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A first principles approach to the theoretical description of the developmentof biological forms, from a fertilized egg to a functioning embryo, remains acentral challenge to applied physics and theoretical biology. Rather than referto principles of self-organization and non-equilibrium statistical mechanics todescribe a developing embryo from its active cellular constituents, a purelygeometric theory is constructed that references the properties of the ambientspace that the embryo occupies. In 1975 the Fields laureate Ren\'{e} Thomdeveloped a system of techniques and local dynamical models that are capable ofreconstructing the local dynamic of an embryo at each new growth event of thesystem. Each new growth event (the development of a limb, for example) is atopological change in the dynamic of the system that can be classified onlyaccording to the properties of space. The local models can be non-conservativeflows with robust attractor behavior that serve as organizing centers forsystems development. Hamiltonian flows can also be considered with novel,self-reproducing vague attractor behavior. The set of growth events become anunfolding space related to the differentiable manifold of states. The set ofgrowth events, which Thom refers to as the catastrophe set, has specialalgebraic properties which permit these models to be low dimensional--the localmodel contains few parameters. We present Thom's work as a research programoutlining a framework for the construction of these local models, and,ultimately, the synthesis of these models into a full theoretical descriptionof a developing biological organism. We give examples of the application ofselected models to key growth events in the process of gastrulation.

Reexamining a Geometric Theory of Biological Growth