The flow and mixing of granular materials occur during handling of a wide variety of substances, from pharmaceuticals to cement to cereal grains(1,2). The understanding of such flows is, however, considerably more limited than it is for fluids(3); even basic processes such as tumbling(4,5), simple shear(6-8) and shaking(9-12) give rise to unexpected results. A case in point is granular pattern formation. A rich variety of patterns, including stripes, squares, hexagons and solitary structures, has been observed in vertically shaken, shallow granular beds(13,14). The vertical dynamics responsible for these patterns have been explored(15-19), but the role of horizontal motions of the grains is less well understood. Here I present a model of these motions that identifies two aspects as central to pattern formation : the randomization of horizontal velocities(20) by shaking, and the inelastic nature of grain collisions. These two elements alone, even without the influence of gravity, are sufficient to produce organized patterns in the horizontal plane-both those observed and others not yet seen experimentally.