  We introduce and study an infinite random triangulation of the unit disk that
arises as the limit of several recursive models. This triangulation is
generated by throwing chords uniformly at random in the unit disk and keeping
only those chords that do not intersect the previous ones. After throwing
infinitely many chords and taking the closure of the resulting set, one gets a
random compact subset of the unit disk whose complement is a countable union of
triangles. We show that this limiting random set has Hausdorff dimension
$\beta^*+1$, where $\beta^*=(\sqrt{17}-3)/2$, and that it can be described as
the geodesic lamination coded by a random continuous function which is
H\"{o}lder continuous with exponent $\beta^*-\varepsilon$, for every
$\varepsilon>0$. We also discuss recursive constructions of triangulations of
the $n$-gon that give rise to the same continuous limit when $n$ tends to
infinity.
