9-demicube

Demienneract (9-demicube)
Petrie polygon
Type	Uniform 9-polytope
Family	demihypercube
Coxeter symbol	161
Schläfli symbol	{3,3^6,1} = h{4,3^7} s{2^1,1,1,1,1,1,1,1}
Coxeter-Dynkin diagram	=
8-faces	274	18 {3^1,5,1} 256 {3^7}
7-faces	2448	144 {3^1,4,1} 2304 {3^6}
6-faces	9888	672 {3^1,3,1} 9216 {3^5}
5-faces	23520	2016 {3^1,2,1} 21504 {3^4}
4-faces	36288	4032 {3^1,1,1} 32256 {3^3}
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^+,4,3^7] [2^8]^+
Dual	?
Properties	convex

In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices truncated. 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 an 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
    • (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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Richard Klitzing, 9D uniform polytopes (polyyotta), x3o3o *b3o3o3o3o3o3o - henne

External links

• Olshevsky, George, Demienneract at Glossary for Hyperspace.
• Multi-dimensional Glossary