Chapter 8: Symmetry in Crystallography

The mathematical foundation relies on linear algebra and coordinate transformation matrices to describe how points move within different reference systems, including Cartesian and hexagonal frameworks. A central distinction separates isometric transformations—operations that preserve distances and angles—into proper motions, which maintain the original handedness of an object and yield transformation matrices with determinants of positive one, and improper motions, which reverse handedness and possess determinants of negative one. Proper motions encompass rotations and translations, while improper motions include reflections and inversions. The chapter introduces two standardized classification systems: Hermann-Mauguin notation, which predominates in crystallographic literature, and Schoenflies notation, preferred in chemistry and physics contexts. A substantial section addresses the mathematical relationship between rotational symmetry and translational periodicity, demonstrating through rigorous proof why periodic crystal structures permit only one-, two-, three-, four-, and six-fold rotation axes while excluding five-fold symmetry in conventional crystals, with the notable exception of quasicrystalline materials. The text further explores compound symmetry operations including roto-inversions that combine rotation with inversion through a center, screw axes that pair rotation with parallel translation, and glide planes that unite reflection with translation along the mirror plane. The chapter concludes by applying Euler's theorem to analyze point symmetry and the geometric relationships that emerge when multiple symmetry elements intersect, providing essential background for classifying and identifying the finite point groups that characterize crystals.