  A first principles approach to the theoretical description of the development
of biological forms, from a fertilized egg to a functioning embryo, remains a
central challenge to applied physics and theoretical biology. Rather than refer
to principles of self-organization and non-equilibrium statistical mechanics to
describe a developing embryo from its active cellular constituents, a purely
geometric theory is constructed that references the properties of the ambient
space that the embryo occupies. In 1975 the Fields laureate Ren\'{e} Thom
developed a system of techniques and local dynamical models that are capable of
reconstructing the local dynamic of an embryo at each new growth event of the
system. Each new growth event (the development of a limb, for example) is a
topological change in the dynamic of the system that can be classified only
according to the properties of space. The local models can be non-conservative
flows with robust attractor behavior that serve as organizing centers for
systems development. Hamiltonian flows can also be considered with novel,
self-reproducing vague attractor behavior. The set of growth events become an
unfolding space related to the differentiable manifold of states. The set of
growth events, which Thom refers to as the catastrophe set, has special
algebraic properties which permit these models to be low dimensional--the local
model contains few parameters. We present Thom's work as a research program
outlining a framework for the construction of these local models, and,
ultimately, the synthesis of these models into a full theoretical description
of a developing biological organism. We give examples of the application of
selected models to key growth events in the process of gastrulation.
