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

1  {3,3,3,3,4} 6orthoplex Hexacontatetrapeton (gee) 

2  t_{1}{3,3,3,3,4} Rectified 6orthoplex Rectified hexacontatetrapeton (rag) 

3  t_{2}{3,3,3,3,4} Birectified 6orthoplex Birectified hexacontatetrapeton (brag) 

4  t_{2}{4,3,3,3,3} Birectified 6cube Birectified hexeract (brox) 

5  t_{1}{4,3,3,3,3} Rectified 6cube Rectified hexeract (rax) 

6  {4,3,3,3,3} 6cube Hexeract (ax) 

64  h{4,3,3,3,3} 6demicube Hemihexeract 

7  t_{0,1}{3,3,3,3,4} Truncated 6orthoplex Truncated hexacontatetrapeton (tag) 

8  t_{0,2}{3,3,3,3,4} Cantellated 6orthoplex Small rhombated hexacontatetrapeton (srog) 

9  t_{1,2}{3,3,3,3,4} Bitruncated 6orthoplex Bitruncated hexacontatetrapeton (botag) 

10  t_{0,3}{3,3,3,3,4} Runcinated 6orthoplex Small prismated hexacontatetrapeton (spog) 

11  t_{1,3}{3,3,3,3,4} Bicantellated 6orthoplex Small birhombated hexacontatetrapeton (siborg) 

12  t_{2,3}{4,3,3,3,3} Tritruncated 6cube Hexeractihexacontitetrapeton (xog) 

13  t_{0,4}{3,3,3,3,4} Stericated 6orthoplex Small cellated hexacontatetrapeton (scag) 

14  t_{1,4}{4,3,3,3,3} Biruncinated 6cube Small biprismatohexeractihexacontitetrapeton (sobpoxog) 

15  t_{1,3}{4,3,3,3,3} Bicantellated 6cube Small birhombated hexeract (saborx) 

16  t_{1,2}{4,3,3,3,3} Bitruncated 6cube Bitruncated hexeract (botox) 

17  t_{0,5}{4,3,3,3,3} Pentellated 6cube Small terihexeractihexacontitetrapeton (stoxog) 

18  t_{0,4}{4,3,3,3,3} Stericated 6cube Small cellated hexeract (scox) 

19  t_{0,3}{4,3,3,3,3} Runcinated 6cube Small prismated hexeract (spox) 

20  t_{0,2}{4,3,3,3,3} Cantellated 6cube Small rhombated hexeract (srox) 

21  t_{0,1}{4,3,3,3,3} Truncated 6cube Truncated hexeract (tox) 

22  t_{0,1,2}{3,3,3,3,4} Cantitruncated 6orthoplex Great rhombated hexacontatetrapeton (grog) 

23  t_{0,1,3}{3,3,3,3,4} Runcitruncated 6orthoplex Prismatotruncated hexacontatetrapeton (potag) 

24  t_{0,2,3}{3,3,3,3,4} Runcicantellated 6orthoplex Prismatorhombated hexacontatetrapeton (prog) 

25  t_{1,2,3}{3,3,3,3,4} Bicantitruncated 6orthoplex Great birhombated hexacontatetrapeton (gaborg) 

26  t_{0,1,4}{3,3,3,3,4} Steritruncated 6orthoplex Cellitruncated hexacontatetrapeton (catog) 

27  t_{0,2,4}{3,3,3,3,4} Stericantellated 6orthoplex Cellirhombated hexacontatetrapeton (crag) 

28  t_{1,2,4}{3,3,3,3,4} Biruncitruncated 6orthoplex Biprismatotruncated hexacontatetrapeton (boprax) 

29  t_{0,3,4}{3,3,3,3,4} Steriruncinated 6orthoplex Celliprismated hexacontatetrapeton (copog) 

30  t_{1,2,4}{4,3,3,3,3} Biruncitruncated 6cube Biprismatotruncated hexeract (boprag) 

31  t_{1,2,3}{4,3,3,3,3} Bicantitruncated 6cube Great birhombated hexeract (gaborx) 

32  t_{0,1,5}{3,3,3,3,4} Pentitruncated 6orthoplex Teritruncated hexacontatetrapeton (tacox) 

33  t_{0,2,5}{3,3,3,3,4} Penticantellated 6orthoplex Terirhombated hexacontatetrapeton (tapox) 

34  t_{0,3,4}{4,3,3,3,3} Steriruncinated 6cube Celliprismated hexeract (copox) 

35  t_{0,2,5}{4,3,3,3,3} Penticantellated 6cube Terirhombated hexeract (topag) 

