D6 polytope
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (February 2023) |
![]() 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
[edit]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
[edit]- 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)".