B8 polytope

Orthographic projections in the B8 Coxeter plane

8-cube
8-orthoplex
8-demicube

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[edit]

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
Schläfli 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}
Quadritruncated 8-cube
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