9-demicube

Demienneract (9-demicube)
Petrie polygon
Type	Uniform 9-polytope
Family	demihypercube
Coxeter symbol	1₆₁
Schläfli symbol	{3,3^6,1} = h{4,3⁷} s{2^{1,1,1,1,1,1,1,1}}
Coxeter-Dynkin diagram	=
8-faces	274	18 {3^1,5,1} 256 {3⁷}
7-faces	2448	144 {3^1,4,1} 2304 {3⁶}
6-faces	9888	672 {3^1,3,1} 9216 {3⁵}
5-faces	23520	2016 {3^1,2,1} 21504 {3⁴}
4-faces	36288	4032 {3^1,1,1} 32256 {3³}
Cells	37632	5376 {3^1,0,1} 32256 {3,3}
Faces	21504	{3}
Edges	4608
Vertices	256
Vertex figure	Rectified 8-simplex
Symmetry group	D₉, [3^6,1,1] = [1⁺,4,3⁷] [2⁸]⁺
Dual	?
Properties	convex

In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₉ for a 9-dimensional half measure polytope.

Coxeter named this polytope as 1₆₁ from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol $\left\{3{\begin{array}{l}3,3,3,3,3,3\\3\end{array}}\right\}$ or {3,3^6,1}.

Cartesian coordinates

Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract:

(±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₉	D₉	D₈
Graph
Dihedral symmetry	[18]⁺ = [9]	[16]	[14]
Graph
Coxeter plane	D₇	D₆
Dihedral symmetry	[12]	[10]
Coxeter group	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₇	A₅	A₃
Graph
Dihedral symmetry	[8]	[6]	[4]

References

H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
- 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]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o *b3o3o3o3o3o3o - henne".

External links

Olshevsky, George. "Demienneract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Multi-dimensional Glossary

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 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 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