Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
Overview
In Chapter 2, further numerical studies on the quasiperiodic surface are carried out after the necessary formalism for quasiperiodic functions, minimal surfaces, and tilings has been developed. The numerical characterization of the surface supports many of Sheng's original conclusions and more closely approaches the desired limit where a true minimal surface exists in physical space. Beyond the numerics of modelling the minimal surface, the critical surface's evolution as the limit is approached is examined, and its appearance is compared with the previous studies. Finally, the construction of a cell complex with the same topology as the minimal surface in the conjectured limit is studied in order to characterize the topology of the hypersurface and illuminate the physical space structure of the surface and its correspondence to a tiling.
Chapter 3 describes a novel family of quasiperiodic tilings that arose from studying the connection between topological jumps of the surface and phason jumps in tilings. This ensemble of tilings contains a great deal of freedom in its construction analagous to the freedom inherent in the construction of quasiperiodic, 12-fold symmetric square-triangle tilings. The construction of these 3D tilings is examined in detail from the viewpoint of tile and vertex-based substitution rules, and the topology of the tilings' acceptance domains is examined.
Chapter 4 treats triangulated surfaces as approximations to minimal surfaces and gives numerical characterizations of their properties for a variety of periodic and quasiperiodic minimal surfaces. The evolution of a triangulated surface patch to a minimal surface may be accomplished by minimizing the integral of the squared mean curvature over the surface. The validity and numerical stability of such an evolution as it applies to surface patches forming quasiperiodic minimal surfaces is studied in detail using surface patches associated with the tiling in Chapter 3. These surface patches may be extended face-to-face across tiles to construct larger surface patches reproducing the observed topology of the conjectured limiting surface studied in Chapter 2.
Chris Kimmer 2011-06-01