(a) Distribution of the distances

*r* of each individual from the swarm centre, normalised by the swarm size

*R*_{s}. Each curve shows data for one swarm, and these volumetric probabilities

*P* satisfy

. (b) Swarm size as a function of mean swarm population. Each data point is computed as the average over the entire time of observation, and the ellipses show the standard deviation. Note that the number of individuals in each swarm is not fixed, since midges may enter or leave the swarm during the measurement period. Marker colours correspond to curve colours in (a). The dashed curve is a

*R*_{s} ∝ 〈

*N*〉

^{1/3} fit, as would be expected if the number density were independent of the swarm size. For the largest swarms, some of the midges flew outside the region imaged by the cameras; in these cases, the markers are outlined in grey. (c) Swarm aspect ratio as a function of swarm size. (d) Bulk swarm orientation. One axis of each swarm nearly aligns with gravity; for large swarms, it is the axis along which the swarm is longest,

*e*_{1}. (e) Spatial variation of swarm density. Slices through the three-dimensional probability density function of midge position are shown in colour on a logarithmic scale for the swarm marked with a black arrow in (b).

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