6-cube Hexeract | |
---|---|

Orthogonal projection inside Petrie polygon Orange vertices are doubled, and the center yellow has 4 vertices | |

Type | Regular 6-polytope |

Family | hypercube |

Schläfli symbol | {4,3^{4}} |

Coxeter diagram | |

5-faces | 12 {4,3,3,3} |

4-faces | 60 {4,3,3} |

Cells | 160 {4,3} |

Faces | 240 {4} |

Edges | 192 |

Vertices | 64 |

Vertex figure | 5-simplex |

Petrie polygon | dodecagon |

Coxeter group | B_{6}, [3^{4},4] |

Dual | 6-orthoplex |

Properties | convex, Hanner polytope |

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.

- Related polytopes
- As a configuration
- Cartesian coordinates
- Construction
- Projections
- Related polytopes 2
- References
- External links

It has Schläfli symbol {4,3^{4}}, being composed of 3 5-cubes around each 4-face. It can be called a **hexeract**, a portmanteau of tesseract (the *4-cube*) with *hex* for six (dimensions) in Greek. It can also be called a regular **dodeca-6-tope** or **dodecapeton**, being a 6-dimensional polytope constructed from 12 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes.

Applying an * alternation * operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.

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

Cartesian coordinates for the vertices of a 6-cube centered at the origin and edge length 2 are

- (±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}) with −1 < x_{i} < 1.

There are three Coxeter groups associated with the 6-cube, one regular, with the C_{6} or [4,3,3,3,3] Coxeter group, and a half symmetry (D_{6}) or [3^{3,1,1}] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.

Name | Coxeter | Schläfli | Symmetry | Order |
---|---|---|---|---|

Regular 6-cube | {4,3,3,3,3} | [4,3,3,3,3] | 46080 | |

Quasiregular 6-cube | [3,3,3,3^{1,1}] | 23040 | ||

hyperrectangle | {4,3,3,3}×{} | [4,3,3,3,2] | 7680 | |

{4,3,3}×{4} | [4,3,3,2,4] | 3072 | ||

{4,3}^{2} | [4,3,2,4,3] | 2304 | ||

{4,3,3}×{}^{2} | [4,3,3,2,2] | 1536 | ||

{4,3}×{4}×{} | [4,3,2,4,2] | 768 | ||

{4}^{3} | [4,2,4,2,4] | 512 | ||

{4,3}×{}^{3} | [4,3,2,2,2] | 384 | ||

{4}^{2}×{}^{2} | [4,2,4,2,2] | 256 | ||

{4}×{}^{4} | [4,2,2,2,2] | 128 | ||

{}^{6} | [2,2,2,2,2] | 64 |

Coxeter plane | B_{6} | B_{5} | B_{4} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [12] | [10] | [8] |

Coxeter plane | Other | B_{3} | B_{2} |

Graph | |||

Dihedral symmetry | [2] | [6] | [4] |

Coxeter plane | A_{5} | A_{3} | |

Graph | |||

Dihedral symmetry | [6] | [4] |

3D Projections | |

6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D. | 6-cube quasicrystal structure orthographically projected to 3D using the golden ratio. |

The *6-cube* is 6th in a series of hypercube:

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

This polytope is one of 63 uniform 6-polytopes generated from the B_{6} Coxeter plane, including the regular 6-cube or 6-orthoplex.

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.

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

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-orthoplex**, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell *4-faces*, and 64 *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 **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.

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.

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 five-dimensional geometry, a **rectified 5-orthoplex** is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

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

In five-dimensional geometry, a **rectified 5-cube** is a convex uniform 5-polytope, being a rectification 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 five-dimensional geometry, a **runcinated 5-orthoplex** is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

In six-dimensional geometry, a **pentic 6-cube** is a convex uniform 6-polytope.

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, H.S.M.
*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) - Klitzing, Richard. "6D uniform polytopes (polypeta) o3o3o3o3o4x - ax".

- 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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