This chapter elucidates three primary topics: (1) The convergence of surfaces with zero charge density in crystals to triply periodic minimal surfaces (TPMS); (2) The topological continuity of martensitic phase transformations; and (3) The differences between shape memory alloys and ordinary martensites. The first question is addressed using the Boltzmann equation, and the following two questions are resolved using the geometric entropy model. The geometric entropy proposed in this work includes two terms: (1) a term based on the Helfrich Hamiltonian, determined by mean curvature and Gaussian curvature, to characterise the bending energy of a surface with zero charge density, and (2) the energy of lattice points moving on that surface. Martensitic phase transformations necessitate the vanishing of the K-related part in the Helfrich term due to the continuity requirement. Through the analysis of NiTi phase transformations, this work shows that shape memory alloys undergo a phase transformation that lives within a space of TPMS. Thus, the shape memory effect requires the geometric entropy to remain invariant. To aid comprehension, the Helfrich Hamiltonian and the energy of points moving on a manifold are thoroughly explained.