Crystalline symmetries : an informal mathematical introduction

Mathematical crystals: What is mathematical crystallography? Un peu d'histoire. Models of crystal structure. Symmetry and point groups: Symmetries and isometries. Symmetry groups. Point groups in 3-space. Lattices: Lattices and symmetry. Unit cells. The Voronoi cell. The reciprocal lattice. The space groups: Chronology. The orbits of a space group. Cosets and normal subgroups. Constructing the space groups. Symmorphic and nonsymmorphic space groups. Subgroups of space groups. Color symmetry: Why colors: Coloring finite figures. Coloring infinite patterns. Color groups, color symmetry, and colorings. Classification and the international tables: The classification problem. Why symmetry? Crystallographic classifications. Translating a page of the international tables. N-dimensional crystallography?: The view from N-dimensional space. Projections. The Penrose tiles. De Bruijn's interpretation. Generalized crystallography. Further reading and appendices.