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B6 polytope

{{DISPLAYTITLE:B6 polytope}}

class=wikitable align=right width=320

|+ Orthographic projections in the B6 Coxeter plane

align=center

|160px
6-cube
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}

|160px
6-orthoplex
{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}

|160px
6-demicube
{{CDD|node_h|4|node|3|node|3|node|3|node|3|node}}

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.

__TOC__

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.

class="wikitable"

!rowspan=2|#

!colspan=7|Coxeter plane graphs

!rowspan=2|Coxeter-Dynkin diagram
Schläfli symbol
Names

B6
[12]|| B5 / D4 / A4
[10]|| B4
[8]|| B3 / A2
[6]|| B2
[4]|| A5
[6]|| A3
[4]
align=center BGCOLOR="#f0e0e0"

!1

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}
{3,3,3,3,4}
6-orthoplex
Hexacontatetrapeton (gee)
align=center BGCOLOR="#f0e0e0"

!2

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node|3|node|3|node_1|3|node}}
t1{3,3,3,3,4}
Rectified 6-orthoplex
Rectified hexacontatetrapeton (rag)
align=center BGCOLOR="#f0e0e0"

!3

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node|3|node_1|3|node|3|node}}
t2{3,3,3,3,4}
Birectified 6-orthoplex
Birectified hexacontatetrapeton (brag)
align=center BGCOLOR="#e0e0f0"

!4

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node|3|node|3|node}}
t2{4,3,3,3,3}
Birectified 6-cube
Birectified hexeract (brox)
align=center BGCOLOR="#e0e0f0"

!5

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node|3|node|3|node}}
t1{4,3,3,3,3}
Rectified 6-cube
Rectified hexeract (rax)
align=center BGCOLOR="#e0e0f0"

!6

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
{4,3,3,3,3}
6-cube
Hexeract (ax)
align=center BGCOLOR="#d0f0f0"

!64

80px80px80px80px80px80px80px{{CDD|node_h|4|node|3|node|3|node|3|node|3|node}}
h{4,3,3,3,3}
6-demicube
Hemihexeract
align=center BGCOLOR="#f0e0e0"

!7

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1}}
t0,1{3,3,3,3,4}
Truncated 6-orthoplex
Truncated hexacontatetrapeton (tag)
align=center BGCOLOR="#f0e0e0"

!8

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1}}
t0,2{3,3,3,3,4}
Cantellated 6-orthoplex
Small rhombated hexacontatetrapeton (srog)
align=center BGCOLOR="#f0e0e0"

!9

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node}}
t1,2{3,3,3,3,4}
Bitruncated 6-orthoplex
Bitruncated hexacontatetrapeton (botag)
align=center BGCOLOR="#f0e0e0"

!10

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1}}
t0,3{3,3,3,3,4}
Runcinated 6-orthoplex
Small prismated hexacontatetrapeton (spog)
align=center BGCOLOR="#f0e0e0"

!11

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node}}
t1,3{3,3,3,3,4}
Bicantellated 6-orthoplex
Small birhombated hexacontatetrapeton (siborg)
align=center BGCOLOR="#e0f0e0"

!12

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node}}
t2,3{4,3,3,3,3}
Tritruncated 6-cube
Hexeractihexacontitetrapeton (xog)
align=center BGCOLOR="#f0e0e0"

!13

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1}}
t0,4{3,3,3,3,4}
Stericated 6-orthoplex
Small cellated hexacontatetrapeton (scag)
align=center BGCOLOR="#e0f0e0"

!14

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node}}
t1,4{4,3,3,3,3}
Biruncinated 6-cube
Small biprismato-hexeractihexacontitetrapeton (sobpoxog)
align=center BGCOLOR="#e0e0f0"

!15

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node}}
t1,3{4,3,3,3,3}
Bicantellated 6-cube
Small birhombated hexeract (saborx)
align=center BGCOLOR="#e0e0f0"

!16

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node}}
t1,2{4,3,3,3,3}
Bitruncated 6-cube
Bitruncated hexeract (botox)
align=center BGCOLOR="#e0f0e0"

!17

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1}}
t0,5{4,3,3,3,3}
Pentellated 6-cube
Small teri-hexeractihexacontitetrapeton (stoxog)
align=center BGCOLOR="#e0e0f0"

!18

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node}}
t0,4{4,3,3,3,3}
Stericated 6-cube
Small cellated hexeract (scox)
align=center BGCOLOR="#e0e0f0"

!19

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node}}
t0,3{4,3,3,3,3}
Runcinated 6-cube
Small prismated hexeract (spox)
align=center BGCOLOR="#e0e0f0"

