# Demicube

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n -demicube | |
---|---|

Rank | n |

Type | Uniform |

Notation | |

Coxeter diagram | ... |

Elements | |

Vertices | |

Vertex figure | Rectified (n -1)-simplex, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | Demicube-demicube-demicube: 90° |

Demicube-simplex-simplex: | |

Height | |

Central density | 1 |

Number of external pieces | |

Level of complexity | |

Related polytopes | |

Conjugate | None |

Abstract & topological properties | |

Flag count | |

Euler characteristic | 0 if n even 2 if n odd |

Orientable | Yes |

Properties | |

Symmetry | D_{n }, order |

Convex | Yes |

Nature | Tame |

A **demicube**, **demihypercube**, or **half measure polytope** is one of an infinite family of convex uniform polytopes. The n -dimensional demicube, or simply the **n -demicube**, has vertices. It can be formed by alternation of the n -hypercube, making its facets into (n -1)-demicube facets and half of its vertex figures into (n -1)-simplex facets.

## Examples[edit | edit source]

The demicubes up to 10D are the following:

Rank | Name | Picture | Rank | Name | Picture | |
---|---|---|---|---|---|---|

3 | Tetrahedron | 7 | Demihepteract | |||

4 | Hexadecachoron | 8 | Demiocteract | |||

5 | Demipenteract | 9 | Demienneract | |||

6 | Demihexeract | 10 | Demidekeract |

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of an n -demicube with edge length 1 are given by all even sign changes of:

- (√2/4, √2/4, ..., √2/4).

## Measures[edit | edit source]

- The circumradius of an n -demicube of unit edge length is .
- Its height from a demicube facet to the opposite demicube facet is , regardless of n .
- The angle between two demicube facet hyperplanes is 90°, and the angle between a demicube and a simplex facet hyperplane is .

## External links[edit | edit source]

- Wikipedia contributors. "Demihypercube".
- Klitzing, Richard. Demihypercube D
_{n}