9-demicube

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.

Demienneract (9-demicube)
Petrie polygon
Type	Uniform 9-polytope
Family	demihypercube
Coxeter symbol	161
Schläfli symbol	{3,36,1} = h{4,37} s{21,1,1,1,1,1,1,1}
Coxeter-Dynkin diagram	=
8-faces	274	18 {31,5,1} 256 {37}
7-faces	2448	144 {31,4,1} 2304 {36}
6-faces	9888	672 {31,3,1} 9216 {35}
5-faces	23520	2016 {31,2,1} 21504 {34}
4-faces	36288	4032 {31,1,1} 32256 {33}
Cells	37632	5376 {31,0,1} 32256 {3,3}
Faces	21504	{3}
Edges	4608
Vertices	256
Vertex figure	Rectified 8-simplex
Symmetry group	D9, [36,1,1] = [1+,4,37] [28]+
Dual	?
Properties	convex

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 ${\displaystyle \left\{3{\begin{array}{l}3,3,3,3,3,3\\3\end{array}}\right\}}$ 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:
  • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5), ISBN 0-486-61480-8
  • 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, 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]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things, 2008, Chapter 26, p. 409, Hemicubes: 1_n1, ISBN 978-1-56881-220-5
• 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	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations