  Spherical embryos of the algal genus $Volvox$ must turn themselves inside out
to complete their embryogenesis. This `inversion', which shares important
features with morphological events such as gastrulation in animals, is perhaps
the simplest example of a topological transition in developmental biology.
Waves of cell shape changes are believed to play a major role in the process,
but quantification of the dynamics and formulation of a mathematical
description of the process have been lacking. Here, we use selective plane
illumination microscopy on $V. globator$ to obtain the first quantitative
three-dimensional visualizations of inversion $in~vivo$. A theory is formulated
for inversion based on local variations of intrinsic curvature and stretching
of an elastic shell, and for the mechanics of an elastic snap-through resisted
by the surrounding fluid.
