A dual class of Lorentz transformations (LT), coupling time and space in linear and Brownian motions, is presented. It follows from two LT groups, for self-diffusion in simple liquids (BLT) and geometry (GLT), which are suitable to deal with scaling concepts in polymer and statistical physics. Dual LT exhibit a rich and complex phenomenology, promising however in view of proceeding former investigations on macromolecular dynamics and statistical mechanics defined at different length scales. When only BLT are concerned, time dilation and length contraction occur upon ordering the originally indistinguishable molecular disorder (i.e., by a diffusion coefficient decrease). Ether can be identified with a Brownian medium that invalidates the axiom of covariance. An uncertainty principle for intensive and extensive motion properties, depending on length scale, is finally deduced from GLT. The Brownian movement of a colloidal particle turns out corresponding to the picture of motion encountered in quantum mechanics.