Chapter 16: Periodic and Aperiodic Tilings

The foundation rests on regular monohedral tilings—the triangular, square, and hexagonal arrangements—which are formally characterized using Schlafli symbols that encode polygon types and vertex configurations. The discussion extends to semi-regular tilings, or Archimedean tilings, which combine multiple regular polygon types while preserving uniform vertex environments, with complementary Laves tilings serving as mathematical duals. A critical conceptual shift introduces quasi-periodic tilings, exemplified by the Penrose tiling, which exhibits five-fold rotational symmetry and demonstrates self-similarity properties connected to the golden ratio, serving as theoretical models for quasicrystalline materials. The chapter then transitions into three-dimensional geometry by examining Platonic solids as the five regular polyhedra and generalizing this framework to polytopes in higher dimensional spaces. The practical application of these geometric concepts becomes evident in the analysis of close-packed crystal structures, where successive layers of close-packed triangles generate three-dimensional lattices. This stacking produces face-centered cubic and hexagonal close-packed arrangements, each featuring tetrahedral and octahedral interstitial voids that become crucial for understanding alloy chemistry and defect physics. Finally, the chapter provides rigorous treatment of multiple notational systems for describing stacking sequences and polytypic variations, including classical ABC nomenclature, Ramsdell notation for polymorphic distinctions like 4H and 6H silicon carbide polytypes, Zhdanov numbers for characterizing repeating layer patterns, and h-c notation distinguishing layers by local symmetry environments. The concluding material addresses planar crystallographic defects including stacking fault formation and classification.