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{28}
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.

Coxeter named this polytope as 161 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length Coxeter-Dynkin diagram branches.

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]