  Embryonic development leads to the reproducible and ordered appearance of
complexity from egg to adult. The successive differentiation of different cell
types, that elaborates this complexity, result from the activity of gene
networks and was likened by Waddington to a flow through a landscape in which
valleys represent alternative fates. Geometric methods allow the formal
representation of such landscapes and codify the types of behaviors that result
from systems of differential equations. Results from Smale and coworkers imply
that systems encompassing gene network models can be represented as potential
gradients with a Riemann metric, justifying the Waddington metaphor. Here, we
extend this representation to include parameter dependence and enumerate all
3-way cellular decisions realisable by tuning at most two parameters, which can
be generalized to include spatial coordinates in a tissue. All diagrams of cell
states vs model parameters are thereby enumerated. We unify a number of
standard models for spatial pattern formation by expressing them in potential
form. Turing systems appear non-potential yet in suitable variables the
dynamics are low dimensional, potential, and a time independent embedding
recovers the biological variables. Lateral inhibition is described by a saddle
point with many unstable directions. A model for the patterning of the
Drosophila eye appears as relaxation in a bistable potential. Geometric
reasoning provides intuitive dynamic models for development that are well
adapted to fit time-lapse data.
