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Studies of fate patterning during development typically emphasize cell-cell communication via diffusible chemical signals. Recent experiments on stem cell colonies, however, suggest that in some cases mechanical stresses, rather than secreted chemicals, enable long-ranged cell-cell interactions that specify positional information and pattern cell fates. These findings inspire a model of mechanical patterning: fate affects cell contractility, and pressure in the cell layer biases fate. Cells at the colony edge, more contractile than cells at the center, seed a pattern that propagates via force transmission. Strikingly, our model implies that the width of the outer fate domain varies nonmonotonically with substrate stiffness, a prediction that we confirm experimentally; we argue that a similar dependence on substrate stiffness can be achieved by a chemical morphogen only if strong constraints on the signaling pathway's mechanobiology are met. Our findings thus support the idea that mechanical stress can mediate patterning in the complete absence of chemical morphogens, even in nonmotile cell layers, thus expanding the repertoire of possible roles for mechanical signals in development and morphogenesis. Future tests of additional model predictions, like the effect of anisotropic substrate rigidity, will further broaden the range of achievable fate patterns. Published by the American Physical Society2025