# 9-demicube

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-faces27418 {31,5,1}
256 {37}
7-faces2448144 {31,4,1}
2304 {36}
6-faces9888672 {31,3,1}
9216 {35}
5-faces235202016 {31,2,1}
21504 {34}
4-faces362884032 {31,1,1}
32256 {33}
Cells376325376 {31,0,1}
32256 {3,3}
Faces21504{3}
Edges4608
Vertices256
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 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 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 or {3,36,1}.

## 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: 1n1)
• Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o *b3o3o3o3o3o3o - henne".

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds