A team from the University of Arkansas has discovered the first shape that can cover a wall without creating a repeating pattern.
This property is known as “acyclic tiling” and so far it has only been achieved using more than one shape.
But this “cap” is able to open by itself, creating infinitely extended patterns. It even retains its non-periodic tiling ability as the 13 sides of the figure change in length, allowing more patterns to be created.
Tiling refers to covering a flat surface with shapes that fit together without any gaps or overlaps.
A non-periodic tiling is defined as a special type of tiling in which the pattern of shapes used to cover the surface does not repeat.
This is different from periodic tiling, which uses molds to cover a surface in a regularly repeating pattern, such as triangles and squares.
The first set of figures that together could create infinitely different patterns was discovered in 1963 by the American mathematician Robert Berger.
It consisted of 20,426 unique shapes, but this discovery led to further research into non-periodic tessellations to see if this number could be reduced.
The best-known group of non-periodic tiles is known as the “Penrose tiles”, which consist of two different rhombic shapes and were first published in 1974.
Since then, mathematicians have been searching for the elusive “Einstein”. A figure that by itself can achieve an aperiodic tiling.
In their 89-page study, published in arXiv, the Fayetteville researchers sought to discover the real Einstein, which means “one stone” in German.
“For a long time, the question remained open whether such tiles exist,” they wrote.
The team was the first to use computers to sieve hundreds of different shapes.
They then looked closely at the figures cast as possible Einsteins and attempted to prove mathematically that they would create a non-periodic tiling.
Lead author, Dr. Chaim Goodman-Strauss, said: “You are literally looking for one thing in a million. And you filter, and then you have something strange, and then it’s worth investigating further. Then you start to manually explore it and try it. to figure it out and you start pulling out the structure. And this is where you come into play. “A computer would be useless, since a person would have to participate in the creation of evidence that a person could understand.”
This was the only “hat” they succeeded in, and in fact they succeeded twice in proving it to be non-periodic.
Mathematicians hope that knowledge of their unique shape will lead to the creation of new materials that are stronger or have other useful properties.
In the molecular structures of crystalline materials, repeating patterns can often be seen, which makes them easy to destroy.
Source: Daily Mail
