 6-demicube     |
 6-orthoplex   |
In 6-dimensional geometry, there are 47 uniform polytopes with D6 symmetry, of which 16 are unique and 31 are shared with the B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-demicube with 12 and 32 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the D6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 16 polytopes can be made in the D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B6 is also included although only half of its [12] symmetry exists in these polytopes.
These 16 polytopes are each shown in these 7 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | Coxeter plane graphs | Coxeter diagram Names | ||||||
|---|---|---|---|---|---|---|---|---|
| B6 [12/2] | D6 [10] | D5 [8] | D4 [6] | D3 [4] | A5 [6] | A3 [4] | ||
| 1 |  | |  |  |  |  |  | =   6-demicube Hemihexeract (hax) |
| 2 |  |  |  |  |  |  |  | = cantic 6-cube Truncated hemihexeract (thax) |
| 3 |  |  |  |  |  |  |  | = runcic 6-cube Small rhombated hemihexeract (sirhax) |
| 4 |  |  |  |  |  |  |  | = steric 6-cube Small prismated hemihexeract (sophax) |
| 5 |  |  |  |  |  |  |  | = pentic 6-cube Small cellated demihexeract (sochax) |
| 6 |  |  |  |  |  |  |  | = runcicantic 6-cube Great rhombated hemihexeract (girhax) |
| 7 |  |  |  |  |  |  |  | = stericantic 6-cube Prismatotruncated hemihexeract (pithax) |
| 8 |  |  |  |  |  |  |  | = steriruncic 6-cube Prismatorhombated hemihexeract (prohax) |
| 9 |  |  |  |  |  |  |  | = Stericantic 6-cube Cellitruncated hemihexeract (cathix) |
| 10 |  |  |  |  |  |  |  | = Pentiruncic 6-cube Cellirhombated hemihexeract (crohax) |
| 11 |  |  |  |  |  |  |  | = Pentisteric 6-cube Celliprismated hemihexeract (cophix) |
| 12 |  |  |  |  |  |  |  | = Steriruncicantic 6-cube Great prismated hemihexeract (gophax) |
| 13 |  |  |  |  |  |  |  | = Pentiruncicantic 6-cube Celligreatorhombated hemihexeract (cagrohax) |
| 14 |  |  |  |  |  |  |  | = Pentistericantic 6-cube Celliprismatotruncated hemihexeract (capthix) |
| 15 |  |  |  |  |  |  |  | = Pentisteriruncic 6-cube Celliprismatorhombated hemihexeract (caprohax) |
| 16 |  |  |  |  |  |  |  | = Pentisteriruncicantic 6-cube Great cellated hemihexeract (gochax) |
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)".