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
R
n
,
{\displaystyle \mathbb {R} ^{n},}
and the study of these lattices provides fundamental information on algebraic numbers.[ 1] The geometry of numbers was initiated by Hermann Minkowski (1910 ).
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 .[ 2]
^ MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html , Classification 11HXX.
^ 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