Geometry of numbers

Not to be confused with Numerical algebraic geometry.

Geometry of numbers, also known as geometric number theory, 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 $\mathbb {R} ^{n},$ and the study of these lattices provides fundamental information on algebraic numbers. Hermann Minkowski (1896) initiated this line of research at the age of 26 in his work The Geometry of Numbers.

Best rational approximants for irrational numbers $\pi$ (green circle), $e$ (blue diamond), $\phi$ (pink oblong), ${\sqrt {3}}/2$ (grey hexagon), $1/{\sqrt {2}}$ (red octagon) and $1/{\sqrt {3}}$ (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.