# it can be repeated indefinitely without ever repeating the same pattern – time.news

Of Paolo Virtuani

an irregular polygon with 13 sides. The solution, explained in 89 pages, had been sought by topologists for decades

If you want to tile a wall of infinite length, this shape will ensure that no two patterns are ever the same. And without leaving uncovered gaps, not even the smallest. Seeing is believing. This was discovered by four mathematicians who released an 89-page study full of geometric formulas, solving a problem which topologists and scientists who analyze the geometric shape of space (reductive definition because the concept is much more complex) have been racking their brains for decades ). The magic tile is an irregular polygon with 13 sides, formed by a regular hexagon from which a third of the surface in the shape of an irregular pentagon has been removed but to which two other pentagons equal to the one removed have been joined. In short, a strange shape.

##### Aperiodic tiling

The exact term aperiodic tiling, until now no one knew that you could get a pattern that never repeats infinitely using just one geometric shape (the 13-sided irregular polygon mentioned above). Nature prefers highly symmetrical and regular shapes, which are also associated with the least energy needed to build them. Let’s think, for example, of crystals in minerals: we are fascinated by their regularity and symmetry even at the atomic level. Indeed, the macroscopic form is the sum of innumerable identical forms that repeat themselves at the microscopic level. Only for about forty years have scientists discovered that there are crystals formed by lattices that are never identical. They were called quasicrystals and the scientist who discovered them in 1984, Dan Shechtman, was awarded the Nobel Prize in chemistry in 2011.

##### Applications

Pure mathematical-geometric speculation? Not only. Applications in lasers are being studied with quasicrystals to make them more powerful using the same energy. Not to mention the tile industry: perhaps some manufacturers will have an idea for launching the magic tile on the market.

March 23, 2023 (change March 23, 2023 | 14:44)

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