Science & Technology

The one-of-a-kind shape never creates a pattern.

Mathematicians have tried for nearly five decades to design a shape that would never repeat a pattern, no matter how large the surface. This elusive configuration is named “Einstein,” but not for reasons you might think. The words “ein stein” mean “one stone” in German, making the name perfect for the one-of-a-kind shape. The long-awaited breakthrough became a reality in November 2022 when a retired engineer stumbled across the wonky 13-sided shape. This breakthrough is rocking the world of mathematics.

Many Tried to Discover the Einstein

For years, scientists tried to reduce the number of tiles – or shapes – necessary to cover a plain with no gaps or repeating patterns (also called aperiodic tiling). The first to successfully do this was mathematician Robert Berger in 1966. However, Berger’s creation required 20,426 different colored tiles. Shortly after, Berger beat his own record with 104 tiles. In the following years, the number of tiles was reduced to 92 by Donald Knuth, then to six by Rafael Robinson, and finally down to two in 1974 by physicist Roger Penrose.

Since then, the goal has been to find a single shape that could create a pattern with no gaps and no repeating patterns. Being quite a challenging task, it wasn’t accomplished until late fall of 2022 when a retired print technician, David Smith, revealed a tile he had formed that he called “the hat.” Smith believed it could be an Einstein and a significant discovery. After calling in colleagues for help and some rigorous testing, “the hat” was determined to be an aperiodic tile. Smith, and a group of others, have received notoriety for their research and findings.

What Does It Look Like?

The shape has taken on the nickname “the hat” or “top hat,” as it does resemble a tall hat in many ways. Mathematicians refer to it as a “polykite” because it is made up of several kite shapes. Others feel it looks more like a t-shirt. Either way, the surprisingly simplistic figure has 13 jagged sides of varying lengths. No matter the number of hat tiles placed on a surface, a pattern will never form, and the tiles will push up tight against one another.

Interesting Development

Though “the hat” is due for more testing to confirm that it is an Einstein, questions have arisen about the figure’s authenticity. The tiles are installed using a mirror image or reflection of itself, probing experts to question whether the hat is one or two shapes. Most experts agree it is still just one, though this must be proven for it to truly be considered an Einstein. It will be interesting to see what more testing will reveal about the new mathematical shape.