  Martensitic transformations, viewed as continuous transformations between
triply periodic minimal surfaces (TPMS), as originally proposed by Hyde and
Andersson [Z. Kristallogr. 174, 225 (1986)], is extended to include paths
between the initial and final phases. Bravais lattices correspond to particular
TPMS whose lattice points are flat points, where the Gaussian curvature
vanishes. Reversible transformations, which correspond to shape memory
materials, occur only if lattice points remain at flat points on a TPMS
throughout a continuous deformation. For the shape memory material NiTi,
density-functional theory (DFT) yields irreversible and reversible paths with
and without energy barriers, respectively. Although there are TPMS for
face-centered gamma-Fe) and body-centered (alpha-Fe) cubic lattices, gamma to
alpha deformation paths are not reversible, in agreement with non-vanishing
energy barriers obtained from DFT.
