9-demicube

Demienneract (9-demicube)
File:Demienneract ortho petrie.svg 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	Template:CDD = Template:CDD Template:CDD
8-faces	274	18 {31,5,1} File:Demiocteract ortho petrie.svg 256 {37} File:8-simplex t0.svg
7-faces	2448	144 {31,4,1} File:Demihepteract ortho petrie.svg 2304 {36} File:7-simplex t0.svg
6-faces	9888	672 {31,3,1} File:Demihexeract ortho petrie.svg 9216 {35} File:6-simplex t0.svg
5-faces	23520	2016 {31,2,1} File:Demipenteract graph ortho.svg 21504 {34} File:5-simplex t0.svg
4-faces	36288	4032 {31,1,1} File:Cross graph 4.svg 32256 {33} File:4-simplex t0.svg
Cells	37632	5376 {31,0,1} File:3-simplex t0.svg 32256 {3,3} File:3-simplex t0.svg
Faces	21504	{3} File:2-simplex t0.svg
Edges	4608
Vertices	256
Vertex figure	Rectified 8-simplex File:8-simplex t1.svg
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, Template:CDD and Schläfli symbol $\left\{3\begin{array}{c}3,3,3,3,3,3\\ 3\end{array}\right\}$ or {3,3^6,1}.
Acronym: henne

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	File:9-demicube t0 B9.svg	File:9-demicube t0 D9.svg	File:9-demicube t0 D8.svg
Dihedral symmetry	[18]+ = [9]	[16]	[14]
Coxeter plane	D7	D6
Graph	File:9-demicube t0 D7.svg	File:9-demicube t0 D6.svg
Dihedral symmetry	[12]	[10]
Coxeter plane	D5	D4	D3
Graph	File:9-demicube t0 D5.svg	File:9-demicube t0 D4.svg	File:9-demicube t0 D3.svg
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A7	A5	A3
Graph	File:9-demicube t0 A7.svg	File:9-demicube t0 A5.svg	File:9-demicube t0 A3.svg
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), Template:Isbn
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:Isbn
    • (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, Template:Isbn
• Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o *b3o3o3o3o3o3o - henne".

External links

• Template:GlossaryForHyperspace
• Multi-dimensional Glossary