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

TL;DR

The paper develops the Extended Edge Bending (EEB) framework to classify soft tilings that share vertices and combinatorics with a given polyhedral tiling, focusing on second-order geometry defined by half-tangent edges. By applying EEB to the Dirichlet-Voronoi $(e2)$ cell of the $bcc$ lattice under octahedral and tetrahedral symmetries, it proves that exactly two second-order soft tilings exist in the first-order class and exactly four in the tetrahedral case, with Schwarz P and D unit cells corresponding to soft tilings $(g2)$ and $(i2)$. It further shows that one-parameter families connect Schwarz P/D tilings to Kelvin foams via planar-face constraints, revealing deep links between minimal surfaces, tilings, and natural foams. The work thereby provides a geometric framework connecting TPMS, their Voronoi partitions, and complex soft-cell tilings, with potential implications for modeling natural structures and materials science.

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.

Figures (7)

• Figure 1: The DV-$bcc$ cell (called the (e2) cell in softcells1) and its four soft versions
• Figure 2: The DV-$bcc$ cell
• Figure 3: All solutions of the softening equations for the polyhedral (e2) cell shown in Figure \ref{['fig:2']}, plotted on the plane $(\phi, \theta)$ of Euler angles ($\theta=0, \phi=0$ coinciding respectively with the $x$ and $z$ axes of the Cartesian system shown on Figure \ref{['fig:2']}). Red lines: great circles solving complete sets of softening equations. Blue lines: great circles solving constraint equations which keep faces planar. Green lines:great circles connection non-standard soft cells. Blue dots with red fill: standard soft cells identified by the intersection of two red lines and one blue line. Red dots with yellow fill: non-standard soft cells defined by the intersection of one blue and one red line.Yellow dots with white fill: non-soft cells. For detailed data on cells (e2),(f2),(g2),(h2),(i2) see Table \ref{['tab:allsolutions']}. For data on the Kelvin cell and the $PD$ cell see Table \ref{['tab:KelvinPD']}. For better visibility, nodal set $\mathbf{a},\mathbf{b}, \mathbf{c}, \mathbf{d}$ is shown on vertex 21 instead on vertex 1.
• Figure 4: Discrete structures on triply periodic minimal surfaces. Upper row, (a1)-(b1): Unit cells and skeletal graphs. Bottom row, (a2)-(b2): Unit cells carrying the vertices of the skew polyhedron $\{6|4,4\}$. Left column, (a1)-(a2): Schwarz P surface. Right column, (b1)-(b2): Schwarz D surface.
• Figure 5: The Kelvin Cell

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Definition 3
• Remark 1
• proof
• Proposition 1
• proof