  We determine the intrinsic, 3-dimensional shape distribution of star-forming
galaxies at 0<z<2.5, as inferred from their observed projected axis ratios. In
the present-day universe star-forming galaxies of all masses 1e9 - 1e11 Msol
are predominantly thin, nearly oblate disks, in line with previous studies. We
now extend this to higher redshifts, and find that among massive galaxies (M* >
1e10 Msol) disks are the most common geometric shape at all z < 2. Lower-mass
galaxies at z>1 possess a broad range of geometric shapes: the fraction of
elongated (prolate) galaxies increases toward higher redshifts and lower
masses. Galaxies with stellar mass 1e9 Msol (1e10 Msol) are a mix of roughly
equal numbers of elongated and disk galaxies at z~1 (z~2). This suggests that
galaxies in this mass range do not yet have disks that are sustained over many
orbital periods, implying that galaxies with present-day stellar mass
comparable to that of the Milky Way typically first formed such sustained
stellar disks at redshift z~1.5-2. Combined with constraints on the evolution
of the star formation rate density and the distribution of star formation over
galaxies with different masses, our findings imply that, averaged over cosmic
time, the majority of stars formed in disks.
