In the last decade mathematical crystallography has found increasing interest. Siginificant results have been obtained by algebraic, geometric, and group theoretic methods. Also classical crystallography in three-dimen- sional Euclidean space has been extended to higher dimen- sions in order to understand better the dimension independent crystallographic properties. The aim of this note is to introduce the reader to the fascinating and rich world of geometric crystallography. The prerequisites for reading it are elementary geometry and topological notations, and basic knowledge of group theory and linear algebra. Crystallography is geometric by its nature. In many cases, geometric arguments are the most appropriate and can thus best be understood. Thus the geometric point of view is emphasized here. The approach is axiomatic start- ing from discrete point sets in Euclidean space. Symmetry comes in very soon and plays a central role. Each chapter starts with the necessary definitions and then the subject is treated in two- and three-dimensional space. Subsequent sections give an extension to higher dimensions.
Short historical remarks added at the end of the chapters will show the development of the theory. The chapters are main- ly self-contained. Frequent cross references, as well as an extended subject index, will help the reader who is only interested in a particular subject.
`The book will be welcomed by crystallographers who want to improve their knowledge of theoretical and mathematical crystallography.' Acta Crystallographica, 1987
1. Basic definitions.- 1.1. Axioms of geometric crystallography.- 1.2. Euclidean vector space.- 1.3. Rigid motions.- 1.4. Symmetry operations.- 1.5. Classifications.- 1.6. Historical remarks.- 2. Dirichlet domains.- 2.1. Definition of the Dirichlet domain.- 2.2. Some properties of Dirichlet domains.- 2.3. Dirichlet domain partition.- 2.4. A practical method to calculate.- 3. Lattices.- 3.1. The theorem of Bieberbach.- 3.2. Lattice bases.- 3.3. Orthogonal basis.- 3.4. Lattice planes.- 3.5. Dirichlet parallelotopes.- 4. Reduction of quadratic forms.- 4.1. Definition of the ?-reduced form.- 4.2. The reduction scheme of Lagrange.- 4.3. The reduction scheme of Seeber.- 4.4. The reduction scheme of Selling.- 4.5. The reduction scheme of Minkowski.- 4.6. Historical remarks.- 5. Crysta1lographic symmetry operations.- 5.1. Defini11ons.- 5.2. Rotations in E2.- 5.3. Rotations in En.- 5.4. Symmetry support.- 5.5. General symmetry operations in En.- 6. Crvstallographic point groups.- 6.1. Definitions.- 6.2. Point groups in E2.- 6.3. Point groups in E3.- 6.4. Point groups in En.- 6.5. Root classes.- 6.6. Isomorphsm types of point groups.- 6.7. Historical remarks.- 7. Lattice symmetries.- 7.1. Definitions.- 7.2. Bravais point groups.- 7.3. Bravais types of lattices.- 7.4. Arithmetic crystal classes.- 7.5. Crystal forms.- 7.6. Historical remarks.- 8. Space groups.- 8.1. Definitions.- 8.2. Derivation of space groups.- 8.3. Normalizers of symmetry groups.- 8.4. Subgroups of space groups.- 8.5. Crystallographic orbits.- 8.6. Colour groups and colourings.- 8.7. Subperiodic groups.- 8.8. Historical remarks.- 9. Space partitions.- 9.1. Definitions.- 9.2. Dirichlet domain partitions.- 9.3. Parallelotopes.- 9.4. The regularity condition.- 9.5. Dissections of polytopes.- 9.6. Historical remarks.- 10. Packings of balls.- 10.1. Definitions.- 10.2. Packings of disks into E2.- 10.3. Packings of balls into E3.- 10.4. Lattice packings of balls in En.- 10.5. Historical remarks.- References.