## Twisted Honeycombs |

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### Contents

Introduction | 1 |

Regular tessellations | 2 |

Finite groups of rotations | 3 |

Groups generated by two or three reflections | 5 |

The Pelrie polygon of a tessellation | 8 |

The regular tessellations ip q | 11 |

Threedimensional honeycombs | 14 |

The simplex cube cross polytope and 24cell | 17 |

The 120cell and the 600cell Imagine a pillar formed by ten solid dodecahedra D D | 19 |

Groups generated by four reflections | 24 |

The Petrie polygon of a honeycomb | 25 |

The regular honeycombs p q r | 28 |

Finite groups of quaternions The onecelled honeycombs | 36 |

Quaternions and Petrie polygons | 42 |

### Common terms and phrases

ABCD Abraham Sinkov adjacent alternating group Analogously antipodal points BCDE central inversion characteristic tetrahedron circles of symmetry compound rotation cosets cross polytope cyclic group denote dihedral angle dihedron dodecahedral cells equator Euclidean 4-space Euclidean plane finite groups glide reflection group of order groups of quaternions groups of rotations H. S. M. COXETER half-turn hedron hexahedra icosahedral group imply infinite inscribed integers interchanges isometry left screw left-handed Petrie polygons line-segment lines of symmetry LMNM midpoints normal subgroup obtain octahedron one-celled honeycomb opposite faces order 60 p-gons permutations Platonic solids polyhedron quotient group R R R R Reflexible honeycombs regular honeycomb ip regular tessellation ip representation right screw right-handed Petrie polygon rotatory reflection Schlafli symbol Similarly simplex Sinkov skew sphere spherical tessellations symmetry group Table three-dimensional tion torus transforms triangle p q Twisted honeycombs unit quaternion vectors vertex figure