The (standard) continued fraction is an algorithm that produces for any real number $x$ a fraction of the form $a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \ldots}}$ where $a_0,a_1,\ldots$ are integers. The finite truncation of this fraction are rational numbers that are exactly the so called best approximations of $x$.

It is well known that the standard continued fraction can also be seen more geometrically as a diagonal changes on the space of tori: it encodes the geodesic flow on $\operatorname{SL}(2,\mathbb{R}) / \operatorname{SL}(2,\mathbb{Z})$ with respect to some fundamental domain.

One possible generalization to higher dimensions is via the diagonal actions on the homogeneous spaces $\operatorname{SL}(d,\mathbb{R}) / \operatorname{SL}(d,\mathbb{Z})$. This approach has been proved fruitful to study diophantine properties of vectors. One difficulty to do some codings in this settings is the absence of simple fundamental domains.

We will present another generalization to surfaces of higher genus (that are so called translation surfaces). This generalization provides a nice coding of Teichmüller geodesics and gives some insight on the geometry and dynamics of translation surfaces.