36  t_{0,2,4}{4,3,3,3,3} Stericantellated 6cube Cellirhombated hexeract (crax) 

37  t_{0,2,3}{4,3,3,3,3} Runcicantellated 6cube Prismatorhombated hexeract (prox) 

38  t_{0,1,5}{4,3,3,3,3} Pentitruncated 6cube Teritruncated hexeract (tacog) 

39  t_{0,1,4}{4,3,3,3,3} Steritruncated 6cube Cellitruncated hexeract (catax) 

40  t_{0,1,3}{4,3,3,3,3} Runcitruncated 6cube Prismatotruncated hexeract (potax) 

41  t_{0,1,2}{4,3,3,3,3} Cantitruncated 6cube Great rhombated hexeract (grox) 

42  t_{0,1,2,3}{3,3,3,3,4} Runcicantitruncated 6orthoplex Great prismated hexacontatetrapeton (gopog) 

43  t_{0,1,2,4}{3,3,3,3,4} Stericantitruncated 6orthoplex Celligreatorhombated hexacontatetrapeton (cagorg) 

44  t_{0,1,3,4}{3,3,3,3,4} Steriruncitruncated 6orthoplex Celliprismatotruncated hexacontatetrapeton (captog) 

45  t_{0,2,3,4}{3,3,3,3,4} Steriruncicantellated 6orthoplex Celliprismatorhombated hexacontatetrapeton (coprag) 

46  t_{1,2,3,4}{4,3,3,3,3} Biruncicantitruncated 6cube Great biprismatohexeractihexacontitetrapeton (gobpoxog) 

47  t_{0,1,2,5}{3,3,3,3,4} Penticantitruncated 6orthoplex Terigreatorhombated hexacontatetrapeton (togrig) 

48  t_{0,1,3,5}{3,3,3,3,4} Pentiruncitruncated 6orthoplex Teriprismatotruncated hexacontatetrapeton (tocrax) 

49  t_{0,2,3,5}{4,3,3,3,3} Pentiruncicantellated 6cube Teriprismatorhombihexeractihexacontitetrapeton (tiprixog) 

50  t_{0,2,3,4}{4,3,3,3,3} Steriruncicantellated 6cube Celliprismatorhombated hexeract (coprix) 

51  t_{0,1,4,5}{4,3,3,3,3} Pentisteritruncated 6cube Tericellihexeractihexacontitetrapeton (tactaxog) 

52  t_{0,1,3,5}{4,3,3,3,3} Pentiruncitruncated 6cube Teriprismatotruncated hexeract (tocrag) 

53  t_{0,1,3,4}{4,3,3,3,3} Steriruncitruncated 6cube Celliprismatotruncated hexeract (captix) 

54  t_{0,1,2,5}{4,3,3,3,3} Penticantitruncated 6cube Terigreatorhombated hexeract (togrix) 

55  t_{0,1,2,4}{4,3,3,3,3} Stericantitruncated 6cube Celligreatorhombated hexeract (cagorx) 

56  t_{0,1,2,3}{4,3,3,3,3} Runcicantitruncated 6cube Great prismated hexeract (gippox) 

57  t_{0,1,2,3,4}{3,3,3,3,4} Steriruncicantitruncated 6orthoplex Great cellated hexacontatetrapeton (gocog) 

58  t_{0,1,2,3,5}{3,3,3,3,4} Pentiruncicantitruncated 6orthoplex Terigreatoprismated hexacontatetrapeton (tagpog) 

59  t_{0,1,2,4,5}{3,3,3,3,4} Pentistericantitruncated 6orthoplex Tericelligreatorhombated hexacontatetrapeton (tecagorg) 

60  t_{0,1,2,4,5}{4,3,3,3,3} Pentistericantitruncated 6cube Tericelligreatorhombated hexeract (tocagrax) 

61  t_{0,1,2,3,5}{4,3,3,3,3} Pentiruncicantitruncated 6cube Terigreatoprismated hexeract (tagpox) 

62  t_{0,1,2,3,4}{4,3,3,3,3} Steriruncicantitruncated 6cube Great cellated hexeract (gocax) 

63  t_{0,1,2,3,4,5}{4,3,3,3,3} Omnitruncated 6cube Great terihexeractihexacontitetrapeton (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, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
(Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
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  A_{n}  B_{n}  I_{2}(p) / D_{n}  E_{6} / E_{7} / E_{8} / F_{4} / G_{2}  H_{n}  
Regular polygon  Triangle  Square  pgon  Hexagon  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Uniform npolytope  nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds 
Hellenica World  Scientific Library
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