The limit shapes for the height function in the celebrated six-vertex model are still unknown in general situations. However, we recently found ways to compute them in a degeneration, which leads to a low density of corners of paths (or, equivalently, of c-type vertices; or, equivalently, of horizontal/vertical molecules of water). I will report on the progress in this direction emphasizing various new features (as compared to, say, dimers): appearance of hyperbolic PDEs; discontinuities in densities; connections to random permutations.