From Friezes to Quasicrystals: A History of Symmetry Groups

Even if, since many thousands of years, humans were fascinated by symmetry, which is reflected in many preserved ornaments on buildings, paintings, pottery, etc., the mathematical study of symmetrical patterns is just a few hundred years old. In fact, until the nineteenth century, when it received an impulse from crystallography, no systematical approaches were developed. In this chapter, we follow the history of (Euclidean) symmetry from its first appearances and connections with art, over geometric classifications by nineteenth-century crystallographers, the coalescence of this crystallographic-geometrical approach with theory of groups in the second part of the nineteenth century, until various twentieth-century generalizations to symmetry groups in higher dimensions, magnetic, and colored groups.