  We bound the time it takes for a group of birds to reach steady state in a
standard flocking model. We prove that (i) within single exponential time
fragmentation ceases and each bird settles on a fixed flying direction; (ii)
the flocking network converges only after a number of steps that is an iterated
exponential of height logarithmic in the number of birds. We also prove the
highly surprising result that this bound is optimal. The model directs the
birds to adjust their velocities repeatedly by averaging them with their
neighbors within a fixed radius. The model is deterministic, but we show that
it can tolerate a reasonable amount of stochastic or even adversarial noise.
Our methods are highly general and we speculate that the results extend to a
wider class of models based on undirected flocking networks, whether defined
metrically or topologically. This work introduces new techniques of broader
interest, including the "flight net," the "iterated spectral shift," and a
certain "residue-clearing" argument in circuit complexity.
