The most remarkable characteristic of Min-protein dynamics in E. coli is the degree of robustness with which oscillatory patterns adapt to cell size and shape. However, as the models proposed so far could neither account for transitions between patterns, nor the observed temperature-dependence, the actual origin of Min-oscillations remains elusive. With an analytical approach in elliptical geometry, we introduce a robust minimal model based on experimental data, consistently explaining all transitions between patterns in wild type, filamentous, and nearly spherical cells. Moreover, we explain how an Arrhenius law for the hydrolysis rate accounts for the temperature-dependence of Min-oscillations. We find that pattern formation does not adapt to spatial templates of periodically restricted MinD attachment, indicating that Min-protein patterns are defined by recruitment alone. The underlying mechanism is the transient sequestration of MinE proteins by MinD, and a highly efficient transfer of MinD from old to new polar zones.