D6 polytope

Orthographic projections in the D₆ Coxeter plane

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. 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		80px	80px	80px	80px	80px	80px	= 6-demicube Hemihexeract (hax)
2		80px	80px	80px	80px	80px	80px	= cantic 6-cube Truncated hemihexeract (thax)
3		80px	80px	80px	80px	80px		= runcic 6-cube Small rhombated hemihexeract (sirhax)
4		80px	80px	80px	80px	80px		= steric 6-cube Small prismated hemihexeract (sophax)
5		80px	80px	80px	80px	80px		= pentic 6-cube Small cellated demihexeract (sochax)
6		80px	80px	80px	80px	80px		= runcicantic 6-cube Great rhombated hemihexeract (girhax)
7		80px	80px	80px	80px	80px		= stericantic 6-cube Prismatotruncated hemihexeract (pithax)
8		80px	80px	80px	80px	80px		= steriruncic 6-cube Prismatorhombated hemihexeract (prohax)
9		80px	80px	80px	80px	80px		= Stericantic 6-cube Cellitruncated hemihexeract (cathix)
10		80px	80px	80px	80px	80px		= Pentiruncic 6-cube Cellirhombated hemihexeract (crohax)
11		80px	80px	80px	80px	80px		= Pentisteric 6-cube Celliprismated hemihexeract (cophix)
12		80px	80px	80px	80px	80px		= Steriruncicantic 6-cube Great prismated hemihexeract (gophax)
13		80px	80px	80px	80px	80px		= Pentiruncicantic 6-cube Celligreatorhombated hemihexeract (cagrohax)
14		80px	80px	80px	80px	80px		= Pentistericantic 6-cube Celliprismatotruncated hemihexeract (capthix)
15		80px	80px	80px	80px	80px		= Pentisteriruncic 6-cube Celliprismatorhombated hemihexeract (caprohax)
16		80px	80px	80px	80px	80px		= 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
  • (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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.