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Porosity and topological properties of triply periodic minimal surfaces

Sergei Ermolenko, Pavel Snopov

Abstract

Triple periodic minimal surfaces (TPMS) have garnered significant interest due to their structural efficiency and controllable geometry, making them suitable for a wide range of applications. This paper investigates the relationships between porosity and persistence entropy with the shape factor of TPMS. We propose conjectures suggesting that these relationships are polynomial in nature, derived through the application of machine learning techniques. This study exemplifies the integration of machine learning methodologies in pure mathematical research. Besides the conjectures, we provide the mathematical models that might have the potential implications for the design and modeling of TPMS structures in various practical applications.

Porosity and topological properties of triply periodic minimal surfaces

Abstract

Triple periodic minimal surfaces (TPMS) have garnered significant interest due to their structural efficiency and controllable geometry, making them suitable for a wide range of applications. This paper investigates the relationships between porosity and persistence entropy with the shape factor of TPMS. We propose conjectures suggesting that these relationships are polynomial in nature, derived through the application of machine learning techniques. This study exemplifies the integration of machine learning methodologies in pure mathematical research. Besides the conjectures, we provide the mathematical models that might have the potential implications for the design and modeling of TPMS structures in various practical applications.

Paper Structure

This paper contains 14 sections, 19 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical model of minimal surfaces
  4. Persistence
  5. Filtration Methods
  6. Persistence Entropy
  7. Porosity calculations of TPMS models with different shape factor
  8. Topology estimation of TPMS
  9. Schwarz
  10. Gyroid
  11. Exploring Relationships in TPMS
  12. Shape factor predicts persistence
  13. Shape factor predicts porosity
  14. Conclusion

Figures (8)

  • Figure 1: The examples of the persistent homology visualizations gudhi
  • Figure 2: Dependence of TPMS porosity on shape factor
  • Figure 3: TPMS surfaces: the number at the bottom-left corner corresponds to the number in the Table \ref{['tab:porosity']}.
  • Figure 4: Persistence diagrams for Schwarz surfaces
  • Figure 5: Persistence diagrams for Gyroid surfaces
  • ...and 3 more figures

Theorems & Definitions (1)

  • Conjecture