Stem cells are identical in many scales, they share the same molecular composition, DNA, genes, and genetic networks, yet they should acquire different properties to form a functional tissue. Therefore, they must interact and get some external information from their environment, either spatial (dynamical fields) or temporal (lineage). In this paperwe test towhat extent coupled chemical and physical fields can underlie the cell's positional information during development. We choose the root apical meristem of Arabidopsis thaliana to model the emergence of cellular patterns. We built a model to study the dynamics and interactions between the cell divisions, the local auxin concentration, and physical elastic fields. Our model recovers important aspects of the self-organized and resilient behavior of the observed cellular patterns in the Arabidopsis root, in particular, the reverse fountain pattern observed in the auxin transport, the PIN-FORMED (protein family of auxin transporters) polarization pattern and the accumulation of auxin near the region of maximum curvature in a bent root. Our model may be extended to predict altered cellular patterns that are expected under various applied auxin treatments or modified physical growth conditions.