I will report on some recent progress on the problem of characterizing sets which lie in the image of a Lipschitz map from the plane into $3$-dimensional Euclidean space. The new construction of Lipschitz maps is based on a simple observation about square packings. As an application, in any complete Ahlfors $q$-regular metric space with $q>m-1$, we construct an abundance of $m$-rectifiable doubling measures that are purely $(m-1)$-unrectifiable. Moreover, it is possible to prescribe the lower and upper Hausdorff and packing dimensions of the measures. This is joint work with Raanan Schul.