9-cube

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.

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 {4,3,3}
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

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 < x_i < 1.

Projections

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