Optimal mapping is one of the longest standing problems in computational mathematics. It is natural to measure relative length error to assess map quality. Maximum of such error is called quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We present physics-based quasi-isometric stiffening (QIS) algorithm for max-norm minimization of hyperelastic distortion. QIS perfectly equidistributes distortion over all the domain for the ground truth test (unit hemisphere flattenning) and, when it is not possible, tends to create zones where all cells have the same distortion. Each such zone corresponds to a fragment of elastic material that became rigid under stiffening, reaching the deformation limit. As such, maps built by QIS are related to de Boor equidistribution principle that asks an integral of certain error indicator function to be the same over each mesh cell. We prove that our method can build in a finite number of steps a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum. We performed a massive testing: more than 75'000 maps were computed, and QIS won 100\% of the tests. To sum up, we reliably build 2D and 3D mesh deformations with smallest known distortion estimates for very stiff problems.