# B6 polytope

Orthographic projections in the B$_6$ Coxeter plane :
| 6-cube | 6-orthoplex | 6-demicube |
In 6-dimensional geometry, there are 64 uniform polytopes with B$_6$ 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 B$_6$ Coxeter group, and other subgroups.

## Graphs

Symmetric orthographic projections of these 64 polytopes can be made in the B$_6$, B$_5$, B$_4$, B$_3$, B$_2$, A$_5$, A$_3$, Coxeter planes. A$_k$ has [k+1] symmetry, and B$_k$ 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 plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter plane graphs | Coxeter-Dynkin diagram Schläfli symbol Names |
| # | B$_6$ [12] | B$_5$ / D$_4$ / A$_4$ [10] | B$_4$ [8] | B$_3$ / A$_2$ [6] | B$_2$ [4] | A$_5$ [6] | A$_3$ [4] | Coxeter-Dynkin diagram Schläfli symbol Names |
| 1 | | | | | | | | {3,3,3,3,4} 6-orthoplex Hexacontatetrapeton (gee) |
| 2 | | | | | | | | t$_1${3,3,3,3,4} Rectified 6-orthoplex Rectified hexacontatetrapeton (rag) |
| 3 | | | | | | | | t$_2${3,3,3,3,4} Birectified 6-orthoplex Birectified hexacontatetrapeton (brag) |
| 4 | | | | | | | | t$_2${4,3,3,3,3} Birectified 6-cube Birectified hexeract (brox) |
| 5 | | | | | | | | t$_1${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 | | | | | | | | t$_{0,1}${3,3,3,3,4} Truncated 6-orthoplex Truncated hexacontatetrapeton (tag) |
| 8 | | | | | | | | t$_{0,2}${3,3,3,3,4} Cantellated 6-orthoplex Small rhombated hexacontatetrapeton (srog) |
| 9 | | | | | | | | t$_{1,2}${3,3,3,3,4} Bitruncated 6-orthoplex Bitruncated hexacontatetrapeton (botag) |
| 10 | | | | | | | | t$_{0,3}${3,3,3,3,4} Runcinated 6-orthoplex Small prismated hexacontatetrapeton (spog) |
| 11 | | | | | | | | t$_{1,3}${3,3,3,3,4} Bicantellated 6-orthoplex Small birhombated hexacontatetrapeton (siborg) |
| 12 | | | | | | | | t$_{2,3}${4,3,3,3,3} Tritruncated 6-cube Hexeractihexacontitetrapeton (xog) |
| 13 | | | | | | | | t$_{0,4}${3,3,3,3,4} Stericated 6-orthoplex Small cellated hexacontatetrapeton (scag) |
| 14 | | | | | | | | t$_{1,4}${4,3,3,3,3} Biruncinated 6-cube Small biprismato-hexeractihexacontitetrapeton (sobpoxog) |
| 15 | | | | | | | | t$_{1,3}${4,3,3,3,3} Bicantellated 6-cube Small birhombated hexeract (saborx) |
| 16 | | | | | | | | t$_{1,2}${4,3,3,3,3} Bitruncated 6-cube Bitruncated hexeract (botox) |
| 17 | | | | | | | | t$_{0,5}${4,3,3,3,3} Pentellated 6-cube Small teri-hexeractihexacontitetrapeton (stoxog) |
| 18 | | | | | | | | t$_{0,4}${4,3,3,3,3} Stericated 6-cube Small cellated hexeract (scox) |
| 19 | | | | | | | | t$_{0,3}${4,3,3,3,3} Runcinated 6-cube Small prismated hexeract (spox) |
| 20 | | | | | | | | t$_{0,2}${4,3,3,3,3} Cantellated 6-cube Small rhombated hexeract (srox) |
| 21 | | | | | | | | t$_{0,1}${4,3,3,3,3} Truncated 6-cube Truncated hexeract (tox) |
| 22 | | | | | | | | t$_{0,1,2}${3,3,3,3,4} Cantitruncated 6-orthoplex Great rhombated hexacontatetrapeton (grog) |
| 23 | | | | | | | | t$_{0,1,3}${3,3,3,3,4} Runcitruncated 6-orthoplex Prismatotruncated hexacontatetrapeton (potag) |
| 24 | | | | | | | | t$_{0,2,3}${3,3,3,3,4} Runcicantellated 6-orthoplex Prismatorhombated hexacontatetrapeton (prog) |
| 25 | | | | | | | | t$_{1,2,3}${3,3,3,3,4} Bicantitruncated 6-orthoplex Great birhombated hexacontatetrapeton (gaborg) |
| 26 | | | | | | | | t$_{0,1,4}${3,3,3,3,4} Steritruncated 6-orthoplex Cellitruncated hexacontatetrapeton (catog) |
| 27 | | | | | | | | t$_{0,2,4}${3,3,3,3,4} Stericantellated 6-orthoplex Cellirhombated hexacontatetrapeton (crag) |
| 28 | | | | | | | | t$_{1,2,4}${3,3,3,3,4} Biruncitruncated 6-orthoplex Biprismatotruncated hexacontatetrapeton (boprax) |
| 29 | | | | | | | | t$_{0,3,4}${3,3,3,3,4} Steriruncinated 6-orthoplex Celliprismated hexacontatetrapeton (copog) |
| 30 | | | | | | | | t$_{1,2,4}${4,3,3,3,3} Biruncitruncated 6-cube Biprismatotruncated hexeract (boprag) |
