke)
19
t0,5{3,3,3,3,3,3,4}
Pentellated 8-orthoplex
Small terated diacosipentacontahexazetton (setek)
20
t1,5{3,3,3,3,3,3,4}
Bistericated 8-orthoplex
Small bicellated diacosipentacontahexazetton (sibcak)
21
t2,5{4,3,3,3,3,3,3}
Triruncinated 8-cube
Small triprismato-octeractidiacosipentacontahexazetton (sitpoke)
22
t2,4{4,3,3,3,3,3,3}
Tricantellated 8-cube
Small trirhombated octeract (satro)
23
t2,3{4,3,3,3,3,3,3}
Tritruncated 8-cube
Tritruncated octeract (tato)
24
t0,6{3,3,3,3,3,3,4}
Hexicated 8-orthoplex
Small petated diacosipentacontahexazetton (supek)
25
t1,6{4,3,3,3,3,3,3}
Bipentellated 8-cube
Small biteri-octeractidiacosipentacontahexazetton (sabtoke)
26
t1,5{4,3,3,3,3,3,3}
Bistericated 8-cube
Small bicellated octeract (sobco)
27
t1,4{4,3,3,3,3,3,3}
Biruncinated 8-cube
Small biprismated octeract (sabepo)
28
t1,3{4,3,3,3,3,3,3}
Bicantellated 8-cube
Small birhombated octeract (subro)
29
t1,2{4,3,3,3,3,3,3}
Bitruncated 8-cube
Bitruncated octeract (bato)
30
t0,7{4,3,3,3,3,3,3}
Heptellated 8-cube
Small exi-octeractidiacosipentacontahexazetton (saxoke)
31
t0,6{4,3,3,3,3,3,3}
Hexicated 8-cube
Small petated octeract (supo)
32
t0,5{4,3,3,3,3,3,3}
Pentellated 8-cube
Small terated octeract (soto)
33
t0,4{4,3,3,3,3,3,3}
Stericated 8-cube
Small cellated octeract (soco)
34
t0,3{4,3,3,3,3,3,3}
Runcinated 8-cube
Small prismated octeract (sopo)
35
t0,2{4,3,3,3,3,3,3}
Cantellated 8-cube
Small rhombated octeract (soro)
36
t0,1{4,3,3,3,3,3,3}
Truncated 8-cube
Truncated octeract (tocto)
37
t0,1,2{3,3,3,3,3,3,4}
Cantitruncated 8-orthoplex
Great rhombated diacosipentacontahexazetton
38
t0,1,3{3,3,3,3,3,3,4}
Runcitruncated 8-orthoplex
Prismatotruncated diacosipentacontahexazetton
39
t0,2,3{3,3,3,3,3,3,4}
Runcicantellated 8-orthoplex
Prismatorhombated diacosipentacontahexazetton
40
t1,2,3{3,3,3,3,3,3,4}
Bicantitruncated 8-orthoplex
Great birhombated diacosipentacontahexazetton
41
t0,1,4{3,3,3,3,3,3,4}
Steritruncated 8-orthoplex
Cellitruncated diacosipentacontahexazetton
42
t0,2,4{3,3,3,3,3,3,4}
Stericantellated 8-orthoplex
Cellirhombated diacosipentacontahexazetton
43
t1,2,4{3,3,3,3,3,3,4}
Biruncitruncated 8-orthoplex
Biprismatotruncated diacosipentacontahexazetton
44
t0,3,4{3,3,3,3,3,3,4}
Steriruncinated 8-orthoplex
Celliprismated diacosipentacontahexazetton
45
t1,3,4{3,3,3,3,3,3,4}
Biruncicantellated 8-orthoplex
Biprismatorhombated diacosipentacontahexazetton
46
t2,3,4{3,3,3,3,3,3,4}
Tricantitruncated 8-orthoplex
Great trirhombated diacosipentacontahexazetton
47
t0,1,5{3,3,3,3,3,3,4}
Pentitruncated 8-orthoplex
Teritruncated diacosipentacontahexazetton
48
t0,2,5{3,3,3,3,3,3,4}
Penticantellated 8-orthoplex
Terirhombated diacosipentacontahexazetton
49
t1,2,5{3,3,3,3,3,3,4}
Bisteritruncated 8-orthoplex
Bicellitruncated diacosipentacontahexazetton
50
t0,3,5{3,3,3,3,3,3,4}
Pentiruncinated 8-orthoplex
Teriprismated diacosipentacontahexazetton
51
t1,3,5{3,3,3,3,3,3,4}
Bistericantellated 8-orthoplex
