In 6-dimensional geometry, there are 64 uniform polytopes with B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices respectively. The 6-demicube is added with half the symmetry.

They can be visualized as symmetric orthographic projections in Coxeter planes of the B6 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 64 polytopes can be made in the B6, B5, B4, B3, B2, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.

These 64 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter plane graphs Coxeter-Dynkin diagram
Schläfli symbol
Names
B6
[12]
B5 / D4 / A4
[10]
B4
[8]
B3 / A2
[6]
B2
[4]
A5
[6]
A3
[4]
1
{3,3,3,3,4}
6-orthoplex
Hexacontatetrapeton (gee)
2
t1{3,3,3,3,4}
Rectified 6-orthoplex
Rectified hexacontatetrapeton (rag)
3
t2{3,3,3,3,4}
Birectified 6-orthoplex
Birectified hexacontatetrapeton (brag)
4
t2{4,3,3,3,3}
Birectified 6-cube
Birectified hexeract (brox)
5
t1{4,3,3,3,3}
Rectified 6-cube
Rectified hexeract (rax)
6
{4,3,3,3,3}
6-cube
Hexeract (ax)
64
h{4,3,3,3,3}
6-demicube
Hemihexeract
7
t0,1{3,3,3,3,4}
Truncated 6-orthoplex
Truncated hexacontatetrapeton (tag)
8
t0,2{3,3,3,3,4}
Cantellated 6-orthoplex
Small rhombated hexacontatetrapeton (srog)
9
t1,2{3,3,3,3,4}
Bitruncated 6-orthoplex
Bitruncated hexacontatetrapeton (botag)
10
t0,3{3,3,3,3,4}
Runcinated 6-orthoplex
Small prismated hexacontatetrapeton (spog)
11
t1,3{3,3,3,3,4}
Bicantellated 6-orthoplex
Small birhombated hexacontatetrapeton (siborg)
12
t2,3{4,3,3,3,3}
Tritruncated 6-cube
Hexeractihexacontitetrapeton (xog)
13
t0,4{3,3,3,3,4}
Stericated 6-orthoplex
Small cellated hexacontatetrapeton (scag)
14
t1,4{4,3,3,3,3}
Biruncinated 6-cube
Small biprismato-hexeractihexacontitetrapeton (sobpoxog)
15
t1,3{4,3,3,3,3}
Bicantellated 6-cube
Small birhombated hexeract (saborx)
16
t1,2{4,3,3,3,3}
Bitruncated 6-cube
Bitruncated hexeract (botox)
17
t0,5{4,3,3,3,3}
Pentellated 6-cube
Small teri-hexeractihexacontitetrapeton (stoxog)
18
t0,4{4,3,3,3,3}
Stericated 6-cube
Small cellated hexeract (scox)
19
t0,3{4,3,3,3,3}
Runcinated 6-cube
Small prismated hexeract (spox)
20
t0,2{4,3,3,3,3}
Cantellated 6-cube
Small rhombated hexeract (srox)
21
t0,1{4,3,3,3,3}
Truncated 6-cube
Truncated hexeract (tox)
22
t0,1,2{3,3,3,3,4}
Cantitruncated 6-orthoplex
Great rhombated hexacontatetrapeton (grog)
23
t0,1,3{3,3,3,3,4}
Runcitruncated 6-orthoplex
Prismatotruncated hexacontatetrapeton (potag)
24
t0,2,3{3,3,3,3,4}
Runcicantellated 6-orthoplex
Prismatorhombated hexacontatetrapeton (prog)
25
t1,2,3{3,3,3,3,4}
Bicantitruncated 6-orthoplex
Great birhombated hexacontatetrapeton (gaborg)
26
t0,1,4{3,3,3,3,4}
Steritruncated 6-orthoplex
Cellitruncated hexacontatetrapeton (catog)
27
t0,2,4{3,3,3,3,4}
Stericantellated 6-orthoplex
Cellirhombated hexacontatetrapeton (crag)
28
t1,2,4{3,3,3,3,4}
Biruncitruncated 6-orthoplex
Biprismatotruncated hexacontatetrapeton (boprax)
29
t0,3,4{3,3,3,3,4}
Steriruncinated 6-orthoplex
Celliprismated hexacontatetrapeton (copog)
30
t1,2,4{4,3,3,3,3}
Biruncitruncated 6-cube
Biprismatotruncated hexeract (boprag)
31
t1,2,3{4,3,3,3,3}
Bicantitruncated 6-cube
Great birhombated hexeract (gaborx)
32
t0,1,5{3,3,3,3,4}
Pentitruncated 6-orthoplex
