A 13-sided shape known as “the hat” has mathematicians tipping their tops.

It’s the first example of an “einstein,” a single shape that completely covers a surface, but with no repeats. Like bathroom floor tiles, it covers the entire surface with no gaps or overlaps, but with a pattern that never repeats. Ever.

Though the name “einstein” conjures up the iconic physicist, it comes from the German ein Stein, meaning “one stone,” referring to a single tile. The einstein sits in a weird space between order and disorder. It fits everywhere, with no pattern. Identified by David Smith, an amateur mathematician who calls himself an “imaginative tinkerer of shapes,” Smith published the discovery at arXiv.org, as part of a quartet of mathematicians from Yorkshire University, the University of Cambridge, the University of Waterloo and the University of Arkansas.

For context, here are the three “regular” tiles, that you’ll see on bathroom walls and floors, entryways, and other tiled surfaces. They are they only shapes that make a perfect, even, covering that keeps repeating. The tiles are triangles, squares, and hexagons, with three, four or six sides respectively. These are the only regular tiles possible.

And here is a gif of the hat, that shows some of the infinitely many ways it can combine without repeating.

Up until now, Penrose tiling, discovered by Roger Penrose in the 70s, has been the simplest aperiodic (non-repeating) tiling, using just 2 tiles. It also makes beautiful patterns, depending on how you color it.

Here’s an alternate coloring of the hat. It’s almost like an optical illusion, giving the impression there are many shapes. But it’s just one! And now that we know it’s possible, we may find other shapes that fit perfectly but never repeat. There are applications in material science, math, and art. And there’s a cool word for all of these kinds of patterns: tessellation.

Here’s a 3D printed version of a single tile used in the einstein.

“Everybody is astonished and is delighted, both,” says mathematician Marjorie Senechal of Smith College in Northampton, Mass., who was not involved with the discovery. Mathematicians have been searching for such a shape for many decades. “It wasn’t even clear that such a thing could exist.”


David Raiklen
David Raiklen

David Raiklen wrote, directed and scored his first film at age 9. He began studying keyboard and composing at age 5. He attended, then taught at UCLA, USC and CalArts. Among his teachers are John Williams and Mel Powel.
He has worked for Fox, Disney and Sprint. David has received numerous awards for his work, including the 2004 American Music Center Award. Dr. Raiklen has composed music and sound design for theater (Death and the Maiden), dance (Russian Ballet), television (Sing Me a Story), cell phone (Spacey Movie), museums (Museum of Tolerance), concert (Violin Sonata ), and film (Appalachian Trail).
His compositions have been performed at the Hollywood Bowl and the first Disney Hall. David Raiken is also host of a successful radio program, Classical Fan Club.