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

2024-11-23

arXiv.org

AI Summary

This work addresses how triply periodic minimal surfaces (TPMS)—specifically the Schwarz P, D, and Gyroid—naturally generate non-standard soft tilings. Method: We propose an extended edge-bending algorithm that continuously parametrizes the Schwarz P/D and Gyroid surfaces into families of deformable soft-cell tilings. A unified geometric framework is established, linking TPMS-based soft tilings, Kelvin foams, and Dirichlet–Voronoi tilings on the body-centered cubic (bcc) lattice. Contributions/Results: We rigorously define and prove the first- and second-order equivalence classes of soft tilings, revealing exactly two fully symmetric second-order classes. We construct a single-parameter soft tiling family P↔D that embeds the Kelvin foam as an intermediate state in the continuous transition between standard and non-standard tilings. An explicit construction of Gyroid-induced soft tilings is provided. These results deepen the fundamental geometric connection between soft-matter architectures and naturally occurring morphologies.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Introduces soft cells for gap-free space filling with minimal sharp corners
• Extends edge-bending algorithm to transform Schwarz P and D surfaces into soft tilings
• Classifies soft tilings into equivalence classes based on geometric and symmetry properties

Innovation

• Edge-bending algorithm deforms polyhedral tilings
• Soft tilings derived from Schwarz surfaces
• One-parameter family bridges standard tilings