Teritruncated hexacontatetrapeton (tacox)
33
t0,2,5{3,3,3,3,4}
Penticantellated 6-orthoplex
Terirhombated hexacontatetrapeton (tapox)
34
t0,3,4{4,3,3,3,3}
Steriruncinated 6-cube
Celliprismated hexeract (copox)
35
t0,2,5{4,3,3,3,3}
Penticantellated 6-cube
Terirhombated hexeract (topag)
36
t0,2,4{4,3,3,3,3}
Stericantellated 6-cube
Cellirhombated hexeract (crax)
37
t0,2,3{4,3,3,3,3}
Runcicantellated 6-cube
Prismatorhombated hexeract (prox)
38
t0,1,5{4,3,3,3,3}
Pentitruncated 6-cube
Teritruncated hexeract (tacog)
39
t0,1,4{4,3,3,3,3}
Steritruncated 6-cube
Cellitruncated hexeract (catax)
40
t0,1,3{4,3,3,3,3}
Runcitruncated 6-cube
Prismatotruncated hexeract (potax)
41
t0,1,2{4,3,3,3,3}
Cantitruncated 6-cube
Great rhombated hexeract (grox)
42
t0,1,2,3{3,3,3,3,4}
Runcicantitruncated 6-orthoplex
Great prismated hexacontatetrapeton (gopog)
43
t0,1,2,4{3,3,3,3,4}
Stericantitruncated 6-orthoplex
Celligreatorhombated hexacontatetrapeton (cagorg)
44
t0,1,3,4{3,3,3,3,4}
Steriruncitruncated 6-orthoplex
Celliprismatotruncated hexacontatetrapeton (captog)
45
t0,2,3,4{3,3,3,3,4}
Steriruncicantellated 6-orthoplex
Celliprismatorhombated hexacontatetrapeton (coprag)
46
t1,2,3,4{4,3,3,3,3}
Biruncicantitruncated 6-cube
Great biprismato-hexeractihexacontitetrapeton (gobpoxog)
47
t0,1,2,5{3,3,3,3,4}
Penticantitruncated 6-orthoplex
Terigreatorhombated hexacontatetrapeton (togrig)
48
t0,1,3,5{3,3,3,3,4}
Pentiruncitruncated 6-orthoplex
Teriprismatotruncated hexacontatetrapeton (tocrax)
49
t0,2,3,5{4,3,3,3,3}
Pentiruncicantellated 6-cube
Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)
50
t0,2,3,4{4,3,3,3,3}
Steriruncicantellated 6-cube
Celliprismatorhombated hexeract (coprix)
51
t0,1,4,5{4,3,3,3,3}
Pentisteritruncated 6-cube
Tericelli-hexeractihexacontitetrapeton (tactaxog)
52
t0,1,3,5{4,3,3,3,3}
Pentiruncitruncated 6-cube
Teriprismatotruncated hexeract (tocrag)
53
t0,1,3,4{4,3,3,3,3}
Steriruncitruncated 6-cube
Celliprismatotruncated hexeract (captix)
54
t0,1,2,5{4,3,3,3,3}
Penticantitruncated 6-cube
Terigreatorhombated hexeract (togrix)
55
t0,1,2,4{4,3,3,3,3}
Stericantitruncated 6-cube
Celligreatorhombated hexeract (cagorx)
56
t0,1,2,3{4,3,3,3,3}
Runcicantitruncated 6-cube
Great prismated hexeract (gippox)
57
t0,1,2,3,4{3,3,3,3,4}
Steriruncicantitruncated 6-orthoplex
Great cellated hexacontatetrapeton (gocog)
58
t0,1,2,3,5{3,3,3,3,4}
Pentiruncicantitruncated 6-orthoplex
Terigreatoprismated hexacontatetrapeton (tagpog)
59
t0,1,2,4,5{3,3,3,3,4}
Pentistericantitruncated 6-orthoplex
Tericelligreatorhombated hexacontatetrapeton (tecagorg)
60
t0,1,2,4,5{4,3,3,3,3}
Pentistericantitruncated 6-cube
Tericelligreatorhombated hexeract (tocagrax)
61
t0,1,2,3,5{4,3,3,3,3}
Pentiruncicantitruncated 6-cube
Terigreatoprismated hexeract (tagpox)
62
t0,1,2,3,4{4,3,3,3,3}
Steriruncicantitruncated 6-cube
Great cellated hexeract (gocax)
63
t0,1,2,3,4,5{4,3,3,3,3}
Omnitruncated 6-cube
Great teri-hexeractihexacontitetrapeton (gotaxog)

References

H.S.M. Coxeter:
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 [1]
(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. "6D uniform polytopes (polypeta)".

Notes

Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds