The Symmetries of Things has three major sections, subdivided into 26 chapters. The first of the sections discusses the symmetries of geometric objects. It includes both the symmetries of finite objects in two and three dimensions, and two-dimensional infinite structures such as frieze patterns and tessellations, and develops a new notation for these symmetries based on work of Alexander Murray MacBeath that, as proven by the authors using a simplified form of the Riemann–Hurwitz formula, covers all possibilities. Other topics include Euler's polyhedral formula and the classification of two-dimensional surfaces. It is heavily illustrated with both artworks and objects depicting these symmetries, such as the art of M. C. Escher and Bathsheba Grossman, as well as new illustrations created by the authors using custom software.