# Mucube

Mucube | |
---|---|

Rank | 3 |

Space | Euclidean |

Notation | |

Schläfli symbol | {4, 6 | 4} |

Elements | |

Faces | 3N squares |

Edges | 6N |

Vertices | 2N |

Vertex figure | Skew hexagon |

Related polytopes | |

Dual | Muoctahedron |

Petrie dual | Petrial mucube |

Convex hull | Cubic honeycomb |

Abstract properties | |

Schläfli type | {4,6} |

Topological properties | |

Orientable | Yes |

Genus | ∞ |

Properties | |

Convex | No |

The **mucube** or **muc,** short for multiple cube, is one of the three regular skew apeirohedra in Euclidean 3-space. It's an infinite polyhedron that consists solely of squares, with 6 meeting at each vertex.

The mucube is based on the cubic honeycomb. Its faces are a subset of the faces of the cubic honeycomb, but with some removed (square holes) such that each set of coplanar faces turns into a checkerboard pattern. In fact, when the cubic honeycomb is being given as x4o3o4x, then the here solely being used squares are the ones of type x . . x.

It can also be formed as a modwrap of the order-6 square tiling by identifying every 4th vertex on each hole.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a mucube with unit edge length are given by (i, j, k), where i, j, k take on any integer value.

## Derivatives[edit | edit source]

## External links[edit | edit source]

- Wikipedia Contributors. "Regular skew apeirohedron".
- Klitzing, Richard. "Skew polytopes".
- Klitzing, Richard. "hisquat".

## References[edit | edit source]

- Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008).
*The Symmetries of Things*. pp. 333–335. - Coxeter, H.S.M. (1999).
*The Beauty of Geometry: Twelve essays*. Dover Publications, Inc. pp. 157*ff*. - McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space".
*Discrete Computational Geometry*.**17**: 449–478. - jan Misali (2020). "there are 48 regular polyhedra".