Bicellirhombated diacosipentacontahexazetton
52
t2,3,5{3,3,3,3,3,3,4}
Triruncitruncated 8-orthoplex
Triprismatotruncated diacosipentacontahexazetton
53
t0,4,5{3,3,3,3,3,3,4}
Pentistericated 8-orthoplex
Tericellated diacosipentacontahexazetton
54
t1,4,5{3,3,3,3,3,3,4}
Bisteriruncinated 8-orthoplex
Bicelliprismated diacosipentacontahexazetton
55
t2,3,5{4,3,3,3,3,3,3}
Triruncitruncated 8-cube
Triprismatotruncated octeract
56
t2,3,4{4,3,3,3,3,3,3}
Tricantitruncated 8-cube
Great trirhombated octeract
57
t0,1,6{3,3,3,3,3,3,4}
Hexitruncated 8-orthoplex
Petitruncated diacosipentacontahexazetton
58
t0,2,6{3,3,3,3,3,3,4}
Hexicantellated 8-orthoplex
Petirhombated diacosipentacontahexazetton
59
t1,2,6{3,3,3,3,3,3,4}
Bipentitruncated 8-orthoplex
Biteritruncated diacosipentacontahexazetton
60
t0,3,6{3,3,3,3,3,3,4}
Hexiruncinated 8-orthoplex
Petiprismated diacosipentacontahexazetton
61
t1,3,6{3,3,3,3,3,3,4}
Bipenticantellated 8-orthoplex
Biterirhombated diacosipentacontahexazetton
62
t1,4,5{4,3,3,3,3,3,3}
Bisteriruncinated 8-cube
Bicelliprismated octeract
63
t0,4,6{3,3,3,3,3,3,4}
Hexistericated 8-orthoplex
Peticellated diacosipentacontahexazetton
64
t1,3,6{4,3,3,3,3,3,3}
Bipenticantellated 8-cube
Biterirhombated octeract
65
t1,3,5{4,3,3,3,3,3,3}
Bistericantellated 8-cube
Bicellirhombated octeract
66
t1,3,4{4,3,3,3,3,3,3}
Biruncicantellated 8-cube
Biprismatorhombated octeract
67
t0,5,6{3,3,3,3,3,3,4}
Hexipentellated 8-orthoplex
Petiterated diacosipentacontahexazetton
68
t1,2,6{4,3,3,3,3,3,3}
Bipentitruncated 8-cube
Biteritruncated octeract
69
t1,2,5{4,3,3,3,3,3,3}
Bisteritruncated 8-cube
Bicellitruncated octeract
70
t1,2,4{4,3,3,3,3,3,3}
Biruncitruncated 8-cube
Biprismatotruncated octeract
71
t1,2,3{4,3,3,3,3,3,3}
Bicantitruncated 8-cube
Great birhombated octeract
72
t0,1,7{3,3,3,3,3,3,4}
Heptitruncated 8-orthoplex
Exitruncated diacosipentacontahexazetton
73
t0,2,7{3,3,3,3,3,3,4}
Hepticantellated 8-orthoplex
Exirhombated diacosipentacontahexazetton
74
t0,5,6{4,3,3,3,3,3,3}
Hexipentellated 8-cube
Petiterated octeract
75
t0,3,7{3,3,3,3,3,3,4}
Heptiruncinated 8-orthoplex
Exiprismated diacosipentacontahexazetton
76
t0,4,6{4,3,3,3,3,3,3}
Hexistericated 8-cube
Peticellated octeract
77
t0,4,5{4,3,3,3,3,3,3}
Pentistericated 8-cube
Tericellated octeract
78
t0,3,7{4,3,3,3,3,3,3}
Heptiruncinated 8-cube
Exiprismated octeract
79
t0,3,6{4,3,3,3,3,3,3}
Hexiruncinated 8-cube
Petiprismated octeract
80
t0,3,5{4,3,3,3,3,3,3}
Pentiruncinated 8-cube
Teriprismated octeract
81
t0,3,4{4,3,3,3,3,3,3}
Steriruncinated 8-cube
Celliprismated octeract
82
t0,2,7{4,3,3,3,3,3,3}
Hepticantellated 8-cube
Exirhombated octeract
83
t0,2,6{4,3,3,3,3,3,3}
Hexicantellated 8-cube
Petirhombated octeract
84
t0,2,5{4,3,3,3,3,3,3}
Penticantellated 8-cube
Terirhombated octeract
85
t0,2,4{4,3,3,3,3,3,3}
Stericantellated 8-cube
Cellirhombated octeract
86
t0,2,3{4,3,3,3,3,3,3}
Runcicantellated 8-cube
Prismatorhombated octeract
