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Demihepteract (7-demicube) | ||
---|---|---|

Petrie polygon projection | ||

Type | Uniform 7-polytope | |

Family | demihypercube | |

Coxeter symbol | 1_{41} | |

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

Coxeter diagrams | ||

6-faces | 78 | 14 {3^{1,3,1}} 64 {3 ^{5}} |

5-faces | 532 | 84 {3^{1,2,1}} 448 {3 ^{4}} |

4-faces | 1624 | 280 {3^{1,1,1}} 1344 {3 ^{3}} |

Cells | 2800 | 560 {3^{1,0,1}} 2240 {3,3} |

Faces | 2240 | {3} |

Edges | 672 | |

Vertices | 64 | |

Vertex figure | Rectified 6-simplex | |

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

Dual | ? | |

Properties | convex |

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.

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

A **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{7} for a 7-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 _{41}** 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 demihepteract centered at the origin are alternate halves of the hepteract:

- (±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

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

Graph | |||

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

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

Graph | |||

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

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

Graph | |||

Dihedral symmetry | [6] | [4] |

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element.^{ [1] }^{ [2] }

In geometry, H. S. M. Coxeter called a regular polytope a special kind of **configuration**.

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] }

In geometry, a **Wythoff construction**, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.

D_{7} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | k-figures | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{6} | ( ) | f_{0} | 64 | 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | 0_{41} | D_{7}/A_{6} = 64*7!/7! = 64 | |

A_{4}A_{1}A_{1} | { } | f_{1} | 2 | 672 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | { }×{3,3,3} | D_{7}/A_{4}A_{1}A_{1} = 64*7!/5!/2/2 = 672 | |

A_{3}A_{2} | 1_{00} | f_{2} | 3 | 3 | 2240 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}v( ) | D_{7}/A_{3}A_{2} = 64*7!/4!/3! = 2240 | |

A_{3}A_{3} | 1_{01} | f_{3} | 4 | 6 | 4 | 560 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | D_{7}/A_{3}A_{3} = 64*7!/4!/4! = 560 | |

A_{3}A_{2} | 1_{10} | 4 | 6 | 4 | * | 2240 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D_{7}/A_{3}A_{2} = 64*7!/4!/3! = 2240 | ||

D_{4}A_{2} | 1_{11} | f_{4} | 8 | 24 | 32 | 8 | 8 | 280 | * | 3 | 0 | 3 | 0 | {3} | D_{7}/D_{4}A_{2} = 64*7!/8/4!/2 = 280 | |

A_{4}A_{1} | 1_{20} | 5 | 10 | 10 | 0 | 5 | * | 1344 | 1 | 2 | 2 | 1 | { }v( ) | D_{7}/A_{4}A_{1} = 64*7!/5!/2 = 1344 | ||

D_{5}A_{1} | 1_{21} | f_{5} | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 84 | * | 2 | 0 | { } | D_{7}/D_{5}A_{1} = 64*7!/16/5!/2 = 84 | |

A_{5} | 1_{30} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 448 | 1 | 1 | D_{7}/A_{5} = 64*7!/6! = 448 | |||

D_{6} | 1_{31} | f_{6} | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | 14 | * | ( ) | D_{7}/D_{6} = 64*7!/32/6! = 14 | |

A_{6} | 1_{40} | 7 | 21 | 35 | 0 | 35 | 0 | 21 | 0 | 7 | * | 64 | D_{7}/A_{6} = 64*7!/7! = 64 |

There are 95 uniform polytopes with D_{6} symmetry, 63 are shared by the B_{6} symmetry, and 32 are unique:

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

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 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 seven-dimensional geometry, a **rectified 7-cube** is a convex uniform 7-polytope, being a rectification 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 six-dimensional geometry, a **steric 6-cube** is a convex uniform 6-polytope. There are unique 4 steric forms of 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.

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 five-dimensional geometry, a **steric 5-cube** or is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.

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.

- ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
- ↑ Coxeter, Complex Regular Polytopes, p.117
- ↑ Klitzing, Richard. "x3o3o *b3o3o3o - hax".

- 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. "7D uniform polytopes (polyexa) x3o3o *b3o3o3o3o - hesa".

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

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