9-cube

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 {4,32} File:4-cube graph.svg
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 was given acronym enne by J. Bowers.

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 < x_i < 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

Orthographic projections

B9		B8		B7
File:9-cube t0.svg		File:9-cube t0 B8.svg		File:9-cube t0 B7.svg
[18]		[16]		[14]
B6			B5
File:9-cube t0 B6.svg			File:9-cube t0 B5.svg
[12]			[10]
B4		B3		B2
File:9-cube t0 B4.svg		File:9-cube t0 B3.svg		File:9-cube t0 B2.svg
[8]		[6]		[4]
A7		A5		A3
File:9-cube t0 A7.svg		File:9-cube t0 A5.svg		File:9-cube t0 A3.svg
[8]		[6]		[4]

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.

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5), Template:Isbn
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:Isbn
    • (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)
• Klitzing, Richard. "9D uniform polytopes (polyyotta) o3o3o3o3o3o3o3o4x - enne".

External links

• Template:GlossaryForHyperspace
• Multi-dimensional Glossary: hypercube Garrett Jones