!20

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node}}
t0,2{4,3,3,3,3}
Cantellated 6-cube
Small rhombated hexeract (srox)
align=center BGCOLOR="#e0e0f0"

!21

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node}}
t0,1{4,3,3,3,3}
Truncated 6-cube
Truncated hexeract (tox)
align=center BGCOLOR="#f0e0e0"

!22

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1}}
t0,1,2{3,3,3,3,4}
Cantitruncated 6-orthoplex
Great rhombated hexacontatetrapeton (grog)
align=center BGCOLOR="#f0e0e0"

!23

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1}}
t0,1,3{3,3,3,3,4}
Runcitruncated 6-orthoplex
Prismatotruncated hexacontatetrapeton (potag)
align=center BGCOLOR="#f0e0e0"

!24

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1}}
t0,2,3{3,3,3,3,4}
Runcicantellated 6-orthoplex
Prismatorhombated hexacontatetrapeton (prog)
align=center BGCOLOR="#f0e0e0"

!25

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node}}
t1,2,3{3,3,3,3,4}
Bicantitruncated 6-orthoplex
Great birhombated hexacontatetrapeton (gaborg)
align=center BGCOLOR="#f0e0e0"

!26

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1}}
t0,1,4{3,3,3,3,4}
Steritruncated 6-orthoplex
Cellitruncated hexacontatetrapeton (catog)
align=center BGCOLOR="#f0e0e0"

!27

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1}}
t0,2,4{3,3,3,3,4}
Stericantellated 6-orthoplex
Cellirhombated hexacontatetrapeton (crag)
align=center BGCOLOR="#f0e0e0"

!28

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node}}
t1,2,4{3,3,3,3,4}
Biruncitruncated 6-orthoplex
Biprismatotruncated hexacontatetrapeton (boprax)
align=center BGCOLOR="#f0e0e0"

!29

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1}}
t0,3,4{3,3,3,3,4}
Steriruncinated 6-orthoplex
Celliprismated hexacontatetrapeton (copog)
align=center BGCOLOR="#e0e0f0"

!30

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node}}
t1,2,4{4,3,3,3,3}
Biruncitruncated 6-cube
Biprismatotruncated hexeract (boprag)
align=center BGCOLOR="#e0e0f0"

!31

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node}}
t1,2,3{4,3,3,3,3}
Bicantitruncated 6-cube
Great birhombated hexeract (gaborx)
align=center BGCOLOR="#f0e0e0"

!32

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1}}
t0,1,5{3,3,3,3,4}
Pentitruncated 6-orthoplex
Teritruncated hexacontatetrapeton (tacox)
align=center BGCOLOR="#f0e0e0"

!33

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1}}
t0,2,5{3,3,3,3,4}
Penticantellated 6-orthoplex
Terirhombated hexacontatetrapeton (tapox)
align=center BGCOLOR="#e0e0f0"

!34

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node}}
t0,3,4{4,3,3,3,3}
Steriruncinated 6-cube
Celliprismated hexeract (copox)
align=center BGCOLOR="#e0e0f0"

!35

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1}}
t0,2,5{4,3,3,3,3}
Penticantellated 6-cube
Terirhombated hexeract (topag)
align=center BGCOLOR="#e0e0f0"

!36

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node}}
t0,2,4{4,3,3,3,3}
Stericantellated 6-cube
Cellirhombated hexeract (crax)
align=center BGCOLOR="#e0e0f0"

!37

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node}}
t0,2,3{4,3,3,3,3}
Runcicantellated 6-cube
Prismatorhombated hexeract (prox)
align=center BGCOLOR="#e0e0f0"

!38

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1}}
t0,1,5{4,3,3,3,3}
Pentitruncated 6-cube
Teritruncated hexeract (tacog)
align=center BGCOLOR="#e0e0f0"

!39

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node}}
t0,1,4{4,3,3,3,3}
Steritruncated 6-cube
Cellitruncated hexeract (catax)
align=center BGCOLOR="#e0e0f0"

!40

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node}}
t0,1,3{4,3,3,3,3}
Runcitruncated 6-cube
Prismatotruncated hexeract (potax)
align=center BGCOLOR="#e0e0f0"

!41

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node}}
t0,1,2{4,3,3,3,3}
Cantitruncated 6-cube
Great rhombated hexeract (grox)
align=center BGCOLOR="#f0e0e0"

!42

|80px

80px80px80px80px80px80px{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}
t0,1,2,3{3,3,3,3,4}
Runcicantitruncated 6-orthoplex
Great prismated hexacontatetrapeton (gopog)
align=center BGCOLOR="#f0e0e0"

