9-cube

9-cube Enneract
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
8-faces	18 {4,36}
7-faces	144 {4,35}
6-faces	672 {4,34}
5-faces	2016 {4,33}
4-faces	4032 {4,32}
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

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.¹

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 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.

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 numbers of vertices in each column are a row of Pascal's triangle: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1.

Orthographic projections

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

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.