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Ballistic deposition is a growth model from statistical physics. A version of the model operates according to this rule: particles fall down at random over each point of a multidimensional integer lattice, and stick to the growing cluster at first point of contact. A complicated porous structure results, with sideways growing overhangs that shade parts of the cluster from the incoming stream of new particles. However, the height function of the growing cluster has relatively simple dynamics. We prove that under suitable scaling of time and space, the random height function converges to a deterministic limiting function that satisfies a Hamilton-Jacobi equation.