# Trihelical square tiling

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Trihelical square tiling | |
---|---|

Rank | 3 |

Space | Euclidean |

Notation | |

Schläfli symbol | ^{[1]} |

Elements | |

Faces | N square helices |

Edges | 3M×N |

Vertices | 2M×N |

Vertex figure | Triangle |

Related polytopes | |

Regiment | Trihelical square tiling |

Petrie dual | Tetrahelical triangular tiling |

Abstract properties | |

Schläfli type | {∞,3} |

The **trihelical square tiling** or petrial facetted halved mucube is a regular skew apeirohedron that consists of square helices, with three meeting at each vertex. All adjacent square helices in the trihelical square tiling that share a vertex are perpendicular to each other.

The trihelical square tiling is the Petrie dual of the tetrahelical triangular tiling. It also is the second-order facetting of the petrial halved mucube^{[2]}, so the edges and vertices of the trihelical square tiling are a subset of those found in the halved mucube.

## Vertex coordinates[edit | edit source]

The vertex coordinates of a trihelical square tiling of edge length 1 are given by all permutations of:

where range over the integers.

## External links[edit | edit source]

- jan Misali (2020). "there are 48 regular polyhedra".

## References[edit | edit source]

- ↑ McMullen & Schulte (1997:465)
- ↑ McMullen & Schulte (1997:465)

## Bibliography[edit | edit source]

- McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF).
*Discrete Computational Geometry*(47): 449–478. doi:10.1007/PL00009304.