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August2001

QUASIPERIODIC MINIMAL SURFACES AND TILINGS

Christopher James Kimmer

Abstract:

This work is concerned with the study of quasiperiodic minimal surfaces and their relationship with quasiperiodic tilings. 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. A limiting procedure is used to produce a 5D surface that is locally the product of the 2D minimal surface of interest in 3D physical space and a 3-plane in the orthogonal 3D subspace. This limiting procedure may be interpreted as the stretching of the three unphysical dimensions relative to the physical ones, and a lower dimensional example of this technique is illustrated.

The 2D minimal surface of interest is formed by cutting the 5D surface with a 3-plane representing physical space; moving the cut plane in the orthogonal dimensions leads to topology jumps of the surface in physical space. These jumps are a result of the cut-plane passing through the two-dimensional critical submanifold of the 5D surface, defined by the surface normal being orthogonal to physical space. The form of the critical submanifold is severely constrained by the minimality of the surface in physical space. The submanifold's structure is examined using data obtained by computing the free-energy functional on a grid and by a discrete construction that reproduces the topology of the order parameter field using the Voronoi cells of the order parameter's extrema.

The critical submanifold is to the minimal surface as the boundary of an acceptance domain is to a quasiperiodic tiling, and an ensemble of tilings that are possibly related to the minimal surface is described. These tilings are composed of four prototiles and can be constructed using tile- and vertex-based substitution rules. The form of these substitution rules is studied in detail.

Triangulated surface patches can be associated with each of the four prototiles, and the tiles can be assembled face-to-face to produce arbitrarily large triangulated surfaces. Surface patches assembled in this manner reproduce the topology of the minimal surface and can be relaxed to a near-minimal surface to quantify the curvature properties of the interface obtained from the free-energy minimization.




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Next: Introduction
Chris Kimmer 2011-06-01

cjkimmer -at- ius.edu