Demihexeract (6-demicube) | ||
---|---|---|

Petrie polygon projection | ||

Type | Uniform 6-polytope | |

Family | demihypercube | |

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

Coxeter diagrams | = = | |

Coxeter symbol | 1_{31} | |

5-faces | 44 | 12 {3^{1,2,1}} 32 {3 ^{4}} |

4-faces | 252 | 60 {3^{1,1,1}} 192 {3 ^{3}} |

Cells | 640 | 160 {3^{1,0,1}} 480 {3,3} |

Faces | 640 | {3} |

Edges | 240 | |

Vertices | 32 | |

Vertex figure | Rectified 5-simplex | |

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

Petrie polygon | decagon | |

Properties | convex |

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.

- Cartesian coordinates
- As a configuration
- Images
- Related polytopes
- Skew icosahedron
- References
- External links

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{6} for a 6-dimensional *half measure* polytope.

Coxeter named this polytope as **1 _{31}** from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol or {3,3

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:

- (±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

This configuration matrix represents the 6-demicube. 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-demicube. 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] }

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

A_{4} | ( ) | f_{0} | 32 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | r{3,3,3,3} | D_{6}/A_{4} = 32*6!/5! = 32 | |

A_{3}A_{1}A_{1} | { } | f_{1} | 2 | 240 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | {}x{3,3} | D_{6}/A_{3}A_{1}A_{1} = 32*6!/4!/2/2 = 240 | |

A_{3}A_{2} | {3} | f_{2} | 3 | 3 | 640 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D_{6}/A_{3}A_{2} = 32*6!/4!/3! = 640 | |

A_{3}A_{1} | h{4,3} | f_{3} | 4 | 6 | 4 | 160 | * | 3 | 0 | 3 | 0 | {3} | D_{6}/A_{3}A_{1} = 32*6!/4!/2 = 160 | |

A_{3}A_{2} | {3,3} | 4 | 6 | 4 | * | 480 | 1 | 2 | 2 | 1 | {}v( ) | D_{6}/A_{3}A_{2} = 32*6!/4!/3! = 480 | ||

D_{4}A_{1} | h{4,3,3} | f_{4} | 8 | 24 | 32 | 8 | 8 | 60 | * | 2 | 0 | { } | D_{6}/D_{4}A_{1} = 32*6!/8/4!/2 = 60 | |

A_{4} | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 192 | 1 | 1 | D_{6}/A_{4} = 32*6!/5! = 192 | |||

D_{5} | h{4,3,3,3} | f_{5} | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 12 | * | ( ) | D_{6}/D_{5} = 32*6!/16/5! = 12 | |

A_{5} | {3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 32 | D_{6}/A_{5} = 32*6!/6! = 32 |

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

Graph | ||

Dihedral symmetry | [12/2] | |

Coxeter plane | D_{6} | D_{5} |

Graph | ||

Dihedral symmetry | [10] | [8] |

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

Graph | ||

Dihedral symmetry | [6] | [4] |

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

Graph | ||

Dihedral symmetry | [6] | [4] |

There are 47 uniform polytopes with D_{6} symmetry, 31 are shared by the B_{6} symmetry, and 16 are unique:

The 6-demicube, 1_{31} is third in a dimensional series of uniform polytopes, expressed by Coxeter as k_{31} series. The fifth figure is a Euclidean honeycomb, 3_{31}, and the final is a noncompact hyperbolic honeycomb, 4_{31}. Each progressive uniform polytope is constructed from the previous as its vertex figure.

n | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|

Coxeter group | A_{3}A_{1} | A_{5} | D_{6} | E_{7} | = E_{7}^{+} | =E_{7}^{++} |

Coxeter diagram | ||||||

Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |

Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |

Graph | - | - | ||||

Name | −1_{31} | 0_{31} | 1_{31} | 2_{31} | 3_{31} | 4_{31} |

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The fourth figure is the Euclidean honeycomb 1_{33} and the final is a noncompact hyperbolic honeycomb, 1_{34}.

Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|

n | 4 | 5 | 6 | 7 | 8 | 9 |

Coxeter group | A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |

Coxeter diagram | ||||||

Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [[3<sup>3,3,1</sup>]] | [3^{4,3,1}] |

Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |

Graph | - | - | ||||

Name | 1_{3,-1} | 1_{30} | 1_{31} | 1_{32} | 1_{33} | 1_{34} |

Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the **regular skew icosahedron**.^{ [4] }^{ [5] }

In six-dimensional geometry, a **uniform polypeton** is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-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 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 five-dimensional geometry, a **rectified 5-simplex** is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In 7-dimensional geometry, **1 _{32}** is a uniform polytope, constructed from the E7 group.

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

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 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.

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 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.

In seven-dimensional Euclidean geometry, the **quarter 7-cubic honeycomb** is a uniform space-filling tessellation. It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb. Its facets are 7-demicubes, pentellated 7-demicubes, and {3^{1,1,1}}×{3,3} duoprisms.

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".
- ↑ Coxeter, H. S. M.
*The beauty of geometry : twelve essays*(Dover ed.). Dover Publications. pp. 450–451. ISBN 9780486409191. - ↑ Deza, Michael; Shtogrin, Mikhael (2000). "Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices".
*Advanced Studies in Pure Mathematics*: 77. doi: 10.2969/aspm/02710073 . Retrieved 4 April 2020.

- 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. "6D uniform polytopes (polypeta) x3o3o *b3o3o3o – hax".

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

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