You can, for instance, use a couple of vertical dominoes while otherwise filling the plane with horizontal dominoes. It’s easy to make tilings that aren’t periodic from tiles that also form periodic tilings. Perhaps hobbyists, unlike mathematicians, are “not burdened with knowing how hard this is,” Senechal said. And then there was Joan Taylor’s discovery of the Socolar-Taylor tile. Marjorie Rice, a California housewife, found a new family of pentagonal tilings in 1975. Robert Ammann, who worked as a mail sorter, discovered one set of Penrose’s tiles independently in the 1970s. “I’m not used to this kind of thing.”īut this is far from the first time a hobbyist has made a serious breakthrough in tiling geometry. The excitement the tiles have generated has felt “a bit surreal,” said Smith, who lives in the coastal town of Bridlington in northern England. In the days since the announcement, mathematicians and tiling hobbyists have rushed to get their hands on the new tiles, making paper cutouts, 3D-printing them, and making hat quilts and cookies. The hat tile, Senechal said, shows that periodic and aperiodic tiles are more closely linked than mathematicians had realized. With just a little work, anyone with a magic marker and a hexagonally tiled bathroom floor can trace out a hat tiling. This divides every hexagon into six “kites.” Each hat is made of eight adjacent kites, combined from neighboring hexagons. To get a hat tiling from a hexagonal tiling, first connect the midpoints of the opposite sides of the hexagons. What its tilings do have is a deep relationship with a particular periodic tiling: the honeycomb lattice of hexagons. The hat, by contrast, has no symmetry and is “almost mundane in its simplicity,” the authors wrote. What’s more, they realized, the hat is one of infinitely many different tiles of this type. The hat tile embodies “enough complexity to forcibly disrupt periodic order at all scales,” the researchers wrote in their paper. Mathematicians call such a tile, or set of tiles, “aperiodic,” in contrast to shapes like squares or hexagons that can cover the plane in a repeating (or periodic) fashion. On March 20, Smith and Kaplan, together with two more researchers, announced that the hat tile was something mathematicians have been seeking for more than five decades: a single tile whose copies can fill the entire plane, but only in patterns that don’t consist of a repeating block of tiles. “It’s a tricky little tile.” He sent a description of his tile to Craig Kaplan, an acquaintance and computer scientist at the University of Waterloo in Canada, who immediately started investigating its properties. “I noticed that it was producing a tessellation that I had not seen before,” he said. Smith cut out 30 copies of the hat on cardstock and assembled them on a table. Usually when he created tiles, they would either settle into some repeating pattern or fail to tile much of the screen. Now he was experimenting to see how much of the screen he could fill with copies of that tile, without overlaps or gaps. Using a software package called the PolyForm Puzzle Solver, he had constructed a humble-looking hat-shaped tile. This work has been selected as an Editor's Highlight in Nature Communications.In mid-November of last year, David Smith, a retired print technician and an aficionado of jigsaw puzzles, fractals and road maps, was doing one of his favorite things: playing with shapes. In addition, the complex tessellations in this work may provide new insights for understanding self-organised systems in biology and nanotechnology." This method can be potentially applied to other molecular systems with multiple types of intermolecular interactions to build even more complex architectures. Prof Loh said, "By considering the symmetry of the molecular building blocks and substrate, as well as introducing multimode interactions, we can open up new routes to construct complex surface tessellations. The geometric similarity between these two molecular phases allows the molecular units to serve as tiles to tessellate and form highly complex molecular tessellations. The high-density phase is formed by halogen bonds, while the low-density phase is formed via a halogen-gold coordination network. The two molecular phases, a high-density phase and a low-density phase, arise from the different intermolecular and molecule-substrate interactions. A research team led by Prof Loh Kian Ping from the Department of Chemistry, NUS has demonstrated that highly complex periodic tessellation can be constructed from the tiling of two molecular phases that possess the same geometric symmetry but different packing densities.
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