9-demicube
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.
Demienneract (9-demicube) | ||
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![]() 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 |
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,
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
Coxeter plane | B9 | D9 | D8 |
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Graph | ![]() |
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Dihedral symmetry | [18]+ = [9] | [16] | [14] |
Graph | ![]() |
![]() | |
Coxeter plane | D7 | D6 | |
Dihedral symmetry | [12] | [10] | |
Coxeter group | D5 | D4 | D3 |
Graph | ![]() |
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![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A7 | A5 | A3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
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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".
External links
- Olshevsky, George. "Demienneract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Multi-dimensional Glossary
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 | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |
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