Imagine shining a torch onto a cube: the cube is 3D, and its shadow is 2D. Depending on the angle and position of the torch, the shape of the shadow will change. We can call the shadow of the cube its 'projection', and define specific projections depending on the direction we point the torch, or the position of the cube.
Projections of 3D cubes are simple, but what happens when we add an extra dimension? The human brain can't imagine what a fourth dimension looks like, but we can project a 4D structure onto dimensional spaces we can understand, for example, 2D. If we carry on this way, we can also visualize 5D cubes by projecting onto a 2D and 3D space, or 6D cubes by projecting onto two 3D spaces.
The extra dimensions can be called higher-dimensional spaces, or hyperspaces. The dimensions we can visualize (2D and 3D) are called real-space. Changing the dimension of the cube or the angle of projection in hyperspace changes the real-space structure we obtain. By projecting hyperspace structures, we can produce simple or complex repeating patterns, or create examples which have no repeating units, but still have long-range order. These special arrangements are called aperiodic, or quasicrystalline structures.
Therefore, unlike a 3D cube projection, projected hyperspace structures can produce a huge variety of structures in real-space. Importantly, we have not even begun to explore all the possible structures and uses hidden in projections of hyperspace. Applied Hyperspace Structures (HYPER) aims to exploit and apply new hyperspace structures in novel fashions, demonstrating their power and utilization in three new ways.
The first will be to help understand the stability of physical quasicrystals. Quasicrystals are typically found as intermetallic alloys, and again, they have no repeating unit (unlike crystals) but retain perfect order. Their structure is very complex but can be understood using hyperspace projections. However, we still do not know exactly why such materials are stable - a fundamental question which is important to understand their existence in nature. Current theories argue between the relative ratio of energetic and entropic contributions of phasons, which are atomic movements in the quasicrystal structure. HYPER will take the novel approach of applying hyperspace projections to analyse phasonic motion at the surfaces of physical quasicrystals, measuring at the atomic level using Scanning Tunnelling Microscopy. In doing so, we can understand their precise energetic or entropic contribution to the surface structure, and therefore find the key to understanding quasicrystal stability.
The second focus of HYPER will be to investigate theoretical hybrid materials which have unique structures designed by hyperspace projections. These hybrids will consist of periodic and aperiodic sections which share rotational symmetry - an avenue of aperiodic research which has so far been relatively overlooked. Such hybrid materials have therefore never been imagined before, let alone explored. We will investigate how waves propagate through such structures and study their effective properties, which, hopefully, can be tuned in future manufactured materials for a range of applications.
Finally, HYPER will produce a piece of easy-to-use software which will allow anyone to generate their own hyperspace structures - even if they're not experts on the subject. This software package will allow users to pick their hyperspace dimension, choose between aperiodic and periodic structures, and visualize their resultant projections.
The proposed research is therefore both fundamental and applied in its nature, with the potential to influence and introduce an interdisciplinary audience to the possibilities of exploring hyperstructures.
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