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Demienneract (9-demicube) | ||
---|---|---|

Petrie polygon | ||

Type | Uniform 9-polytope | |

Family | demihypercube | |

Coxeter symbol | 1_{61} | |

Schläfli symbol | {3,3^{6,1}} = h{4,3^{7}}s{2 ^{1,1,1,1,1,1,1,1}} | |

Coxeter-Dynkin diagram | ||

8-faces | 274 | 18 {3^{1,5,1}} 256 {3 ^{7}} |

7-faces | 2448 | 144 {3^{1,4,1}} 2304 {3 ^{6}} |

6-faces | 9888 | 672 {3^{1,3,1}} 9216 {3 ^{5}} |

5-faces | 23520 | 2016 {3^{1,2,1}} 21504 {3 ^{4}} |

4-faces | 36288 | 4032 {3^{1,1,1}} 32256 {3 ^{3}} |

Cells | 37632 | 5376 {3^{1,0,1}} 32256 {3,3} |

Faces | 21504 | {3} |

Edges | 4608 | |

Vertices | 256 | |

Vertex figure | Rectified 8-simplex | |

Symmetry group | D_{9}, [3^{6,1,1}] = [1^{+},4,3^{7}][2 ^{8}]^{+} | |

Dual | ? | |

Properties | convex |

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.

**Geometry** is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

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, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{9} for a 9-dimensional *half measure* polytope.

**Emanuel Lodewijk Elte** was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher.

Coxeter named this polytope as **1 _{61}** from its Coxeter diagram, with a ring on one of the 1-length branches,

In geometry, the **Schläfli symbol** is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract:

- (±1,±1,±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Coxeter plane | B_{9} | D_{9} | D_{8} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [18]^{+} = [9] | [16] | [14] |

Graph | |||

Coxeter plane | D_{7} | D_{6} | |

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

Coxeter group | D_{5} | D_{4} | D_{3} |

Graph | |||

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

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

Graph | |||

Dihedral symmetry | [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 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 five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos^{−1}(1/5), or approximately 78.46°.

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, 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.

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.

In geometry, a **9-orthoplex** or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells *4-faces*, 5376 5-simplex *5-faces*, 4608 6-simplex *6-faces*, 2304 7-simplex *7-faces*, and 512 8-simplex *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-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 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 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 **stericated 8-simplex** is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

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 six-dimensional geometry, a **runcic 6-cube** is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

- 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}) - Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o *b3o3o3o3o3o3o - henne".

**Harold Scott MacDonald** "**Donald**" **Coxeter**, FRS, FRSC, was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London, received his BA (1929) and PhD (1931) from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.

* Regular Polytopes* is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter. Originally published in 1947, the book was updated and republished in 1963 and 1973.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

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

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