The most common shape memory alloys are monoclinic-I martensite. We study their zero energy states and have two surprising results: First, there is a five-dimensional continuum in which the energy minimising microstructures are T3s, i.e. innite-rank laminates. To our knowledge, this is the first real material in which T3s occur. We discuss some of the consequences of this discovery. Second, there are in fact two types of monoclinic-I martensite, which differ by their convex poytope structure but not by their symmetry properties. It happens that all known materials belong to one of the two types. We explore whether materials belonging to the other type would have superior properties since they have dierent zero-energy states. Our analysis uses algebraic methods, in particular the theory of convex polytopes.