Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
Conclusions
The triangulated surfaces studied here are able to offer reasonable approximations to the quasiperiodic minimal surface since they can reproduce the correct topology in rather large regions of physical space. They offer an effective tool for visualization of the surface and its labyrinths, and the mean curvature is seen to be at least two orders of magnitude smaller than the scale suggested by the Gaussian curvatures. However, the approximations used to compute mean curvatures for a polyhedral surface using the Surface Evolver are unstable and do not allow more general conclusions to be drawn about the existence of the quasiperiodic minimal surface of Chapter 2 or even about the ability of the surfaces patches associated with the tiles of Chapter 3 to support their own minimal surfaces. The difficulties presented by the Evolver are inherent in the algorithms used to study minimal surfaces from triangulated meshes, and these difficulties suggest that further numerical study on approximants to the quasiperiodic minimal surface is best carried out using different algorithms. Nonetheless, the triangulated surfaces used here show good convergence for the area and genus per unit volume of the quasiperiodic surface in successively larger volumes.
Chris Kimmer 2011-06-01