!43

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1}}
t0,1,2,4{3,3,3,3,4}
Stericantitruncated 6-orthoplex
Celligreatorhombated hexacontatetrapeton (cagorg)
align=center BGCOLOR="#f0e0e0"

!44

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1}}
t0,1,3,4{3,3,3,3,4}
Steriruncitruncated 6-orthoplex
Celliprismatotruncated hexacontatetrapeton (captog)
align=center BGCOLOR="#f0e0e0"

!45

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1}}
t0,2,3,4{3,3,3,3,4}
Steriruncicantellated 6-orthoplex
Celliprismatorhombated hexacontatetrapeton (coprag)
align=center BGCOLOR="#e0f0e0"

!46

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node}}
t1,2,3,4{4,3,3,3,3}
Biruncicantitruncated 6-cube
Great biprismato-hexeractihexacontitetrapeton (gobpoxog)
align=center BGCOLOR="#f0e0e0"

!47

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1}}
t0,1,2,5{3,3,3,3,4}
Penticantitruncated 6-orthoplex
Terigreatorhombated hexacontatetrapeton (togrig)
align=center BGCOLOR="#f0e0e0"

!48

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1}}
t0,1,3,5{3,3,3,3,4}
Pentiruncitruncated 6-orthoplex
Teriprismatotruncated hexacontatetrapeton (tocrax)
align=center BGCOLOR="#e0f0e0"

!49

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1}}
t0,2,3,5{4,3,3,3,3}
Pentiruncicantellated 6-cube
Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)
align=center BGCOLOR="#e0e0f0"

!50

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node}}
t0,2,3,4{4,3,3,3,3}
Steriruncicantellated 6-cube
Celliprismatorhombated hexeract (coprix)
align=center BGCOLOR="#e0f0e0"

!51

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1}}
t0,1,4,5{4,3,3,3,3}
Pentisteritruncated 6-cube
Tericelli-hexeractihexacontitetrapeton (tactaxog)
align=center BGCOLOR="#e0e0f0"

!52

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1}}
t0,1,3,5{4,3,3,3,3}
Pentiruncitruncated 6-cube
Teriprismatotruncated hexeract (tocrag)
align=center BGCOLOR="#e0e0f0"

!53

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node}}
t0,1,3,4{4,3,3,3,3}
Steriruncitruncated 6-cube
Celliprismatotruncated hexeract (captix)
align=center BGCOLOR="#e0e0f0"

!54

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1}}
t0,1,2,5{4,3,3,3,3}
Penticantitruncated 6-cube
Terigreatorhombated hexeract (togrix)
align=center BGCOLOR="#e0e0f0"

!55

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node}}
t0,1,2,4{4,3,3,3,3}
Stericantitruncated 6-cube
Celligreatorhombated hexeract (cagorx)
align=center BGCOLOR="#e0e0f0"

!56

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node}}
t0,1,2,3{4,3,3,3,3}
Runcicantitruncated 6-cube
Great prismated hexeract (gippox)
align=center BGCOLOR="#f0e0e0"

!57

|80px

80px80px80px80px80px80px{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
t0,1,2,3,4{3,3,3,3,4}
Steriruncicantitruncated 6-orthoplex
Great cellated hexacontatetrapeton (gocog)
align=center BGCOLOR="#f0e0e0"

!58

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}
t0,1,2,3,5{3,3,3,3,4}
Pentiruncicantitruncated 6-orthoplex
Terigreatoprismated hexacontatetrapeton (tagpog)
align=center BGCOLOR="#f0e0e0"

!59

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1}}
t0,1,2,4,5{3,3,3,3,4}
Pentistericantitruncated 6-orthoplex
Tericelligreatorhombated hexacontatetrapeton (tecagorg)
align=center BGCOLOR="#e0e0f0"

!60

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1}}
t0,1,2,4,5{4,3,3,3,3}
Pentistericantitruncated 6-cube
Tericelligreatorhombated hexeract (tocagrax)
align=center BGCOLOR="#e0e0f0"

!61

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1}}
t0,1,2,3,5{4,3,3,3,3}
Pentiruncicantitruncated 6-cube
Terigreatoprismated hexeract (tagpox)
align=center BGCOLOR="#e0e0f0"

!62

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node}}
t0,1,2,3,4{4,3,3,3,3}
Steriruncicantitruncated 6-cube
Great cellated hexeract (gocax)
align=center BGCOLOR="#e0f0e0"

!63

|80px

80px80px80px80px80px80px{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html 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
  • {{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}}

Notes

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Category:6-polytopes