![]() Tiling that does not have repeating patterns is known as aperiodic tiling and is generally achieved by using multiple tile shapes. Under their scenario, the researchers noted that tiling refers to fitting shapes together such that there are no overlaps or gaps. In this new effort, the research team has discovered a single geometric shape that if used for tiling, will not produce repeating patterns. Sometimes though, people want patterns that do not repeat but that represents a challenge if the same types of shape are used. In other applications, some researchers are even considering the reflective finishes that quasicrystals might create if added to household paint.When people tile their floors, they tend to use simple geometric shapes that lend themselves to repeating patterns, such as squares or triangles. Having defects in the crystal pattern or alternatively using a never-repeating quasicrystal pattern at the output end of a laser, makes it possible to create an efficient laser beam with high peak output power. This is because, when periodic crystal patterns are used in a laser, a low-power laser beam is created by the symmetry of the repeating pattern. However, this research isn’t just a conceptual mathematical idea (although the mathematics behind it is addictive) – it has great promise for many practical applications, including making very efficient quasicrystal lasers. That is why we at the University of Leeds, along with colleagues at other institutions, are fascinated with research into such questions. The mathematics behind how such never-repeating patterns are created is very useful in understanding how they are formed and even in designing them with specific properties. Such a model makes it possible to explore the competition between all these different patterns and to identify the conditions under which quasicrystals will be formed in nature. In addition to the never-repeating quasicrystal patterns, this model can also form other observed regular crystal structures such as hexagons, body-centered cubes and so on. And second that these can influence each other strongly. The first is that patterns at two different sizes (length-scale) which are at an appropriate irrational ratio (like phi) both occur in the system. In our recent publication, we identified the two traits that a system must have in order to form a 3D quasicrystal. In the model, it helps to think of the white balls to be the locations where we would find the particles/atoms of the crystal structure and the red and yellow rods to indicate bonds between particles, that represent the shapes and symmetries of the structure. We see this five-fold symmetry both in the image of the quasicrystal as the ten radial lines around the central red dot (above), and also in the scale model of the central part of the quasicrystal made with Zometool (below). Therefore this graphic is an example of a pattern that has rotational symmetry but no translational symmetry. But in each case these stars are surrounded by different shapes, which implies that the whole pattern never repeats in any direction. ![]() For example in the graphic below, the five-pointed orange star is repeated over and over again. We see that many small patches of patterns are repeated many times in this pattern. This looks the same when rotated through 72-degree angles, meaning that if you turn it 360 degrees full circle, it looks the same from five different angles. In the 1970s, physicist Roger Penrose discovered that it was possible to make a pattern from two different shapes with the angles and sides of a pentagon. Now my colleagues and I have made a model that can help understand how this is expressed. In fact we’ve only known that non-periodic tiling, which creates never-repeating patterns, can exist in crystals for a couple of decades. Among all possible arrangements, these regular arrangements are preferred in nature because they are associated with the least amount of energy required to assemble them. Patterns (made up of tiles) and crystals (made up of atoms or molecules) are typically periodic like a sheet of graph paper and have related symmetries. But imagine tiling a bathroom with pentagons instead of squares – it’s impossible, because the pentagons won’t fit together without leaving gaps or overlapping one another. That’s because in this case, the whole tiling has the same symmetry as a single tile. If we moved the whole pattern by the length of a tile (translated it) or rotated it by 90 degrees, we will get the same pattern. ![]() ![]() Remember the graph paper you used at school, the kind that’s covered with tiny squares? It’s the perfect illustration of what mathematicians call a “periodic tiling of space”, with shapes covering an entire area with no overlap or gap. ![]()
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