The world of materials science was shaken to the core in the mid-eighties by the publication of diffraction data implying the existence of so-called forbidden symmetries in an alloy. Since then, scientists everywhere have been investigating the emergence of this exotic form of matter, now known as quasicrystals, in an effort to understand its chemical, physical, and mathematical properties. In this project, I aim at constructing mathematical models of quasicrystals using a geometric method, which involves covering surfaces in 3 euclidean dimensions with patterns from 2-dimensional curved space. This approach has proven useful in the context of understanding emerging structures in soft matter materials science; and I hope to address several questions regarding the formation, stability, and classification of quasicrystals. Since the discovery of quasicrystals, there has been considerable interest in and demand for further studies from the industry, as the discovery opened up an entirely unexplored branch of materials scientific applications. Currently, metallic quasicrystals are being used in and explored for a wide array of applications: from reinforcing steel and developing corrosion resistant coatings to novel nanomaterials. With projects such as this, I hope to induce similar potential applications for quasicrystals made from e.g. polymeric compounds. Quasicrystals In order to fully appreciate the scope of this project, one must first understand the concept of a quasicrystal. In order to do that, it is sensible to start with the textbook definition of a periodic crystal. In this article, by a periodic crystal I mean a pattern that has so-called translational symmetry; a construction, one usually calls a lattice. In other words, a periodic crystal can be described by a unit cell and infinitely many repetitions of this. One of the important consequences of this definition is that periodic crystals exhibit long-range orientational order, which in turn means that they diffract radiation. Figure 1: One of the electron diffraction images published by Shechtman et al. in 1984 A quasicrystal, which is short for quasiperiodic crystal, is a pattern that exhibits similar long-range order yet lacks the repetitive nature of periodic crystals. Because of this lack of crystalline repetitiveness, quasicrystals can produce diffraction patterns that violate the basic theorems of crystallography. For a long time, these patterns were thought to be mathematical artefacts with no real physical significance. However, as mentioned in the abstract, all of this changed in 1984 when Dan Shechtman and his colleagues published the first ever report of experimental evidence suggesting the emergence of a quasicrystalline state in an alloy of aluminium and manganese . The published diffraction data, an example of which is shown in Figure 1, suggested that the sample had a fivefold rotational symmetry, which is forbidden by the classical mathematical rules of crystallography. Unfazed by this, Shechtman and his colleagues analysed and published the data and summarised their findings in the following quote: “We have observed a metallic solid with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations” (Shechtman et al., 1984). Quasicrystals in Nature In 2011, the first report of a quasicrystal appearing in nature was published by Bindi et al. . The sample, now called icosahedrite, was found by the Khatyrka River, Kamchatka, Russia. Tests have shown it to be a fragment of a meteorite. The article and its contents were highly controversial and divided the materials science community for years after publication. However, the evidence and ideas stood the tests of time and scrutiny, and professor Shechtman was awarded the Nobel Prize in Chemistry in 2011 for the discovery of quasicrystals. While quasicrystals are now considered an established concept in science, they are still regarded as somewhat enigmatic, and much still remains to be understood about their emergence, structure, and physical and mathematical properties. This project aims at developing novel ways of describing quasicrystals mathematically using modern geometric tools, which we shall cover in the following sections. Curvature The differences between flat and curved geometry can be exemplified by considering simple triangles. On a flat surface, the angles of a triangle must sum up to exactly π, whereas the angles of a triangle drawn on the surface of a sphere sum up to more than π. Similarly, in hyperbolic geometry, the angles sum up to less than π. As an example, the triangles in Figure 2 have angles of π/2, π/4, and π/6 Non-Euclidean Geometries By non-euclidean geometries, I mean the scenarios in which Euclid's famous axioms are relaxed. The textbook example of such a geometry is the geometry that takes place on the surface of a sphere. Usually, one refers to spherical geometry as a geometry with positive curvature. In the context of this project, however, our interest belongs to geometries of negative curvature - i.e. so-called hyperbolic geometry. On a sphere, at any given point, the surface curves in the same manner in all directions. Conversely, in hyperbolic geometry, any given point is a saddle-point. Our interest in hyperbolic geometry stems from the fact that the array of symmetries and tiling patterns, one can realise in hyperbolic space, is vastly different than for flat (or spherical) geometries. Similarly, our interest in these hyperbolic tiling patterns stems from their relation to so-called triply-periodic minimal surfaces (TMPS) [3, 4]. Shown in Figure 2 is a textbook example of a TMPS: Schwarz's so-called D-surface . The surface itself can be understood as a thin film separating two compartments of equal volume. This class of surfaces are referred to as