Combinatorics of Sequences and Tilings and its Applications

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One of the intriguing aspects of Nature is the symmetry and order apparent in the world around us, for instance in the shape of crystals, or, at a microscopic level, in the regular arrangement of atoms making up the crystal. We have a surprisingly limited understanding of the origin of order and symmetry; and, maybe even more surprisingly, no clear mathematical definition of the concept of order exists. In the example of the crystal, the underlying order is apparent in the periodic arrangement of its constituents. It is particular interesting to investigate ordered structures that lack periodicity, and disordered systems that still show an apparent degree of order. For instance, a Penrose tiling of the plane consist of two basic shapes, which, when arranged properly, allow for an arbitrary large tiling, but never one that exactly repeats itself. Such structures are not only fascinating from a mathematical point of view, but are physically realised in quasicrystals. These are crystals occurring in particular metal alloys which possess an intricate non-periodic order of atoms. Due to the lack of periodicity in the structure, each atom has its own individual environment, and if one looks far enough around, no two atoms will ever have exactly the same surroundings. Therefore, it is interesting to look at properties of such structures, such as the mean number of neighbours of an atom, or mean numbers of atoms at certain distances. Such quantities are related to the diffraction patters of these materials, which provide the experimental proof of the order in the atomic positions. In a more abstract setting, thinking of a structure represented by a tiling, the corresponding question is that of the mean number of vertices in the tiling that are at a certain distance from a given vertex, averaged over all possible vertices as the centres. These numbers are called the averaged shelling numbers, and they are an example of the type of properties that are investigated in this project. The calculation of these numbers turns out to be related to interesting properties of certain types of numbers, such as factorisation of numbers into prime factors, which is a topic of interest in number theory. Moreover, these numbers and similar combinatorial properties are closely related to models of interest in physics and other sciences. This makes this project interesting from a number of different perspectives, ranging from pure mathematics to applications in physics, crystallography and materials science.