  An $SL_2$-tiling is a bi-infinite matrix of positive integers such that each
adjacent 2 by 2 submatrix has determinant 1. Such tilings are infinite
analogues of Conway-Coxeter friezes, and they have strong links to cluster
algebras, combinatorics, mathematical physics, and representation theory.
  We show that, by means of so-called Conway-Coxeter counting, every
$SL_2$-tiling arises from a triangulation of the disc with two, three or four
accumulation points.
  This improves earlier results which only discovered $SL_2$-tilings with
infinitely many entries equal to 1. Indeed, our methods show that there are
large classes of tilings with only finitely many entries equal to 1, including
a class of tilings with no 1's at all. In the latter case, we show that the
minimal entry of a tiling is unique.
