## Quasicrystals and GeometryQuasicrystals and Geometry brings together for the first time the many strands of contemporary research in quasicrystal geometry and weaves them into a coherent whole. The author describes the historical and scientific context of this work, and carefully explains what has been proved and what is conjectured. This, together with a bibliography of over 250 references, provides a solid background for further study. The discovery in 1984 of crystals with 'forbidden' symmetry posed fascinating and challenging problems in many fields of mathematics, as well as in the solid state sciences. Increasingly, mathematicians and physicists are becoming intrigued by the quasicrystal phenomenon, and the result has been an exponential growth in the literature on the geometry of diffraction patterns, the behaviour of the Fibonacci and other nonperiodic sequences, and the fascinating properties of the Penrose tilings and their many relatives. |

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### Contents

Lattices Voronoi cells and quasicrystals | 34 |

Introduction to diffraction geometry | 74 |

Order on the line | 106 |

Tiles and tilings | 135 |

Penrose tilings of the plane | 170 |

The aperiodic zoo | 207 |

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### Common terms and phrases

algebraic Ammann aperiodic crystals aperiodic protoset aperiodic tilings apertures atlas atomic Bruijn canonical projection Chapter compute configurations congruent construct corresponding countable define Definition defvar Delone set delta dense diagram diffraction condition diffraction patterns dimensional Dirac combs discrete dodecahedron dual tiling edges eigenvalue eigenvectors example facet vectors Fibonacci sequence finite number Fourier transform functions fundamental region geometry Godreche grid infinite integer intersection irrational isometry isomorphism class kite and dart lattice points linear matching rules mathematical crystallography matrix multigrid nonperiodic tilings obtain octagonal tilings one-dimensional orbit orthogonal patch Penrose tilings pentagon pentagrid periodic tiling plane point lattice point set population vector problem proof properties Proposition prototiles quasicrystal radius real numbers regular rhomb tiling rhombic root lattices rotation Section Senechal shapes shown in Figure space staircase string structure subset subspace substitution tilings symmetry group Theorem theory three-dimensional tilings admitted translation triangles vertices Voronoi cell Voronoi tessellation