| 31 | | | | | | | | t$_{1,2,3}${4,3,3,3,3} Bicantitruncated 6-cube Great birhombated hexeract (gaborx) |
| 32 | | | | | | | | t$_{0,1,5}${3,3,3,3,4} Pentitruncated 6-orthoplex Teritruncated hexacontatetrapeton (tacox) |
| 33 | | | | | | | | t$_{0,2,5}${3,3,3,3,4} Penticantellated 6-orthoplex Terirhombated hexacontatetrapeton (tapox) |
| 34 | | | | | | | | t$_{0,3,4}${4,3,3,3,3} Steriruncinated 6-cube Celliprismated hexeract (copox) |
| 35 | | | | | | | | t$_{0,2,5}${4,3,3,3,3} Penticantellated 6-cube Terirhombated hexeract (topag) |
| 36 | | | | | | | | t$_{0,2,4}${4,3,3,3,3} Stericantellated 6-cube Cellirhombated hexeract (crax) |
| 37 | | | | | | | | t$_{0,2,3}${4,3,3,3,3} Runcicantellated 6-cube Prismatorhombated hexeract (prox) |
| 38 | | | | | | | | t$_{0,1,5}${4,3,3,3,3} Pentitruncated 6-cube Teritruncated hexeract (tacog) |
| 39 | | | | | | | | t$_{0,1,4}${4,3,3,3,3} Steritruncated 6-cube Cellitruncated hexeract (catax) |
| 40 | | | | | | | | t$_{0,1,3}${4,3,3,3,3} Runcitruncated 6-cube Prismatotruncated hexeract (potax) |
| 41 | | | | | | | | t$_{0,1,2}${4,3,3,3,3} Cantitruncated 6-cube Great rhombated hexeract (grox) |
| 42 | | | | | | | | t$_{0,1,2,3}${3,3,3,3,4} Runcicantitruncated 6-orthoplex Great prismated hexacontatetrapeton (gopog) |
| 43 | | | | | | | | t$_{0,1,2,4}${3,3,3,3,4} Stericantitruncated 6-orthoplex Celligreatorhombated hexacontatetrapeton (cagorg) |
| 44 | | | | | | | | t$_{0,1,3,4}${3,3,3,3,4} Steriruncitruncated 6-orthoplex Celliprismatotruncated hexacontatetrapeton (captog) |
| 45 | | | | | | | | t$_{0,2,3,4}${3,3,3,3,4} Steriruncicantellated 6-orthoplex Celliprismatorhombated hexacontatetrapeton (coprag) |
| 46 | | | | | | | | t$_{1,2,3,4}${4,3,3,3,3} Biruncicantitruncated 6-cube Great biprismato-hexeractihexacontitetrapeton (gobpoxog) |
| 47 | | | | | | | | t$_{0,1,2,5}${3,3,3,3,4} Penticantitruncated 6-orthoplex Terigreatorhombated hexacontatetrapeton (togrig) |
| 48 | | | | | | | | t$_{0,1,3,5}${3,3,3,3,4} Pentiruncitruncated 6-orthoplex Teriprismatotruncated hexacontatetrapeton (tocrax) |
| 49 | | | | | | | | t$_{0,2,3,5}${4,3,3,3,3} Pentiruncicantellated 6-cube Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog) |
| 50 | | | | | | | | t$_{0,2,3,4}${4,3,3,3,3} Steriruncicantellated 6-cube Celliprismatorhombated hexeract (coprix) |
| 51 | | | | | | | | t$_{0,1,4,5}${4,3,3,3,3} Pentisteritruncated 6-cube Tericelli-hexeractihexacontitetrapeton (tactaxog) |
| 52 | | | | | | | | t$_{0,1,3,5}${4,3,3,3,3} Pentiruncitruncated 6-cube Teriprismatotruncated hexeract (tocrag) |
| 53 | | | | | | | | t$_{0,1,3,4}${4,3,3,3,3} Steriruncitruncated 6-cube Celliprismatotruncated hexeract (captix) |
| 54 | | | | | | | | t$_{0,1,2,5}${4,3,3,3,3} Penticantitruncated 6-cube Terigreatorhombated hexeract (togrix) |
| 55 | | | | | | | | t$_{0,1,2,4}${4,3,3,3,3} Stericantitruncated 6-cube Celligreatorhombated hexeract (cagorx) |
| 56 | | | | | | | | t$_{0,1,2,3}${4,3,3,3,3} Runcicantitruncated 6-cube Great prismated hexeract (gippox) |
| 57 | | | | | | | | t$_{0,1,2,3,4}${3,3,3,3,4} Steriruncicantitruncated 6-orthoplex Great cellated hexacontatetrapeton (gocog) |
| 58 | | | | | | | | t$_{0,1,2,3,5}${3,3,3,3,4} Pentiruncicantitruncated 6-orthoplex Terigreatoprismated hexacontatetrapeton (tagpog) |
| 59 | | | | | | | | t$_{0,1,2,4,5}${3,3,3,3,4} Pentistericantitruncated 6-orthoplex Tericelligreatorhombated hexacontatetrapeton (tecagorg) |
| 60 | | | | | | | | t$_{0,1,2,4,5}${4,3,3,3,3} Pentistericantitruncated 6-cube Tericelligreatorhombated hexeract (tocagrax) |
| 61 | | | | | | | | t$_{0,1,2,3,5}${4,3,3,3,3} Pentiruncicantitruncated 6-cube Terigreatoprismated hexeract (tagpox) |
| 62 | | | | | | | | t$_{0,1,2,3,4}${4,3,3,3,3} Steriruncicantitruncated 6-cube Great cellated hexeract (gocax) |
| 63 | | | | | | | | t$_{0,1,2,3,4,5}${4,3,3,3,3} Omnitruncated 6-cube Great teri-hexeractihexacontitetrapeton (gotaxog) |