Demiocteract (8-demicube) | |
---|---|

Petrie polygon projection | |

Type | Uniform 8-polytope |

Family | demihypercube |

Coxeter symbol | 1_{51} |

Schläfli symbols | {3,3^{5,1}} = h{4,3^{6}}s{2 ^{1,1,1,1,1,1,1}} |

Coxeter diagrams | |

7-faces | 144: 16 {3 ^{1,4,1}} 128 {3 ^{6}} |

6-faces | 112 {3^{1,3,1}} 1024 {3 ^{5}} |

5-faces | 448 {3^{1,2,1}} 3584 {3 ^{4}} |

4-faces | 1120 {3^{1,1,1}} 7168 {3,3,3} |

Cells | 10752: 1792 {3 ^{1,0,1}} 8960 {3,3} |

Faces | 7168 {3} |

Edges | 1792 |

Vertices | 128 |

Vertex figure | Rectified 7-simplex |

Symmetry group | D_{8}, [3^{5,1,1}] = [1^{+},4,3^{6}]A _{1}^{8}, [2^{7}]^{+} |

Dual | ? |

Properties | convex |

In geometry, a **demiocteract** or **8-demicube** is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{8} for an 8-dimensional *half measure* polytope.

Coxeter named this polytope as **1 _{51}** from its Coxeter diagram, with a ring on one of the 1-length branches,

Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube:

- (±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

This polytope is the vertex figure for the uniform tessellation, 2_{51} with Coxeter-Dynkin diagram:

Coxeter plane | B_{8} | D_{8} | D_{7} | D_{6} | D_{5} |
---|---|---|---|---|---|

Graph | |||||

Dihedral symmetry | [16/2] | [14] | [12] | [10] | [8] |

Coxeter plane | D_{4} | D_{3} | A_{7} | A_{5} | A_{3} |

Graph | |||||

Dihedral symmetry | [6] | [4] | [8] | [6] | [4] |

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 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 **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, a **demienneract** or **9-demicube** is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

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 **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-demicube** or **demidekeract** is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

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

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

In geometry of five dimensions or higher, a **cantic 5-cube**, **cantihalf 5-cube**, **truncated 5-demicube** is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

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

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 **runcic 5-cube** or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

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 **cantic 8-cube** or **truncated 8-demicube** is a uniform 8-polytope, being a truncation of the 8-demicube.

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.

In seven-dimensional geometry, a **pentic 7-cube** is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms.

In seven-dimensional geometry, a **hexic 7-cube** is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.

In seven-dimensional geometry, a **stericated 7-cube** is a convex uniform 7-polytope, being a runcination of the uniform 7-demicube. There are 4 unique runcinations for the 7-demicube including truncation and cantellation.

- 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) - H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973, 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,
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_{n1})

- Olshevsky, George. "Demiocteract".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Multi-dimensional Glossary

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