Exploring hyperbolic order in curved materials

Nature employs a range of topologically complex ordered nanostructures that occur in both natural and synthetic materials. A class of these exhibits negative curvature and form periodic saddle-shaped surfaces in three dimensions. Unlike pattern formation on flat or positively curved surfaces, the understanding of these patterns is highly complicated due to the structures being intrinsically intertwined in three dimensions. We have developed a new method for analysis of patterns on such surfaces by mapping to two-dimensional hyperbolic space analogous to spherical projections in cartography thus effectively creating a more accessible 'hyperbolic map' of the pattern. The aim of the stay will be to dissect a large amount of results from this method to understand these utterly novel patterns.