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.

Each of these 16 polytopes is shown as a projection in the 7 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet.

#	Coxeter plane graphs							Coxeter diagram Names (acronym)
#	B₆ [12/2]	D₆ [10]	D₅ [8]	D₄ [6]	D₃ [4]	A₅ [6]	A₃ [4]	Coxeter diagram Names (acronym)
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)

Notes

↑ Klitzing.

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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) with acronyms".

v t e Fundamental convex regular and uniform polytopes
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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 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 • Polytope operations

[FOOTNOTEKlitzing-1] Klitzing.