Daniel Bragg

Quasicrystals and Geometry


 Throughout the years a clear link has been established between the structure of crystals and the mathematical concepts of tiling spaces. Until recently crystals were thought to be periodic and ordered structures which can be related to a set of points in the n-dimensional Euclidean space En known as a lattice, where these points act as vertices to the building blocks of crystals. It was known that crystals could only be periodic and ordered if their corresponding lattice had either one, two, three, four or six fold rotational symmetry. However in the mid-20th century quasicrystals were discovered which were ordered and aperiodic but possessed rotational symmetries not listed above. This sparked into life the new area of study known as quasi-crystallography.

This report delves into the mathematics behind these interesting structures. First considered are the lattice structures related to the structure of a crystal. I consider the different lattice types in various dimensions and then look at the properties such as the Crystallographic Restriction Theorem described above. I then introduce the idea of a dual lattice, Voronoi cell construction and the canonical projection method - all of which are important tools to be used later. Tiling spaces are then considered. In particular I will be exploring the properties of the rhomb Penrose tiling which are derived from the famous kite and dart Penrose tiling. These tilings are aperiodic so therefore provide a good model for quasicrystal structures. With all of this knowledge in place, I will then explain the projection method for constructing tilings of the plane from five dimensional space. I then digress somewhat and explore the mathematics behind the diffraction patterns created by crystals and quasicrystals. This is so that the diffraction condition can be introduced, which defines the diffraction patterns that represent a crystalline structure. A new method of constructing these diffrraction patterns known as topological Bragg peaks is also discussed.

Finally I discuss the possibility of having one dimensional crystals. In particular we look at the Fibonacci sequence and show how they produce crystalline structures.

Daniel Bragg's Thesis

Daniel Bragg's Presentation



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