FIG. 1. Birds trajectories and turning delays.

a, Reconstructed 3d trajectories of three birds belonging to a flock performing a collective turn. Inset: zig-zag due to wing flapping. b, c, Trajectories of all N = 176 birds in a flock. Each trajectory lies approximately on a plane. d, The radial acceleration of a turning bird displays a maximum as a function of time. This is in fact our very definition of a turn. Given two birds, i and j, we define the mutual turning delay τ_ij as the time we have to shift the full curve of the radial acceleration a_j(t) to maximally overlap it with a_i(t) (Methods). e, In the absence of experimental noise we must have, τ_ik + τ_kj = τ_ij: if i turns 20ms before k, and k turns 15ms before j, then i turns 35ms before j. Due to noise, time ordering will not hold strictly, but we still want it to be correct on average for τ_ij to make biological sense. We consider all triplets of birds and plot τ_ik + τ_kj vs. τ_ij. The data fall on the identity line with relatively small spread, confirming the temporal consistency of the turning delays (see also Methods and SI-Fig. S2).