9-cube

9-cube Enneract
Orthogonal projection insidePetrie polygon Orange vertices are doubled, yellow have 4, and the green center has 8
Type	Regular9-polytope
Family	hypercube
Schläfli symbol	{4,37}
Coxeter-Dynkin diagram
8-faces	18{4,36}
7-faces	144{4,35}
6-faces	672{4,34}
5-faces	2016{4,33}
4-faces	4032{4,3,3}
Cells	5376{4,3}
Faces	4608{4}
Edges	2304
Vertices	512
Vertex figure	8-simplex
Petrie polygon	octadecagon
Coxeter group	C9,[37,4]
Dual	9-orthoplex
Properties	convex,Hanner polytope

Ingeometry,a9-cubeis a nine-dimensionalhypercubewith 512vertices,2304edges,4608squarefaces,5376cubiccells,4032tesseract4-faces,20165-cube5-faces,6726-cube6-faces,1447-cube7-faces,and 188-cube8-faces.

It can be named by itsSchläfli symbol{4,3⁷}, being composed of three8-cubesaround each 7-face. It is also called anenneract,aportmanteauoftesseract(the4-cube) andennefor nine (dimensions) inGreek.It can also be called a regularoctadeca-9-topeoroctadecayotton,as anine-dimensional polytopeconstructed with 18 regularfacets.

It is a part of an infinite family of polytopes, called hypercubes. Thedualof a 9-cube can be called a9-orthoplex,and is a part of the infinite family ofcross-polytopes.

Cartesian coordinates

Cartesian coordinatesfor 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

This 9-cube graph is anorthogonal 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 inPascal's triangle,being 1:9:36:84:126:126:84:36:9:1.

Images

orthographic projections

B9		B8		B7
[18]		[16]		[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
[8]		[6]		[4]

Derived polytopes

Applying analternationoperation, deleting alternating vertices of the9-cube,creates anotheruniform polytope,called a9-demicube,(part of an infinite family calleddemihypercubes), which has 188-demicubeand 256 8-simplex facets.

References

• H.S.M. Coxeter:
  • Coxeter,Regular Polytopes,(3rd edition, 1973), Dover edition,ISBN0-486-61480-8,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,ISBN978-0-471-01003-6[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]
• Norman JohnsonUniform 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

• Weisstein, Eric W."Hypercube".MathWorld.
• Olshevsky, George."Measure polytope".Glossary for Hyperspace.Archived fromthe originalon 4 February 2007.
• Multi-dimensional Glossary: hypercubeGarrett Jones