The well tried and tested method of proof using Robinson tiles can be illustrated with six different tiles. The lines drawn on the tiles are similar to the colored edges in Wang’s squares: you just place Robinson tiles next to each other in such a way that lines of the same color continue seamlessly. Following this rule results in a recognizable pattern: the yellow lines form squares with a larger square corner starting in the center, and so on. So you can set a hierarchical structure for tiles: tiles that get bigger and bigger and intersect. So tessellation cannot be periodic. If this is the case, sections of the form will appear elsewhere in exactly the same way. But if you do part of the tiling and move it around, you inevitably destroy the hierarchical structure.
Kaplan, Goodman Strauss, and Myers were able to show something similar to the Einstein hat-shaped tiles proposed by Smith. However, they did not look at the hat itself, but at four different combinations, some of which consisted of several tiles: a hexagonal structure with four hat tiles, a pentagon with two pieces, a parallelogram with two pieces and one tile, which the researchers represent with a triangle. By defining the appropriate rules on how the four cluster polygons combine, the entire plane can be covered. At the same time, cluster polygons form a pattern that, they proved, never repeats itself.
The combination of the four groups together results in one of the four polygons (hexagon, pentagon, parallelogram, or triangle). Myers, Kaplan, and Goodman-Strauss were also able to show that this larger structure is unique: it can only arise from a single arrangement of smaller block polygons. Now you can merge the larger polygons together again to form larger structures: these also result in either a six, five, four, or a triangle – these also consist of a clear arrangement of the smaller polygons. This can be repeated indefinitely, giving a hierarchical structure, as in the case of Robinson’s tiling. An exception is periodicity: larger and larger polygons can be generated, which consist of a clear arrangement of smaller groups. If you simply move sections of it to another location, the superstructure may lose its uniqueness. As it turns out, this proof can be made not only with the “hat” that Smith provided, but also with the various shapes of the tiles.
Two guides are better than one
A lot of detailed calculations were necessary to provide this proof: you have to make many combinations of how the groups are put together and make sure that the resulting structure is always unique. To do this, the three scientists resorted to the help of computers. They have independently developed two programs and Publish freelyso that anyone can check them for any errors.
But Myers wasn’t satisfied with that: He presented a second proof that required no computers at all. He was able to show that the tiling of the hat is related to two other tilings of the so-called polydiamonds (geometric figures consisting of equilateral triangles): if the polydiamond tiling is aperiodic, then the tiling of the hat automatically has the same properties. And in fact, multidiamond systems are even easier to study: if they are periodic, there will be displacement vectors that shift one region of the tessellation into a similar one. And the displacement vectors of the two polydiamond pavements must differ by a rational operator (a rational number), according to Myers. But Myers found a ratio of √2 – an irrational number. Irrational values in surfaces usually indicate non-periodicity: if the pattern were to repeat itself, the ratio of the given volumes would be the same over the entire area and would result in a rational number. So Myers ruled out that the multidiamond surfaces are periodic, and thus proved that the hat slab is also aperiodic.
With this said, not only did Myers, Kaplan, and Goodman Strauss prove that the tiles Smith found were really the Einsteins he was looking for. They also provided a new method for proving the non-periodicity of patterns: by dividing the tessellation into two more parts and working with them. This method may also be useful for other applications, the scientists explain in their paper. Experts excited about the results: The mathematician Colin Adams of Williams College in Massachusetts, for example, agrees “new world”immediately he will line his bathroom with tiles in the shape of a hat.
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