From quartz to quasicrystals: probing nature's geometric patterns in crystalline substances

Abstract

This review traces the evolution of our understanding of the periodic patterns observed in crystals, as described by geometric crystallography and in related group theoretical methods. Succinct introductions are also presented on aperiodic fractal, geometries and quasicrystals, as well as the development of the field of crystal structure analysis, which is concerned with the methods and procedures for establishing crystal structure. No assumption is made as to the level of crystallographic training of the reader. The examination here is divided into four time periods: (1) The Hellenic period to the seventeenth century, which was concerned with the study of polyhedra; (2) The eighteenth century, which gave rise to the birth of modern crystallography from mineralogy; (3) The nineteenth century, wherein the theories of external and internal symmetry arose, along with their relation to physical properties; and (4) The twentieth century, including the development of X-ray crystallography through to the discovery of quasicrystals. The review is a fact-based chronological documentation of events that attempts to bring continuity to numerous prior historical accounts of the various topics contained herein, which have appeared piecewise elsewhere over the last century.