minimal, as they minimise the area for a fixed boundary; we are particularly interested in the structure of the smoothest ones of these. The fundamental message to take away from this section is that one can describe structures in 3-dimensional euclidean space as structures in 2-dimensional hyperbolic space covering the aforementioned surfaces . In the case shown in Figure 2, the structure possesses the symmetry of the crystallographic space group Pn3 ̅m. This is the defining idea behind this project: to develop quasicrystal-like structures on surfaces such as this and investigate their properties. Novel Structures It is important to stress that in materials science, the function and properties of a material is intricately connected to the structure of the material in question on micro- or nanoscale. As an example, properties such as thermal and electric conductivity, structural stability, and optical attributes are all related to the geometry and symmetries of the structure. Due to an extensive list of remarkable developments in synthesis methods and understanding of novel materials during the last several decades, the number of parameters experimentalists are able to manipulate in a given compound, has increased substantially. In a sense, this is our motivation for investigating: to guide synthesis and experiments in interesting and promising directions. This is a common approach in theoretical materials science . In particular, I hope to establish a better understanding of - or perhaps rather a different view on - how a structure might undergo a transition into a quasicrystalline state; specifically by understanding the relationship between the symmetries in the structures. Figure 2: Schwarz's D-surface embedded in three-dimensional euclidean space. The surface has been divided into so-called asymmetric patches. Each pair of these patches are related by a symmetry operation. The structure was given its name, as it can be represented as two intertwined labyrinths, each one of which having the shape of the chemical bond-structure in diamond. Figure by Martin Cramer Pedersen Cross-Disciplinary Gains While this project aims at producing results for the materials science community, it is important to stress the inherent cross-disciplinarity of the project. Aside from these goals, the project will produce algorithms for describing the geometry of and navigating the complicated structures we have been discussing. The algorithms for visualisation and mathematical manipulation of the structures relevant to this project can be and usually are fairly complicated. In addition, developments in these directions are naturally tied to computer science. Fortunately, the host venue of the project, the Australian National University (ANU) in Canberra, hosts the National Supercomputing Facility and has a research group dedicated entirely to visualisation methods. Similarly, mathematical descriptions of structure in the aforementioned curved geometries are highly relevant for polymer science, molecular biophysics, and scattering physics. Impact Theoretical models are a necessity when developing and investigating novel materials on a micro- and nanoscopic scale. And in turn, innovating and evolving the materials we use are issues of vital importance in the challenges that our society is facing - challenges such as clean energy, industrial development, and climate-friendly production. In 2001, the American Department of Energy enumerated five "Grand Challenges" in materials science. Number 3 reads as follows: “How do remarkable properties of matter emerge from complex correlations of the atomic or electronic constituents and how can we control these properties?” (US Department of Energy, 2001). At this point, I hope to have convinced you that quasicrystals are indeed remarkable structures. The significance of the wide array of possible applications of quasicrystalline materials was recognised by the Nobel committee, when professor Shechtman was awarded the prize: “Quasicrystals, while brittle, could reinforce steel like armour.” (Nobel citation, 2011). On a related note, I would like to stress the educational value of the mathematical examples produced by this project and similar ones. Visualisation of complicated aspects from differential geometry is a tremendous help in understanding these abstract concepts. The Support from the Carlsberg Foundation This project is supported by the Carlsberg Foundation with a Carlsberg Foundation’s Internationalisation Fellowship. The host institution, ANU in Canberra, Australia, has contributed to the running costs of this project as well. The project was initiated in April 2015. Denmark hosts a strong research environment in the field of biophysics, soft matter materials science, scattering physics, and the related mathematical and computer scientific disciplines, in order for projects such as this one to be in line with the national trends and strategies. On a personal level, I am very grateful for the support this project has received. My stay at the Department of Applied Mathematics at ANU has been very interesting, very challenging, and certainly very rewarding and it has opened up numerous collaborations with international contacts that I would not have had otherwise. References  Shechtman, Blech, Gratias and Cahn, Phys. Rev. Lett. 53(20), pp. 1951-1954, 1984.  Bindi, Steinhardt, Yao and Lu, Am. Mineral. 96, pp. 928-931, 2011.  Ramsden, Robins and Hyde, Acta Crystallogr. Sect. A 65(2), pp. 81-108, 2009.  Robins, Ramsden and Hyde, Eur. Phys. J. B 39, pp. 365-375, 2004.  Schwarz, Gesammelte mathematische Abhandlungen, AMS Chelsea Publishing, 1890.  Sadoc and Charvolin, Acta Crystallogr. Sect. A 45(1), pp. 10-20, 1989.  Kirkensgaard, Pedersen and Hyde, Soft Matter 10, pp. 7182-7194, 2014.