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]	Coxeter diagram Names
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 ^[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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.

Notes

↑ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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[1] Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter