| This thesis describes two theoretical studies which may shed light on the problem of structural determination in quasicrystals. In order to reconstruct the atomic structure of a quasicrystal using the tiling description, one must determine the symmetry, the atomic decoration of the unit cells, and the unit-cell packing. There is an uncountably infinite number of distinguishable, perfect quasicrystal packings for any given symmetry. A key difficulty is that no method is known for separately determining the correct packing and decoration of the unit cells from diffraction data.; It is proposed that any physically realizable unit-cell packing should satisfy three simple criteria: it should be uniquely described by "matching rules" which fix the allowed unit-cell clusters smaller than some finite bound; there should exist local growth rules which permit the aggregation of unit cells to form a perfect packing; and the growth should be sufficiently fast that macroscopic quasicrystals can be obtained on laboratory time scales. These criteria are applied to a continuum of 2D pentagonal quasicrystal tilings, only one of which--the Penrose tiling--was previously known to meet the criteria. A countably infinite subset of these tilings is shown to satisfy all three criteria. The subset can be ordered according to the range of the matching rules and growth rules. This range is physically plausible in only a handful of cases. If this result were to extend to 3D, it would greatly simplify the determination of atomic structure.; The second study concerns the external morphology of quasicrystals grown under conditions close to thermodynamic equilibrium. On slow cooling from the melt, stable icosahedral quasicrystals have been observed to form large, faceted grains. The allowed zero-temperature equilibrium shapes for icosahedrally symmetric materials are catalogued, and specific examples are calculated for a simple model of quasicrystal faceting. It is proved that one of the experimentally observed shapes, the pentagonal dodecahedron, may be an equilibrium shape only if significant repulsive interactions are present within the material. This result may yield clues about the interactions within real quasicrystals, and thereby help to distinguish among atomic models of the icosahedral phase. |
