I have seen in some engineering departments that they manufacture models of periodic minimal forms (characterised by equal and opposite curvature at every points on the surface). In pure mathematics, they are known as triply periodic minimal surfaces.
If I understand rightly, these have been observed experimentally in crystallography and polymer chemistry but I assume they must have been studied in differential geometry as well. The Wikipedia page mentions the classification of these surfaces as an open problem: has there been any recent progress on this? The physics literature also mentions the possibility of constructing minimal surfaces with the properties of a quasicrystal (ie. minimal surfaces with a quasicrystalline order). Again, has there been any further geometric work on this construction?