Last updated
Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, yellow have 4, and the green center has 8
TypeRegular 9-polytope
Family hypercube
Schläfli symbol {4,37}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-faces18 {4,36} 8-cube.svg
7-faces144 {4,35} 7-cube graph.svg
6-faces672 {4,34} 6-cube graph.svg
5-faces2016 {4,33} 5-cube graph.svg
4-faces4032 {4,3,3}
Cells5376 {4,3} 3-cube graph.svg
Faces4608 {4} 2-cube.svg
Vertex figure 8-simplex 8-simplex graph.svg
Petrie polygon octadecagon
Coxeter group C9, [37,4]
Dual 9-orthoplex 9-orthoplex.svg
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 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.


9-cube column graph.svg
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.


orthographic projections
9-cube t0.svg 9-cube t0 B8.svg 9-cube t0 B7.svg
9-cube t0 B6.svg 9-cube t0 B5.svg
9-cube t0 B4.svg 9-cube t0 B3.svg 9-cube t0 B2.svg
9-cube t0 A7.svg 9-cube t0 A5.svg 9-cube t0 A3.svg

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.


    Related Research Articles

    <span class="mw-page-title-main">Tesseract</span> Four-dimensional analogue of the cube

    In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

    <span class="mw-page-title-main">Cantellated tesseract</span>

    In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

    In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.

    <span class="mw-page-title-main">Rectified tesseract</span>

    In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a runcic tesseract.

    In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

    <span class="mw-page-title-main">5-demicube</span>

    In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

    <span class="mw-page-title-main">6-cube</span> 6-dimensional hypercube

    In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

    <span class="mw-page-title-main">6-demicube</span>

    In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

    <span class="mw-page-title-main">7-cube</span> 7-dimensional hypercube

    In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

    <span class="mw-page-title-main">7-demicube</span>

    In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

    <span class="mw-page-title-main">8-demicube</span>

    In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

    <span class="mw-page-title-main">8-cube</span> 8-dimensional hypercube

    In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

    <span class="mw-page-title-main">10-cube</span> 10-dimensional hypercube

    In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

    <span class="mw-page-title-main">Rectified 6-cubes</span>

    In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.

    <span class="mw-page-title-main">Cantic 5-cube</span>

    In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

    <span class="mw-page-title-main">Runcic 5-cubes</span>

    In six-dimensional geometry, a runcic 5-cube or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

    <span class="mw-page-title-main">Cantic 7-cube</span>

    In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

    <span class="mw-page-title-main">Cantic 8-cube</span> A uniform 8-polytope

    In eight-dimensional geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube.

    <span class="mw-page-title-main">Runcic 6-cubes</span>

    In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

    <span class="mw-page-title-main">Runcic 7-cubes</span>

    In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.


    Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
    Regular polygon Triangle Square p-gon Hexagon Pentagon
    Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
    Uniform polychoron Pentachoron 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