A principal-stress rule for cell division in epithelia
Dense active materials, from cellular tissues to jammed and glassy systems, must continuously relieve internal mechanical stress to remain structurally stable as they are driven far from equilibrium. In epithelial tissues, this relief occurs through cell division, yet what sets the geometry of this structural remodeling event has, for over a century, been attributed to a purely geometric principle: Hertwigs rule, whereby cells divide along their long axis. We show that as epithelial tissues densify and cell shape anisotropy collapses, this geometric rule is superseded by a mechanical one in which dense epithelia relieve anisotropic stress by cells dividing along their principal axis, independent of the tissues isotropic stress state. Using direct force measurement and stress inference, we show that stress orientation, rather than cell shape, governs the axis of cell division across mechanically distinct systems, from fluid-like to jammed monolayers and structurally heterogeneous organoids, remaining predictive precisely where the classical geometric rule fails. This stress-oriented remodeling is reciprocally coupled to the materials mechanical state: anisotropic stress accelerates the underlying remodeling rate, while each remodeling event locally dissipates the stress that triggered it, closing a negative feedback loop. This "principal-stress rule" recasts epithelial cell division as a stress-relief mechanism intrinsic to dense active matter, providing a general mechanical framework linking internal stress, structural remodeling, and homeostasis in living materials.