 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
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
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