The einstein tile can cover an infinite plane only with a nonrepeating pattern. ... known as an aperiodic tiling. Some earlier aperiodic tilings have connections to real-world materials.

Many aperiodic tile sets, including Penrose's, can be shown to tile the plane with substitution systems like these. Credit: Jen Christiansen.

“The hat,” however, is an aperiodic tile, meaning it can still completely cover a surface without any gaps, but you can never identify any cluster that periodically repeats itself to do so.

Such tile sets are called aperiodic. Although Berger and others were able to bring down the size of these aperiodic sets significantly, in the mid-1970s Roger Penrose captured the world’s attention by ...

In the 1960s, mathematicians began to study “aperiodic” tile sets with far richer behavior. Perhaps the most famous is a pair of diamond-shaped tiles discovered in the 1970s by the polymathic ...

The new aperiodic tile could spark further investigations in materials science, Senechal says. Similar tilings have inspired artists, and the hat appears to be no exception.

The polygon Tile(1, 1) is a weakly chiral aperiodic monotile: a shape that tiles aperiodically if only translations and rotations are permitted. Then, by modifying the edges of that polygon, we obtain ...

Here we see the first four iterations of the H metatile and its supertiles. (Image credit: Smith el at. (2023)) Since then, mathematicians around the world have searched for the aperiodic tiling ...

“The hat,” however, is an aperiodic tile, meaning it can still completely cover a surface without any gaps, but you can never identify any cluster that periodically repeats itself to do so.

In the 1960s, mathematicians began to study “aperiodic” tile sets with far richer behavior. Perhaps the most famous is a pair of diamond-shaped tiles discovered in the 1970s by the polymathic ...

"The hat," however, is an aperiodic tile, meaning it can still completely cover a surface without any gaps, but you can never identify any cluster that periodically repeats itself to do so.