Diophantine approximation primarily deals with how closely real numbers can be approximated by rationals. In this talk, I will observe the limit points of these approximations under different normalisations.

In particular, if the real vector together with 1 form a basis for a number field, under the correct normalisation, the set of limit points of the approximations of that vector exhibit well ordered geometric structures. For example when the approximating vector is from a cubic number field, these approximations are countable union of ellipses or pair of hyperbolas depending on whether or not, the number field can be embedded into complex numbers in a non-trivial way.