Soft cells, Kelvin's foam and the minimal surfaces of Schwarz

Abstract

Recently, we introduced a new class of shapes, called soft cells which fill space as soft tilings without gaps and overlaps while minimizing the number of sharp corners. We introduced the edge bending algorithm that deforms a polyhedral tiling into a soft tiling and we proved that an infinite class of polyhedral tilings can be smoothly deformed into standard soft tilings. Here, we demonstrate that certain triply periodic minimal surfaces naturally give rise to non-standard soft tilings. By extending the edge-bending algorithm, we further establish that the soft tilings derived from the Schwarz P and Schwarz D surfaces can be continuously transformed into one another through a one-parameter family of intermediate non-standard soft tilings. Notably, by carrying its combinatorial structure, both resulting tilings belong to the first order equivalence class of the Dirichlet-Voronoi tiling on the body-centered cubic bcc lattice, highlighting a deep geometric connection underlying these minimal surface configurations. By requiring identical end-tangents for edges in a first order class, we also define second order equivalence classes among tilings and prove that there exist exactly two such classes among soft tilings which share the full symmetry group of the DV-bcc tiling. Additionally, we construct a one-parameter family of tilings bridging standard and non-standard soft tilings, explicitly including the classic Kelvin foam structure as an intermediate configuration. This construction highlights that both the soft cells themselves and the geometric methods employed in their generation provide valuable insights into the structural principles underlying natural forms. We also present the soft tiling induced by the gyroid structure.