9-demicube

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Demienneract
(9-demicube)
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 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.png
8-faces 274 18 {31,5,1} Demiocteract ortho petrie.svg
256 {37} 8-simplex t0.svg
7-faces 2448 144 {31,4,1} Demihepteract ortho petrie.svg
2304 {36} 7-simplex t0.svg
6-faces 9888 672 {31,3,1} Demihexeract ortho petrie.svg
9216 {35} 6-simplex t0.svg
5-faces 23520 2016 {31,2,1} Demipenteract graph ortho.svg
21504 {34} 5-simplex t0.svg
4-faces 36288 4032 {31,1,1} Cross graph 4.svg
32256 {33} 4-simplex t0.svg
Cells 37632 5376 {31,0,1} 3-simplex t0.svg
32256 {3,3} 3-simplex t0.svg
Faces 21504 {3} 2-simplex t0.svg
Edges 4608
Vertices 256
Vertex figure Rectified 8-simplex
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 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, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png and Schläfli symbol or {3,36,1}.

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B9 D9 D8
Graph 9-demicube t0 B9.svg 9-demicube t0 D9.svg 9-demicube t0 D8.svg
Dihedral symmetry [18]+ = [9] [16] [14]
Graph 9-demicube t0 D7.svg 9-demicube t0 D6.svg
Coxeter plane D7 D6
Dihedral symmetry [12] [10]
Coxeter group D5 D4 D3
Graph 9-demicube t0 D5.svg 9-demicube t0 D4.svg 9-demicube t0 D3.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A7 A5 A3
Graph 9-demicube t0 A7.svg 9-demicube t0 A5.svg 9-demicube t0 A3.svg
Dihedral symmetry [8] [6] [4]

References[edit]

  • 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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Richard Klitzing, 9D uniform polytopes (polyyotta), x3o3o *b3o3o3o3o3o3o - henne

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / 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