## Crystalline Symmetries, An Informal Mathematical IntroductionCrystalline Symmetries: an informal mathematical introduction is a guided tour through the maze of mathematical models and classifications that are used today to describe the symmetries of crystals. The mathematical basis of crystallography and the interpretation of The International Tables for X-ray Crystallography are explained in a heuristic and accessible way. In addition to discussing standard crystals, a special feature of this book is the chapter on generalised crystals and the Penrose tile model for the kinds of generalised crystals known as quasicrystals. This fruitful interaction between pure mathematics (symmetry, tilings) and physics should prove invaluable to final year undergraduate/graduate physicists and materials scientists; the reader gets a flavour of the powerful coherence of a group theoretical approach to crystallography. Mathematicians interested in applications of group theory to physical science will also find this book useful. |

### Contents

Symmetry and point groups | 22 |

Lattices | 40 |

The space groups | 59 |

Copyright | |

6 other sections not shown

### Common terms and phrases

assigned atomic b₁ Bravais lattices Chapter class-equivalent classification color symmetry configuration congruent conjugate construction correspond crystal structure crystal systems cube d₁ diffraction patterns dimensions dodecahedron dual lattice edges elements of G enumerated equation Escher example faces finite group finite subgroups five-fold symmetry geometry glide reflection group G group of order group theory Haüy hexagons identical infinite integer International Tables inversion isometries Kepler lattice planes lattice point layer left coset mathematical crystallography mathematician nonperiodic normal subgroup octahedron orbit orthogonal Penrose tiles pentagrid point group point set poles polygons prism projected quasicrystals regular rhombi right cosets rotary reflection screw rotation sectors Senechal set of points shown in Figure space groups space-filling polyhedra sphere stabilizer subgroup H subgroups of O(3 symmetry elements symmetry group symmetry operations symmorphic systems of points Theorem three-dimensional lattice three-dimensional space tion translation group translation-equivalent subgroup truncated octahedron unit cell vertices Voronoï cells