9-cube
Template:Short description Script error: No such module "Unsubst".
| 9-cube Enneract | |
|---|---|
| File:9-cube.svg Orthogonal projection inside Petrie polygon Orange vertices are doubled, yellow have 4, and the green center has 8 | |
| Type | Regular 9-polytope |
| Family | hypercube |
| Schläfli symbol | {4,37} |
| Coxeter-Dynkin diagram | Template:CDD |
| 8-faces | 18 {4,36}File:8-cube.svg |
| 7-faces | 144 {4,35}File:7-cube graph.svg |
| 6-faces | 672 {4,34}File:6-cube graph.svg |
| 5-faces | 2016 {4,33}File:5-cube graph.svg |
| 4-faces | 4032 {4,3,3} |
| Cells | 5376 {4,3}File:3-cube graph.svg |
| Faces | 4608 {4}File:2-cube.svg |
| Edges | 2304 |
| Vertices | 512 |
| Vertex figure | 8-simplex File:8-simplex graph.svg |
| Petrie polygon | octadecagon |
| Coxeter group | C9, [37,4] |
| Dual | 9-orthoplex File:9-orthoplex.svg |
| 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
| File:9-cube column graph.svg This 9-cube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:9:36:84:126:126:84:36:9:1. |
Images
Template:B9 Coxeter plane graphs
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
<templatestyles src="Reflist/styles.css" />
Script error: No such module "Check for unknown parameters".
References
- H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, Template:ISBN, 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, Template:ISBN [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]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
- Template:KlitzingPolytopes
External links
- Script error: No such module "Template wrapper".
- Template:GlossaryForHyperspace
- Multi-dimensional Glossary: hypercube Garrett Jones