A ‘Vampire Einstein’ Tile Outdoes Mathematicians’ Latest Feat


Only a few months have passed since scientists announced the discovery of the first “einstein” — a unique tile capable of covering an infinite surface in a non-repetitive manner. The same group has now identified an even more remarkable shape.

While the inaugural einstein, playfully termed “the hat,” formed a design that incorporated both the hat and its inverse image, this newly identified tile also crafts a non-repetitive design but does so without resorting to such mirrored imagery.

The findings were shared on May 28 at arXiv.org. Given that this shape doesn’t involve its mirror counterpart, researchers humorously dubbed it the “vampire einstein.”

The term “einstein” is derived from the German phrase for “one stone” and bears no relation to the renowned physicist. This unique shape is a member of a broader category of vampire einsteins which the team labeled “spectres.”

Mathematicians have a deep-seated interest in understanding how tiles can uniformly cover an infinite surface without any overlap or voids.

While certain tiles can be structured in a manner that prevents them from forming a repetitive design, einsteins are distinct in that they can only tile in this unique manner.

Historically, scholars were aware of tile groupings that could cover surfaces in exclusively non-repetitive designs. However, the existence of a singular tile with this capability remained elusive until recently.

Upon identifying the initial einstein, the team’s curiosity led them to ponder the possibility of a tile that could create a non-repetitive design sans any mirror replicas.

By tweaking a shape reminiscent of the hat and adjusting its edges to prevent the tile’s reflection from aligning with itself, they successfully fashioned the vampire einstein tile.