  Geometrical constructions, such as the tangent construction on the molar free
energy for determining whether a particular composition of a solution, is
stable, are related to similar tangent constructions on the
orientation-dependent interfacial energy for determining stable interface
orientations and on the orientation dependence of the crystal growth rate which
tests whether a particular orientation appears on a growing crystal. Subtle
differences in the geometric constructions for the three fields arise from the
choice of a metric (unit of measure). Using results from studies of extensive
and convex functions we demonstrate that there is a common mathematical
structure for these three disparate topics, and use this to find new uses for
well-known graphical methods for all three topics. Thus the use of chemical
potentials for solution thermodynamics is very similar to known vector
formulations for surface thermodynamics, and the method of characteristics
which tracks the interfaces of growing crystals; the Gibbs-Duhem equation is
analogous to the Cahn-Hoffman equation. The Wulff construction for equilibrium
crystal shapes can be modified to construct a ``phase shape'' from solution
free energies that is a potentially useful method of numerical calculations of
phase diagrams from known thermodynamical data.
