Orthographic projections in the B6 Coxeter plane

6-cube

6-orthoplex

6-demicube

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

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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

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  • 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

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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds