The engaging symmetry of Riemann surfaces: A historical perspective

The historical development of Riemann surfaces, starting in the late 1800’s, was driven in large part by the study of highly symmetrical surfaces. Not only do these surfaces have an engaging beauty, but they have very strong interconnections with other structures such as maps on a surface. In this expository article we first develop the basics of Riemann surfaces and their automorphism groups, laying out the tools for the historical treatment of the highly symmetrical surfaces in the later sections. The main topics of the later sections will be Hurwitz surfaces and groups, maps and dessins d’enfant on surfaces, and platonic and quasiplatonic surfaces and groups. For the novice reader, the introductory material will also be helpful background for reading some of the other papers in this volume, particularly the companion article on future directions in the field.