Platonic bodies, sphere packings and hyperbolic geometry
In the first part of the paper, the author describes more or less the content of his Habilitationsvortrag from 1994, reviewing the theory of polygonal tilings of the sphere and of the hyperbolic plane. The second half introduces to some more recent developments which are closely related to this, in particular, there is a description of some punctured hyperbolic surfaces which are extreme in their moduli space with respect to, e.g., length of the systoles. As a general phenomenon, these extreme surfaces are quotients of the hyperbolic plane by some arithmetically defined group. The fact is stressed that the hyperbolic world is richer than the euclidean one. The Gauss conjecture on the infinity of real quadratic fields with class number one is formulated in a geometric way. Reviewer's remark: The author mainly summarizes results. For proofs the reader is referred to the literature.
scientific article; zbMATH DE number 1508091