# 5-demicube

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

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

Petrie polygon projection | ||

Type | Uniform 5-polytope | |

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

Families (E_{n}) |
k_{21} polytope1 _{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 |

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,

^{2,1}}.

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

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.

## As a configuration

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 |

## Projected images

Perspective projection. |

## Images

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] |

## Related polytopes

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-cells and 16-cells in the case of the rectified 5-cell). 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 | 192 | 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 | 192 | 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} |

## References

- 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".

## External links

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