Christopher J. Kimmer, Ph.D.

IU Southeast Informatics

Next: The Mathematics of Quasiperiodicity Up: QUASIPERIODIC MINIMAL SURFACES AND Previous: Overview

Quasiperiodic Minimal Surfaces

A Landau-Ginzburg free-energy functional in six dimensions can be used to model a quasiperiodic minimal surface as the interface of a binary phase separation. The minimal surface is formed by cutting the 5D zero level set of the order parameter with a 3-plane corresponding to physical space. In order to have an order parameter that yields a minimal surface when cut by physical space, the 5D surface is defined as the limit of an evolution designed to create a minimal surface in the physical space dimenions and a locally planar surface in the orthogonal three dimensions. A geometric interpretation of this limiting procedure is suggested, and an easily-visualizable example is given. The minimal surface in physical space undergoes topology jumps as the cut plane is translated in the orthogonal directions, and the critical surface defining the points where these topology jumps occur is examined. Finally, a discrete structure that reproduces the topology of the order parameter field is described.

• The Mathematics of Quasiperiodicity
• Constructing Quasiperiodic Minimal Surfaces
  • Symmetry and BPMSs
  • Modelling a BPMS
  • Sheng's Numerical Results
  • Sheng's Critical Surface
  
  
• Methodology
• Numerical Results
• The Critical Surface
• Discrete Structure of the Surface
  • The Voronoi Construction
  • The Voronoi Construction at t = 1
  • The t → 0 Voronoi Construction
  • The Critical Surface Obtained from the t → 0 Voronoi Construction
  
  
• The Physical Space Structure of the Surface
• Conclusions

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