Demipenteract (5-demicube) | ||
---|---|---|

Petrie polygon projection | ||

Type | Uniform 5-polytope | |

Family (D_{n}) | 5-demicube | |

Families (E_{n}) | k_{21} polytope 1 _{k2} polytope | |

Coxeter symbol | 1_{21} | |

Schläfli symbols | {3,3^{2,1}} = h{4,3^{3}}s{2,4,3,3} or h{2}h{4,3,3} sr{2,2,4,3} or h{2}h{2}h{4,3} h{2}h{2}h{2}h{4} s{2 ^{1,1,1,1}} or h{2}h{2}h{2}s{2} | |

Coxeter diagrams | ||

4-faces | 26 | 10 {3^{1,1,1}} 16 {3,3,3} |

Cells | 120 | 40 {3^{1,0,1}} 80 {3,3} |

Faces | 160 | {3} |

Edges | 80 | |

Vertices | 16 | |

Vertex figure | rectified 5-cell | |

Petrie polygon | Octagon | |

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

Properties | convex |

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

- Cartesian coordinates
- As a configuration
- Projected images
- Images
- Related polytopes
- References
- External links

It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM_{5} for a 5-dimensional *half measure* polytope.

Coxeter named this polytope as **1 _{21}** from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches,

It exists in the k_{21} polytope family as 1_{21} with the Gosset polytopes: 2_{21}, 3_{21}, and 4_{21}.

The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead.

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:

- (±1,±1,±1,±1,±1)

with an odd number of plus signs.

This configuration matrix represents the 5-demicube. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-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_{5} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | k-figure | notes | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{4} | ( ) | f_{0} | 16 | 10 | 30 | 10 | 20 | 5 | 5 | rectified 5-cell | D_{5}/A_{4} = 16*5!/5! = 16 | |

A_{2}A_{1}A_{1} | { } | f_{1} | 2 | 80 | 6 | 3 | 6 | 3 | 2 | triangular prism | D_{5}/A_{2}A_{1}A_{1} = 16*5!/3!/2/2 = 80 | |

A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 160 | 1 | 2 | 2 | 1 | Isosceles triangle | D_{5}/A_{2}A_{1} = 16*5!/3!/2 = 160 | |

A_{3}A_{1} | h{4,3} | f_{3} | 4 | 6 | 4 | 40 | * | 2 | 0 | { } | D_{5}/A_{3}A_{1} = 16*5!/4!/2 = 40 | |

A_{3} | {3,3} | 4 | 6 | 4 | * | 80 | 1 | 1 | { } | D_{5}/A_{3} = 16*5!/4! = 80 | ||

D_{4} | h{4,3,3} | f_{4} | 8 | 24 | 32 | 8 | 8 | 10 | * | ( ) | D_{5}/D_{4} = 16*5!/8/4! = 10 | |

A_{4} | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | 16 | ( ) | D_{5}/A_{4} = 16*5!/5! = 16 |

Perspective projection. |

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

Graph | ||

Dihedral symmetry | [10/2] | |

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

Graph | ||

Dihedral symmetry | [8] | [6] |

Coxeter plane | D_{3} | A_{3} |

Graph | ||

Dihedral symmetry | [4] | [4] |

It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 Uniform 5-polytopes (uniform 5-polytopes) that can be constructed from the D_{5} symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.

D5 polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

h{4,3,3,3} | h _{2}{4,3,3,3} | h _{3}{4,3,3,3} | h _{4}{4,3,3,3} | h _{2,3}{4,3,3,3} | h _{2,4}{4,3,3,3} | h _{3,4}{4,3,3,3} | h _{2,3,4}{4,3,3,3} |

The 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-simplices and 5-orthoplexes in the case of the 5-demicube). In Coxeter's notation the 5-demicube is given the symbol 1_{21}.

k_{21} figures in n dimensional | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

E_{n} | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

Coxeter group | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 1,920 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||

Graph | - | - | |||||||||

Name | −1_{21} | 0_{21} | 1_{21} | 2_{21} | 3_{21} | 4_{21} | 5_{21} | 6_{21} |

1_{k2} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

Coxeter group | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry (order) | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [[3<sup>2,2,1</sup>]] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 1,920 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ||||

Graph | - | - | |||||||||

Name | 1_{−1,2} | 1_{02} | 1_{12} | 1_{22} | 1_{32} | 1_{42} | 1_{52} | 1_{62} |

In four-dimensional geometry, the **rectified 5-cell** is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

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 geometry, **demihypercubes** are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as *hγ _{n}* for being

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 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 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **4 _{21}** is a semiregular uniform 8-polytope, constructed within the symmetry of the E

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

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 geometry, the **5 _{21} honeycomb** is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5

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 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 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.

- ↑ Coxeter, Regular Polytopes, sec 1.8 Configurations
- ↑ Coxeter, Complex Regular Polytopes, p.117
- ↑ Klitzing, Richard. "x3o3o *b3o3o - hin".

- T. Gosset:
*On the Regular and Semi-Regular Figures in Space of n Dimensions*, Messenger of Mathematics, Macmillan, 1900 - 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. "5D uniform polytopes (polytera) x3o3o *b3o3o - hin".

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

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