Deformations with finitely many gradients

How does the range of the gradient of a lipschitz map look like? Here we consider finite sets of matrices as possible candidates and give a new class of examples - gradients without rank-one connections. The sharp estimate of the minimal number of gradients involved is related to properties of quasiconformal mappings, a geometric approach to rank-one convexity and a new, unifying way how to solve partial differential inclusions.