Demienneract (9-demicube)
Petrie polygon
Type	Uniform9-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

Ingeometry, ademienneract or9-demicube is a uniform9-polytope, constructed from the9-cube, withalternated vertices removed. It is part of a dimensionally infinite family ofuniform polytopes calleddemihypercubes.

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

Coxeter named this polytope as1₆₁ from itsCoxeter diagram, with a ring onone of the 1-length branches, andSchläfli symbol${\displaystyle \left\{3\begin{array}{l}3,3,3,3,3,3\\ 3\end{array}\right\}}$ or {3,3^6,1}.

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

(±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

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]

• 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".

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

• 9-polytopes

• Articles with short description
• Short description matches Wikidata