Self-Similarity, Tiling, and Wavelets

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

ABSTRACT This proposal concerns several interrelated areas of mathematics: self-affine tiles and reptiles, refinable functions and wavelets, spectral sets, and mathematical quasicrystals. Although the subjects may appear to be rather distant from each other, the proposal actually has a very coherent central theme, which is self-similarity and tiling, and using analytical techniques to study them. The theme occurs naturally in self-affine tiles and reptiles, and in the branch of quasicrystal questions the PI proposes here. It occurs, although less obviously, in refinable functions and in spectral sets. The refinement equation satisfied by a refinable function is a generalized self-similarity, and it is closely related to self-affine tiles and self-similar measures. Spectral sets, on the other hand, are closely tied with tiling, as it is conjectured that a set is a spectral set if and only if it tiles by translation. A self-similar tile is one that can be disected into several pieces such that all pieces are identical in shape and are similar to the original tile. The simplest example of such a tile is the square, which can be disected into 4 identical squares half in dimension of the original square. By repeated inflation and disecting of a self-similar tile we can cover a larger and larger area, and this results a self-similar tiling. Self-similar tilings are fascinating, and many of the aperiodic tiles discovered in recent years, such as the Penrose aperiodic tiles, can be viewed as generalized self-similar tiles. Furthermore, self-similar tilings have been linked to quasicrystals, materials whose atomic structures are aperiodic rather than the usual periodic structures. It is not clear how these types of atomic structure are formed, but models have been conceived by physicists using aperiodic tiles.