8-cube

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

CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.png

7-faces16 {4,35} 7-cube graph.svg
6-faces112 {4,34} 6-cube graph.svg
5-faces448 {4,33} 5-cube graph.svg
4-faces1120 {4,32} 4-cube graph.svg
Cells1792 {4,3} 3-cube.png
Faces1792 {4} 2-cube.svg
Edges1024
Vertices256
Vertex figure 7-simplex 7-simplex graph.svg
Petrie polygon hexadecagon
Coxeter group C8, [36,4]
Dual 8-orthoplex 8-orthoplex.svg
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 hexadeca-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]

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]

B8CDel 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 4.pngCDel node 1.pngk-facefkf0f1f2f3f4f5f6f7 k-figure Notes
A7CDel 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 2.pngCDel node x.png( )f0256828567056288 {3,3,3,3,3,3} B8/A7 = 2^8*8!/8! = 256
A6A1CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node 1.png{ }f1210247213535217 {3,3,3,3,3} B8/A6A1 = 2^8*8!/7!/2 = 1024
A5B2CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png {4} f244179261520156 {3,3,3,3} B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A4B3CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png {4,3} f381261792510105 {3,3,3} B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A3B4CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png {4,3,3} f416322481120464 {3,3} B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A2B5CDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png {4,3,3,3} f5328080401044833 {3} B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A1B6CDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png {4,3,3,3,3} f66419224016060121122{ }B8/A1B6 = 2^8*8!/2/2^6/6! = 112
B7CDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png {4,3,3,3,3,3} f7128448672560280841416( )B8/B7 = 2^8*8!/2^7/7! = 16

Projections

8-cube column graph.svg
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
8-cube t0.svg 8-cube t0 B7.svg
[16][14]
B6B5
8-cube t0 B6.svg 8-cube t0 B5.svg
[12][10]
B4B3B2
8-cube t0 B4.svg 8-cube t0 B3.svg 8-cube t0 B2.svg
[8][6][4]
A7A5A3
8-cube t0 A7.svg 8-cube t0 A5.svg 8-cube t0 A3.svg
[8][6][4]

Derived polytopes

Applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called an 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:

Petrie polygon orthographic projections
1-simplex t0.svg 2-cube.svg 3-cube graph.svg 4-cube graph.svg 5-cube graph.svg 6-cube graph.svg 7-cube graph.svg 8-cube.svg 9-cube.svg 10-cube.svg
Line segment Square Cube 4-cube 5-cube 6-cube 7-cube 8-cube 9-cube 10-cube

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 / 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 compoundsPolytope operations