# 8-cube

8-cube
Octeract

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

7-faces 16 {4,35}
6-faces 112 {4,34}
5-faces 448 {4,33}
4-faces 1120 {4,32}
Cells 1792 {4,3}
Faces 1792 {4}
Edges 1024
Vertices 256
Vertex figure 7-simplex
Coxeter group C8, [36,4]
Dual 8-orthoplex
Properties convex

In geometry, an 8-cube is an eight-dimensional hypercube (8-cube). 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.

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 a 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

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[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]

B8 k-face fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure notes
A7 ( ) f0 256 8 28 56 70 56 28 8 {3,3,3,3,3,3} B8/A7 = 2^8*8!/8! = 256
A6A1 { } f1 2 1024 7 21 35 35 21 7 {3,3,3,3,3} B8/A6A1 = 2^8*8!/7!/2 = 1024
A5B2 {4} f2 4 4 1792 6 15 20 15 6 {3,3,3,3} B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A4B3 {4,3} f3 8 12 6 1792 5 10 10 5 {3,3,3} B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A3B4 {4,3,3} f4 16 32 24 8 1120 4 6 4 {3,3} B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A2B5 {4,3,3,3} f5 32 80 80 40 10 448 3 3 {3} B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A1B6 {4,3,3,3,3} f6 64 192 240 160 60 12 112 2 { } B8/A1B6 = 2^8*8!/2/2^6/6!= 112
B7 {4,3,3,3,3,3} f7 128 448 672 560 280 84 14 16 ( ) 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
B8 B7
[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 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.

## References

1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
2. ^ Coxeter, Complex Regular Polytopes, p.117
3. ^ Klitzing, Richard. "o3o3o3o3o3o3o4x - octo".
• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard. "8D uniform polytopes (polyzetta) o3o3o3o3o3o3o4x - octo".