9-cube
{{Short description|9-dimensional hypercube}}
{{No footnotes|date=September 2017}}
| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|9-cube | |
| bgcolor=#ffffff align=center colspan=2|280px Orthogonal projection inside Petrie polygon Orange vertices are doubled, yellow have 4, and the green center has 8 | |
| bgcolor=#e7dcc3|Type | Regular 9-polytope |
| bgcolor=#e7dcc3|Family | hypercube |
| bgcolor=#e7dcc3|Schläfli symbol | {4,37} |
| bgcolor=#e7dcc3|Coxeter-Dynkin diagram | {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} |
| bgcolor=#e7dcc3|8-faces | 18 {4,36}25px |
| bgcolor=#e7dcc3|7-faces | 144 {4,35}25px |
| bgcolor=#e7dcc3|6-faces | 672 {4,34}25px |
| bgcolor=#e7dcc3|5-faces | 2016 {4,33}25px |
| bgcolor=#e7dcc3|4-faces | 4032 {4,3,3} |
| bgcolor=#e7dcc3|Cells | 5376 {4,3}25px |
| bgcolor=#e7dcc3|Faces | 4608 {4}25px |
| bgcolor=#e7dcc3|Edges | 2304 |
| bgcolor=#e7dcc3|Vertices | 512 |
| bgcolor=#e7dcc3|Vertex figure | 8-simplex 25px |
| bgcolor=#e7dcc3|Petrie polygon | octadecagon |
| bgcolor=#e7dcc3|Coxeter group | C9, [37,4] |
| bgcolor=#e7dcc3|Dual | 9-orthoplex 25px |
| bgcolor=#e7dcc3|Properties | convex, Hanner polytope |
In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.
It can be named by its Schläfli symbol {4,37}, being composed of three 8-cubes around each 7-face. It is also called an enneract, a portmanteau of tesseract (the 4-cube) and enne for nine (dimensions) in Greek. It can also be called a regular octadeca-9-tope or octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the infinite family of cross-polytopes.
Cartesian coordinates
Cartesian coordinates for the vertices of a 9-cube centered at the origin and edge length 2 are
: (±1,±1,±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7, x8) with −1 < xi < 1.
Projections
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Images
{{B9 Coxeter plane graphs|t0|200}}
Derived polytopes
Applying an alternation operation, deleting alternating vertices of the 9-cube, creates another uniform polytope, called a 9-demicube, (part of an infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets.
Notes
{{Reflist}}
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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
- (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]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
- {{KlitzingPolytopes|polyyotta.htm|9D uniform polytopes (polyyotta)|o3o3o3o3o3o3o3o4x - enne}}
External links
- {{MathWorld|title=Hypercube|urlname=Hypercube}}
- {{GlossaryForHyperspace | anchor=Measure | title=Measure polytope }}
- [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary: hypercube] Garrett Jones
{{Polytopes}}