87
t0,1,7{4,3,3,3,3,3,3}
Heptitruncated 8-cube
Exitruncated octeract
88
t0,1,6{4,3,3,3,3,3,3}
Hexitruncated 8-cube
Petitruncated octeract
89
t0,1,5{4,3,3,3,3,3,3}
Pentitruncated 8-cube
Teritruncated octeract
90
t0,1,4{4,3,3,3,3,3,3}
Steritruncated 8-cube
Cellitruncated octeract
91
t0,1,3{4,3,3,3,3,3,3}
Runcitruncated 8-cube
Prismatotruncated octeract
92
t0,1,2{4,3,3,3,3,3,3}
Cantitruncated 8-cube
Great rhombated octeract
93
t0,1,2,3{3,3,3,3,3,3,4}
Runcicantitruncated 8-orthoplex
Great prismated diacosipentacontahexazetton
94
t0,1,2,4{3,3,3,3,3,3,4}
Stericantitruncated 8-orthoplex
Celligreatorhombated diacosipentacontahexazetton
95
t0,1,3,4{3,3,3,3,3,3,4}
Steriruncitruncated 8-orthoplex
Celliprismatotruncated diacosipentacontahexazetton
96
t0,2,3,4{3,3,3,3,3,3,4}
Steriruncicantellated 8-orthoplex
Celliprismatorhombated diacosipentacontahexazetton
97
t1,2,3,4{3,3,3,3,3,3,4}
Biruncicantitruncated 8-orthoplex
Great biprismated diacosipentacontahexazetton
98
t0,1,2,5{3,3,3,3,3,3,4}
Penticantitruncated 8-orthoplex
Terigreatorhombated diacosipentacontahexazetton
99
t0,1,3,5{3,3,3,3,3,3,4}
Pentiruncitruncated 8-orthoplex
Teriprismatotruncated diacosipentacontahexazetton
100
t0,2,3,5{3,3,3,3,3,3,4}
Pentiruncicantellated 8-orthoplex
Teriprismatorhombated diacosipentacontahexazetton
101
t1,2,3,5{3,3,3,3,3,3,4}
Bistericantitruncated 8-orthoplex
Bicelligreatorhombated diacosipentacontahexazetton
102
t0,1,4,5{3,3,3,3,3,3,4}
Pentisteritruncated 8-orthoplex
Tericellitruncated diacosipentacontahexazetton
103
t0,2,4,5{3,3,3,3,3,3,4}
Pentistericantellated 8-orthoplex
Tericellirhombated diacosipentacontahexazetton
104
t1,2,4,5{3,3,3,3,3,3,4}
Bisteriruncitruncated 8-orthoplex
Bicelliprismatotruncated diacosipentacontahexazetton
105
t0,3,4,5{3,3,3,3,3,3,4}
Pentisteriruncinated 8-orthoplex
Tericelliprismated diacosipentacontahexazetton
106
t1,3,4,5{3,3,3,3,3,3,4}
Bisteriruncicantellated 8-orthoplex
Bicelliprismatorhombated diacosipentacontahexazetton
107
t2,3,4,5{4,3,3,3,3,3,3}
Triruncicantitruncated 8-cube
Great triprismato-octeractidiacosipentacontahexazetton
108
t0,1,2,6{3,3,3,3,3,3,4}
Hexicantitruncated 8-orthoplex
Petigreatorhombated diacosipentacontahexazetton
109
t0,1,3,6{3,3,3,3,3,3,4}
Hexiruncitruncated 8-orthoplex
Petiprismatotruncated diacosipentacontahexazetton
110
t0,2,3,6{3,3,3,3,3,3,4}
Hexiruncicantellated 8-orthoplex
Petiprismatorhombated diacosipentacontahexazetton
111
t1,2,3,6{3,3,3,3,3,3,4}
Bipenticantitruncated 8-orthoplex
Biterigreatorhombated diacosipentacontahexazetton
112
t0,1,4,6{3,3,3,3,3,3,4}
Hexisteritruncated 8-orthoplex
Peticellitruncated diacosipentacontahexazetton
113
t0,2,4,6{3,3,3,3,3,3,4}
Hexistericantellated 8-orthoplex
Peticellirhombated diacosipentacontahexazetton
114
t1,2,4,6{3,3,3,3,3,3,4}
Bipentiruncitruncated 8-orthoplex
Biteriprismatotruncated diacosipentacontahexazetton
115
t0,3,4,6{3,3,3,3,3,3,4}
Hexisteriruncinated 8-orthoplex
