  We investigate the geometry of polycrystals, showing that for polycrystals
formed of convex grains the interior grains are polyhedral, while for
polycrystals with general grain geometry the set of triple points is small.
Then we investigate possible martensitic morphologies resulting from intergrain
contact. For cubic-to-tetragonal transformations we show that homogeneous
zero-energy microstructures matching a pure dilatation on a grain boundary
necessarily involve more than four deformation gradients. We discuss the
relevance of this result for observations of microstructures involving second
and third-order laminates in various materials. Finally we consider the more
specialized situation of bicrystals formed from materials having two
martensitic energy wells (such as for orthorhombic to monoclinic
transformations), but without any restrictions on the possible microstructure,
showing how a generalization of the Hadamard jump condition can be applied at
the intergrain boundary to show that a pure phase in either grain is impossible
at minimum energy.
