Turbulent flows disperse Lagrangian particles resulting in the growth of pa
irwise separations and, for sets of three or more particles, in a nontrivia
l dynamics of their configuration. The shape of such clusters is controlled
by the competition between coherent straining of the cluster and the indep
endent random motion of the particles due to small scale velocity fluctuati
ons. We introduce a statistical description of the geometry of the Lagrangi
an clusters and predict a self-similar distribution of shapes, which should
be observable in the inertial range of scales in high Reynolds numbers flo
ws.