List of D6 polytopes
6-demicube |
6-orthoplex |
In 6-dimensional geometry, there are 47 uniform polytopes with D6 symmetry, 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 | Truncated 6-demicube Truncated hemihexeract (thax) | |||||||
3 | Cantellated 6-demicube Small rhombated hemihexeract (sirhax) | |||||||
4 | Runcinated 6-demicube Small prismated hemihexeract (sophax) | |||||||
5 | Stericated 6-demicube Small cellated demihexeract (sochax) | |||||||
6 | Cantitruncated 6-demicube Great rhombated hemihexeract (girhax) | |||||||
7 | Runcitruncated 6-demicube Prismatotruncated hemihexeract (pithax) | |||||||
8 | Runcicantellated 6-demicube Prismatorhombated hemihexeract (prohax) | |||||||
9 | Runcitruncated 6-demicube Cellitruncated hemihexeract (cathix) | |||||||
10 | Stericantellated 6-demicube Cellirhombated hemihexeract (crohax) | |||||||
11 | Steriruncinated 6-demicube Celliprismated hemihexeract (cophix) | |||||||
12 | Runcicantitruncated 6-demicube Great prismated hemihexeract (gophax) | |||||||
13 | Stericantitruncated 6-demicube Celligreatorhombated hemihexeract (cagrohax) | |||||||
14 | Steriruncitruncated 6-demicube Celliprismatotruncated hemihexeract (capthix) | |||||||
15 | Steriruncicantellated 6-demicube Celliprismatorhombated hemihexeract (caprohax) | |||||||
16 | Steriruncicantitruncated 6-demicube 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
- Richard Klitzing, 6D, uniform polytopes (polypeta)
Notes
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |