Geometry of numbers

Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers.[1] Hermann Minkowski (1896) initiated this line of research at the age of 26 in his work The Geometry of Numbers.[2]

Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)  

The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.[3]

  1. ^ MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX.
  2. ^ Minkowski, Hermann (2013-08-27). Space and Time: Minkowski's papers on relativity. Minkowski Institute Press. ISBN 978-0-9879871-1-2.
  3. ^ Schmidt's books. Grötschel, Martin; Lovász, László; Schrijver, Alexander (1993), Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, doi:10.1007/978-3-642-78240-4, ISBN 978-3-642-78242-8, MR 1261419