Overview
- New approach to the geometry of numbers, very visual and algorithmic
- Numerous illustrations and examples
- Problems for each chapter
Part of the book series: Algorithms and Computation in Mathematics (AACIM, volume 26)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
About this book
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.
Similar content being viewed by others
Keywords
Table of contents (27 chapters)
-
Front Matter
-
Regular Continued Fractions
-
Front Matter
-
-
Multidimensional Continued Fractions
-
Front Matter
-
Reviews
“There are a modest number of exercises at the end of each chapter; most of these are to work out specific numerical examples. I view this as a monograph on a very specialized subject rather than a textbook.” (Allen Stenger, MAA Reviews, October 30, 2022)
Authors and Affiliations
About the author
Oleg Karpenkov is a mathematician at the University of Liverpool (UK), working in the general area of discrete geometry and its applications. More specifically, his research interests include geometry of numbers, discrete and semi-discrete differential geometry and self-stressed configurations of graphs. Oleg has completed his Ph.D. at Moscow State University under the supervision of Vladimir Arnold in 2005. Further he held several postdoctoral positions in Paris (Fellowship of the Mairie de Paris), Leiden, and Graz (Lise Meitner Fellowship) before arriving in Liverpool in 2012. In 2013 he published a book "Geometry of Continued Fractions" (its extended second edition will be available soon). Currently his Erdos number is 3.
Bibliographic Information
Book Title: Geometry of Continued Fractions
Authors: Oleg N. Karpenkov
Series Title: Algorithms and Computation in Mathematics
DOI: https://doi.org/10.1007/978-3-662-65277-0
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag GmbH Germany, part of Springer Nature 2022
Hardcover ISBN: 978-3-662-65276-3Published: 29 May 2022
Softcover ISBN: 978-3-662-65279-4Published: 30 May 2023
eBook ISBN: 978-3-662-65277-0Published: 28 May 2022
Series ISSN: 1431-1550
Edition Number: 2
Number of Pages: XX, 451
Number of Illustrations: 69 b/w illustrations
Topics: Algebra, Order, Lattices, Ordered Algebraic Structures, Approximations and Expansions, Convex and Discrete Geometry, Number Theory