# 8-cube

Last updated
8-cube
Octeract

Orthogonal projection
inside Petrie polygon
TypeRegular 8-polytope
Family hypercube
Schläfli symbol {4,36}
Coxeter-Dynkin diagrams

7-faces16 {4,35}
6-faces112 {4,34}
5-faces448 {4,33}
4-faces1120 {4,32}
Cells1792 {4,3}
Faces1792 {4}
Edges1024
Vertices256
Vertex figure 7-simplex
Coxeter group C8, [36,4]
Dual 8-orthoplex
Properties convex, Hanner polytope

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.

## Contents

It is represented by Schläfli symbol {4,36}, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the 4-cube) and oct for eight (dimensions) in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8-cube can be called an 8-orthoplex and is a part of the infinite family of cross-polytopes.

## Cartesian coordinates

Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are

(±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) with -1 < xi < 1.

## As a configuration

This configuration matrix represents the 8-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces, and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element. [1] [2]

${\displaystyle {\begin{bmatrix}{\begin{matrix}256&8&28&56&70&56&28&8\\2&1024&7&21&35&35&21&7\\4&4&1792&6&15&20&15&6\\8&12&6&1792&5&10&10&5\\16&32&24&8&1120&4&6&4\\32&80&80&40&10&448&3&3\\64&192&240&160&60&12&112&2\\128&448&672&560&280&84&14&16\end{matrix}}\end{bmatrix}}}$

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time. [3]

B8k-facefkf0f1f2f3f4f5f6f7 k-figure notes
A7( )f0256828567056288 {3,3,3,3,3,3} B8/A7 = 2^8*8!/8! = 256
A6A1{ }f1210247213535217 {3,3,3,3,3} B8/A6A1 = 2^8*8!/7!/2 = 1024
A5B2 {4} f244179261520156 {3,3,3,3} B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A4B3 {4,3} f381261792510105 {3,3,3} B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A3B4 {4,3,3} f416322481120464 {3,3} B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A2B5 {4,3,3,3} f5328080401044833 {3} B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A1B6 {4,3,3,3,3} f66419224016060121122{ }B8/A1B6 = 2^8*8!/2/2^6/6!= 112
B7 {4,3,3,3,3,3} f7128448672560280841416( )B8/B7 = 2^8*8!/2^7/7! = 16

## Projections

 This 8-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 1:8:28:56:70:56:28:8:1.
orthographic projections
B8B7
[16][14]
B6B5
[12][10]
B4B3B2
[8][6][4]
A7A5A3
[8][6][4]

## Derived polytopes

Applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube , (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.

The 8-cube is 8th in an infinite series of hypercube:

## Related Research Articles

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.

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.

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

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.

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.

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.

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.

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

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.

In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-faces.

In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.

In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

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

In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.

In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.

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

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.

## References

1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
3. Klitzing, Richard. "o3o3o3o3o3o3o4x - octo".
Family An Bn I2(p) / Dn E6 / E7 / E8 / / 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