BREAKING NEWS

## Summary

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 Coxeter group C9, [37,4]
Dual 9-orthoplex Properties convex

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 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

  
B6 B5

 
B4 B3 B2

  
A7 A5 A3

  

## 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.