## Abstract Regular Polytopes, Volume 92Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory. |

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### Contents

II | 1 |

III | 7 |

IV | 15 |

V | 17 |

VI | 21 |

VII | 22 |

VIII | 31 |

IX | 39 |

L | 255 |

LI | 259 |

LII | 264 |

LIII | 272 |

LIV | 289 |

LV | 290 |

LVI | 298 |

LVII | 305 |

X | 42 |

XI | 49 |

XII | 60 |

XIII | 64 |

XV | 71 |

XVI | 76 |

XVII | 78 |

XVIII | 83 |

XIX | 95 |

XX | 96 |

XXI | 101 |

XXII | 103 |

XXIII | 106 |

XXIV | 109 |

XXV | 115 |

XXVI | 121 |

XXVIII | 127 |

XXIX | 140 |

XXX | 148 |

XXXII | 152 |

XXXIII | 162 |

XXXIV | 165 |

XXXV | 170 |

XXXVI | 172 |

XXXVII | 175 |

XXXVIII | 177 |

XXXIX | 178 |

XL | 183 |

XLII | 192 |

XLIII | 201 |

XLIV | 206 |

XLV | 217 |

XLVI | 236 |

XLVII | 244 |

XLIX | 247 |

LVIII | 320 |

LIX | 332 |

LX | 347 |

LXI | 355 |

LXII | 360 |

LXIII | 363 |

LXIV | 369 |

LXV | 378 |

LXVI | 383 |

LXVII | 387 |

LXVIII | 392 |

LXIX | 400 |

LXX | 410 |

LXXI | 417 |

LXXII | 423 |

LXXIII | 431 |

LXXIV | 437 |

LXXV | 445 |

LXXVI | 450 |

LXXVII | 459 |

LXXVIII | 462 |

LXXIX | 465 |

LXXX | 471 |

LXXXI | 478 |

LXXXII | 484 |

LXXXIII | 490 |

LXXXIV | 500 |

LXXXV | 502 |

LXXXVI | 509 |

519 | |

LXXXVIII | 539 |

LXXXIX | 543 |

544 | |

### Common terms and phrases

abstract group abstract polytope abstract regular polytopes apeirogon apeirohedra apeirotope base flag branches C-group Chapter circuit combinatorial conjugate construction Corollary corresponding Coxeter group defined denote diagonal diagram discrete discussion dual edges element equivalent example exists extra relation faces facets and vertex-figures fact Figure finite group finite polytopes follows geometric group group G group r(P group W hence hermitian form homomorphism honeycomb hyperplane induced infinite initial vertex integer intersection property involution involutory irreducible isometry isomorphic lattice linear matrix n-polytope nodes normal subgroup notation Note obtain orthogonal pair particular permutation Petrie polygon polyhedra Proof Proposition prove quotient realization Recall reflexion group regular polyhedron regular polytopes regular tessellation representation result Schlafli symbol Section semi-direct product simplex space space-forms spherical string C-group subgroup subspace symmetry group Theorem Let topological toroidal translation triangle trivial unitary universal polytope universal regular vectors vertex-figure vertex-set Wythoff