teichmüller curves are algebraic curves in the moduli space of genus $g$ riemann surfaces that are isometrically immersed for the teichmüller metric. these curves arise from $\mathrm{sl}(2,\mathbb{r})$-orbits of highly symmetric translation surfaces, and the underlying surfaces have remarkable dynamical and algebro-geometric properties. a teichmüller curve is algebraically primitive if the trace field of its affine symmetry group has degree $g$. in genus $2$, calta and mcmullen independently discovered an infinite family of algebraically primitive teichmüller curves. however, in higher genus, such curves seem to be much rarer. we will discuss a result that shows that the regular $14$-gon yields the unique algebraically primitive teichmüller curve in genus $3$ of a particular combinatorial type. all relevant notions will be explained during the talk.