Peticelliprismated diacosipentacontahexazetton
116
t1,3,4,6{4,3,3,3,3,3,3}
Bipentiruncicantellated 8-cube
Biteriprismatorhombi-octeractidiacosipentacontahexazetton
117
t1,3,4,5{4,3,3,3,3,3,3}
Bisteriruncicantellated 8-cube
Bicelliprismatorhombated octeract
118
t0,1,5,6{3,3,3,3,3,3,4}
Hexipentitruncated 8-orthoplex
Petiteritruncated diacosipentacontahexazetton
119
t0,2,5,6{3,3,3,3,3,3,4}
Hexipenticantellated 8-orthoplex
Petiterirhombated diacosipentacontahexazetton
120
t1,2,5,6{4,3,3,3,3,3,3}
Bipentisteritruncated 8-cube
Bitericellitrunki-octeractidiacosipentacontahexazetton
121
t0,3,5,6{3,3,3,3,3,3,4}
Hexipentiruncinated 8-orthoplex
Petiteriprismated diacosipentacontahexazetton
122
t1,2,4,6{4,3,3,3,3,3,3}
Bipentiruncitruncated 8-cube
Biteriprismatotruncated octeract
123
t1,2,4,5{4,3,3,3,3,3,3}
Bisteriruncitruncated 8-cube
Bicelliprismatotruncated octeract
124
t0,4,5,6{3,3,3,3,3,3,4}
Hexipentistericated 8-orthoplex
Petitericellated diacosipentacontahexazetton
125
t1,2,3,6{4,3,3,3,3,3,3}
Bipenticantitruncated 8-cube
Biterigreatorhombated octeract
126
t1,2,3,5{4,3,3,3,3,3,3}
Bistericantitruncated 8-cube
Bicelligreatorhombated octeract
127
t1,2,3,4{4,3,3,3,3,3,3}
Biruncicantitruncated 8-cube
Great biprismated octeract
128
t0,1,2,7{3,3,3,3,3,3,4}
Hepticantitruncated 8-orthoplex
Exigreatorhombated diacosipentacontahexazetton
129
t0,1,3,7{3,3,3,3,3,3,4}
Heptiruncitruncated 8-orthoplex
Exiprismatotruncated diacosipentacontahexazetton
130
t0,2,3,7{3,3,3,3,3,3,4}
Heptiruncicantellated 8-orthoplex
Exiprismatorhombated diacosipentacontahexazetton
131
t0,4,5,6{4,3,3,3,3,3,3}
Hexipentistericated 8-cube
Petitericellated octeract
132
t0,1,4,7{3,3,3,3,3,3,4}
Heptisteritruncated 8-orthoplex
Exicellitruncated diacosipentacontahexazetton
133
t0,2,4,7{3,3,3,3,3,3,4}
Heptistericantellated 8-orthoplex
Exicellirhombated diacosipentacontahexazetton
134
t0,3,5,6{4,3,3,3,3,3,3}
Hexipentiruncinated 8-cube
Petiteriprismated octeract
135
t0,3,4,7{4,3,3,3,3,3,3}
Heptisteriruncinated 8-cube
Exicelliprismato-octeractidiacosipentacontahexazetton
136
t0,3,4,6{4,3,3,3,3,3,3}
Hexisteriruncinated 8-cube
Peticelliprismated octeract
137
t0,3,4,5{4,3,3,3,3,3,3}
Pentisteriruncinated 8-cube
Tericelliprismated octeract
138
t0,1,5,7{3,3,3,3,3,3,4}
Heptipentitruncated 8-orthoplex
Exiteritruncated diacosipentacontahexazetton
139
t0,2,5,7{4,3,3,3,3,3,3}
Heptipenticantellated 8-cube
Exiterirhombi-octeractidiacosipentacontahexazetton
140
t0,2,5,6{4,3,3,3,3,3,3}
Hexipenticantellated 8-cube
Petiterirhombated octeract
141
t0,2,4,7{4,3,3,3,3,3,3}
Heptistericantellated 8-cube
Exicellirhombated octeract
142
t0,2,4,6{4,3,3,3,3,3,3}
Hexistericantellated 8-cube
Peticellirhombated octeract
143
t0,2,4,5{4,3,3,3,3,3,3}
Pentistericantellated 8-cube
Tericellirhombated octeract
144
t0,2,3,7{4,3,3,3,3,3,3}
Heptiruncicantellated 8-cube
Exiprismatorhombated octeract
145
t0,2,3,6{4,3,3,3,3,3,3}
Hexiruncicantellated 8-cube
Petiprismatorhombated octeract
146