9-demicube

Short description: Uniform 9-polytope

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

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 [math]\displaystyle{ \left\{3 \begin{array}{l}3, 3, 3, 3, 3, 3\\3\end{array}\right\} }[/math] 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.

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". https://bendwavy.org/klitzing/dimensions/polyyotta.htm. 

External links

• Olshevsky, George. "Demienneract". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#half. 
• Multi-dimensional Glossary