The republication of Felix Klein's “Lectures on the icosahedron and the solution ofequations of the fifth degree” corresponds to the constantly growing demand for thiswork， which was published in Leipzig more than a hundred years ago. A good deal ofin-terest in Klein's book might be certainly due to the continuous relevance of the “icosa-hedral mathematics”， i.e. the mathematics， in which the geometry and symmetry ofthe icosahedron， as well as the other Platonic solids and the regular polygons， play an essential role. In this regard， the foUowing developments in the last twenty years are mentioned： the study of the so-called Klein singularities， also known as Du Val singu-larities， rational double points or simple singularities （see e.g. Du Val [1934l， M. Artin [1966J， Brieskorn [1968]， [1970]， Arnol'd [1972]or the survey articles ofArnol'd [1974]，Brieskorn [1976]， Durfee [1979]， Slowdowy [1983]， the investigation of certain ellip-tic and Hilbert-Blumenthal modular surfaces （see Hirzebruch [1976]， [1977]， Naruki [1978]， Burns [1983]）， the construction of an indecomposable vector bundle of rank 2 on P4 （Horrocks-Mumford [1973]and the analysis ofits properties （see Barth-Hulek-Moore [1984]， [1987]， Barth-Hulek [1985]， Decker-Schreyer [1986]， Hulek [1986]， [1987]，Hulek-Lange[1988]and the survey article of Hulek [1989]）. A particularly remarkable fact， especially with regard to Klein's thanks to Sophus Lie in the preface of his book，is the relationship between the Platonic solids， or more precisely the finite subgroups of SU（2，C）， and the complex simple Lie groups of types Ar， Dr， Er， which was discov-ered by Grothendieck and Brieskorn （see Brieskorn [1970]）. While this discovery built on deep studies on the theory of resolution and deformation of the singularities of sur-faces mentioned above and the geometry of the conjugation classes ofsimple algebraic groups， a more direct， although more formal derivation of this relationship was given by J. McKay， who showed how the irreducible characters of finite binary groups can be parameterized in a natural way by the vertices of the extended Coxeter-Witt-Dynkin diagrams ofthe corresponding Lie groups （see McKay [1980]， Ford-McKay [1979]）.