Faults in brittle rock are shear fractures formed through the interaction and coalescence of many tensile microcracks. The geometry of these microcracks and their surrounding elastic stress fields control the orientation of the final shear fracture surfaces. The classic CoulombMohr failure criterion5 predicts the development of two conjugate (bimodal) shear planes that are inclined at an acute angle to the axis of maximum compressive stress. This criterion, however, is incapable of explaining the three-dimensional polymodal fault patterns that are widely observed in rocks. Here we show that the elastic stress around tensile microcracks in three dimensions promotes a mutual interaction that produces brittle shear planes oriented obliquely to the remote principal stresses, and can therefore account for observed polymodal fault patterns. Our microcrack interaction model is based on the three-dimensional solution of Eshelby, unlike previous models that employed two-dimensional approximations. Our model predicts that shear fractures formed by the coalescence of interacting mode I cracks will be inclined at a maximum of 26 to the axes of remote maximum and intermediate compression. An improved understanding of brittle shear failure in three dimensions has important implications for earthquake seismology and rock-mass stability, as well as fluid migration in fractured rocks.