The 'Einstein' Tile: Mathematicians Find a Shape That Never Repeats

Einstein tile
The "einstein" tile in this computer-generated image is the new shape found by scientists, arranged in concentric rings around a central, darker "hat." Smith, Myers, Kaplan and Goodman-Strauss (2023)

Imagine a bathroom floor covered in black and white tile. Each tile is the same shape and they fit together perfectly without gaps or overlaps. You're probably imagining rectangles or hexagons. That works, but imagine a shape that would create a pattern that never repeated itself, no matter how big your bathroom was. That is an "einstein" tile.


What Is an Einstein?

Seems improbable; it's hard to imagine a single shape that fits together so many ways it creates a pattern that would never, ever repeat. But a group of mathematicians say they have found such a shape, known as an "aperiodic monotile," or einstein. Einstein might be the name of the most famous scientist of the 20th century, but it's also German for "one stone," or even "one shape."

The einstein is a funny idea because interlocking tiles seem, by definition, to be orderly. But an einstein is also chaotic — no matter how you arrange them, it can't create a periodic pattern.


It took geometry buffs until the 1960s to conceive of even a collection of shapes, an "aperiodic set," that could tile an infinite two-dimensional plane with a pattern that never repeats. The first of these shape collections was discovered in 1966 — it consisted of 20,426 different tiles.

Enter David Smith, "the Hat" and "the Turtle"

Over the next decade, mathematicians tinkered with the concept and got the number in the set down from tens of thousands of shapes to just a couple. But all the time they were searching for The One — an answer to the "einstein problem." Some scientists were working hard just to prove that no such single shape existed.

In November 2022, David Smith, a self-described "shape hobbyist" from East Yorkshire, England, announced that he had likely discovered an einstein in the form of a 13-sided tile he called "the Hat," because it looks vaguely like a fedora.


Smith, a retired printing technician, spent a lot of time at home cutting shapes out of paper and experimenting with them. He knew the mathematical theory behind what finding the einstein would take, and one day he found something very promising. He contacted computer scientists and a mathematician he knew, and they set about trying to prove it.

In March 2023, they posted a preprint, much to the astonishment and excitement of mathematicians the world over. But in the midst of this discovery, Smith found another einstein: "the Turtle."

It turns out, "the Hat" and "the Turtle" are just two in a family of einstein tiles, created by adjusting the lengths of the sides of the shapes in relation to all the others.