Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, yellow have 4, and the green center has 8
TypeRegular 9-polytope
Schläfli symbol {4,37}
Coxeter-Dynkin diagram
8-faces18 {4,36}
7-faces144 {4,35}
6-faces672 {4,34}
5-faces2016 {4,33}
4-faces4032 {4,3,3}
Cells5376 {4,3}
Faces4608 {4}
Vertex figure8-simplex
Petrie polygonoctadecagon
Coxeter groupC9, [37,4]

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


while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7, x8) with −1 < xi < 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 number of vertices in each column represents rows in Pascal's triangle, being 1:9:36:84:126:126:84:36:9:1.

Petrie polygon, skew orthographic projection


orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[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.



    External links

    Fundamental convex regular and uniform polytopes in dimensions 2–10
    Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
    Regular polygon Triangle Square p-gon Hexagon Pentagon
    Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
    Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
    Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
    Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
    Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
    Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
    Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
    Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
    Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
    Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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