An Axiomatic Introduction to Crystallography

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.