# Demicross

The **demicrosses** are a series of self-intersecting uniform polytopes that generalize the 3D tetrahemihexahedron to higher dimensions. They are semiregular (having only regular facets) and quasiregular (representable with Coxeter-Dynkin diagrams with a single ringed node). The vertex figure of a demicross is the demicross of the previous dimension.

n -demicross | |
---|---|

Rank | n |

Type | Uniform |

Notation | |

Coxeter diagram | .../2 (o3o3/2o3o3*a3o...o3x) |

Elements | |

Facets | n (n − 1)-orthoplices, (n − 1)-simplices |

Vertices | 2n |

Vertex figure | n −1-demicross |

Measures (edge length 1) | |

Circumradius | |

Inradius | (for simplex facets) |

Surface area | |

Dihedral angle | |

Height | |

Related polytopes | |

Conjugate | None |

Convex hull | n -orthoplex |

Abstract & topological properties | |

Flag count | |

Orientable | No |

Properties | |

Symmetry | D_{n }, order |

Convex | No |

Nature | Tame |

The *n*-demicross is formed as a faceting of the *n*-orthoplex (the result having lower symmetry), retaining elements of every rank up to and including its ridges. Only the facets are changed: half of the (*n* - 1)-simplex facets are removed so that no two of them share a ridge, and *n* (*n* - 1)-orthoplices are added that pass through the center of the polytope. Thus, the demicrosses are hemipolytopes.

An isogonal bowtie formed from faceting a square could be considered to be the 2D demicross, but it isn't uniform so is conventionally excluded.

## Examples edit

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

3 | Tetrahemihexahedron |

4 | Tesseractihemioctachoron |

5 | Hexadecahemidecateron |

6 | Triacontidihemidodecapeton |

7 | Hexacontatetrahemitetradecaexon |

8 | Hecatonicosoctahemihexadecaexon |

9 | Diacosipentacontahexahemioctadecayotton |

## Vertex coordinates edit

Coordinates for the vertices of an *n*-demicross with edge length 1 are given by all permutations of:

- ,

where the last *n*–1 entries are zeros.

## External links edit

- Klitzing, Richard. "Demicross hO
_{n}".