List of D6 polytopes

Orthographic projections in the D₆ Coxeter plane

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. A_k has [k+1] symmetry, D_k 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