This book introduces a new geometric vision of continued fractions. It
covers several applications to questions related to such areas as
Diophantine approximation, algebraic number theory, and toric geometry.
The second edition now includes a geometric approach to Gauss Reduction
Theory, classification of integer regular polygons and some further new
subjects.
Traditionally a subject of number theory, continued fractions appear in
dynamical systems, algebraic geometry, topology, and even celestial
mechanics. The rise of computational geometry has resulted in renewed
interest in multidimensional generalizations of continued fractions.
Numerous classical theorems have been extended to the multidimensional
case, casting light on phenomena in diverse areas of mathematics.
The reader will find an overview of current progress in the geometric
theory of multidimensional continued fractions accompanied by currently
open problems. Whenever possible, we illustrate geometric constructions
with figures and examples. Each chapter has exercises useful for
undergraduate or graduate courses.