8-cube Octeract | |
---|---|

Orthogonal projection inside Petrie polygon | |

Type | Regular 8-polytope |

Family | hypercube |

Schläfli symbol | {4,3^{6}} |

Coxeter-Dynkin diagrams | |

7-faces | 16 {4,3^{5}} |

6-faces | 112 {4,3^{4}} |

5-faces | 448 {4,3^{3}} |

4-faces | 1120 {4,3^{2}} |

Cells | 1792 {4,3} |

Faces | 1792 {4} |

Edges | 1024 |

Vertices | 256 |

Vertex figure | 7-simplex |

Petrie polygon | hexadecagon |

Coxeter group | C_{8}, [3^{6},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.

- Cartesian coordinates
- As a configuration
- Projections
- Derived polytopes
- Related polytopes
- References
- External links

It is represented by Schläfli symbol {4,3^{6}}, 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 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 (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}) with -1 < x_{i} < 1.

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] }

B_{8} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | k-figure | notes | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{7} | ( ) | f_{0} | 256 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | {3,3,3,3,3,3} | B_{8}/A_{7} = 2^8*8!/8! = 256 | |

A_{6}A_{1} | { } | f_{1} | 2 | 1024 | 7 | 21 | 35 | 35 | 21 | 7 | {3,3,3,3,3} | B_{8}/A_{6}A_{1} = 2^8*8!/7!/2 = 1024 | |

A_{5}B_{2} | {4} | f_{2} | 4 | 4 | 1792 | 6 | 15 | 20 | 15 | 6 | {3,3,3,3} | B_{8}/A_{5}B_{2} = 2^8*8!/6!/4/2 = 1792 | |

A_{4}B_{3} | {4,3} | f_{3} | 8 | 12 | 6 | 1792 | 5 | 10 | 10 | 5 | {3,3,3} | B_{8}/A_{4}B_{3} = 2^8*8!/5!/8/3! = 1792 | |

A_{3}B_{4} | {4,3,3} | f_{4} | 16 | 32 | 24 | 8 | 1120 | 4 | 6 | 4 | {3,3} | B_{8}/A_{3}B_{4} = 2^8*8!/4!/2^4/4! = 1120 | |

A_{2}B_{5} | {4,3,3,3} | f_{5} | 32 | 80 | 80 | 40 | 10 | 448 | 3 | 3 | {3} | B_{8}/A_{2}B_{5} = 2^8*8!/3!/2^5/5! = 448 | |

A_{1}B_{6} | {4,3,3,3,3} | f_{6} | 64 | 192 | 240 | 160 | 60 | 12 | 112 | 2 | { } | B_{8}/A_{1}B_{6} = 2^8*8!/2/2^6/6!= 112 | |

B_{7} | {4,3,3,3,3,3} | f_{7} | 128 | 448 | 672 | 560 | 280 | 84 | 14 | 16 | ( ) | B_{8}/B_{7} = 2^8*8!/2^7/7! = 16 |

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. |

B_{8} | B_{7} | ||||
---|---|---|---|---|---|

[16] | [14] | ||||

B_{6} | B_{5} | ||||

[12] | [10] | ||||

B_{4} | B_{3} | B_{2} | |||

[8] | [6] | [4] | |||

A_{7} | A_{5} | A_{3} | |||

[8] | [6] | [4] |

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:

Line segment | Square | Cube | 4-cube | 5-cube | 6-cube | 7-cube | 8-cube |

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.

- ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
- ↑ Coxeter, Complex Regular Polytopes, p.117
- ↑ 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- (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]

- (Paper 22) H.S.M. Coxeter,

- Coxeter,
- Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. (1966)

- N.W. Johnson:
- Klitzing, Richard. "8D uniform polytopes (polyzetta) o3o3o3o3o3o3o4x - octo".

- Weisstein, Eric W. "Hypercube".
*MathWorld*. - Olshevsky, George. "Measure polytope".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Multi-dimensional Glossary: hypercube Garrett Jones

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