The "Einstein" Tile: Aperiodic Monotile
For nearly 60 years, mathematicians and geometry enthusiasts have hunted for a specific, elusive shape. They were looking for a single tile that could cover an infinite plane without leaving any gaps and without ever repeating a pattern. In early 2023, that search finally ended. A team led by a hobbyist mathematician discovered the “einstein” tile, a 13-sided polygon that solves a problem many thought might be impossible.
The Search for the One Stone
To understand why this discovery is significant, you have to look at how tiling works. Most shapes we use to cover floors, like squares, triangles, or hexagons, create “periodic” tiling. This means you can shift the whole pattern over and it will look exactly the same. It repeats efficiently.
In the 1960s, mathematicians began asking if a shape existed that forced a non-repeating pattern. This is called “aperiodic” tiling. In the 1970s, Nobel Prize winner Roger Penrose famously reduced the number of tiles needed to create such a pattern down to two: a “kite” and a “dart.”
However, the “holy grail” was a monotile. The term “einstein” in this context does not refer to the physicist Albert Einstein. It is a play on the German words ein stein, which translates literally to “one stone.” The goal was to find one single stone that could do what Penrose did with two.
Discovery of "The Hat"
In March 2023, the mathematical world changed. David Smith, a retired print technician and hobbyist mathematician from East Yorkshire, UK, found the shape. He had been using paper cutouts to experiment with different geometries when he stumbled upon a 13-sided shape that he called “The Hat.”
Smith teamed up with computer scientists and mathematicians to verify the find. The team included:
- Craig S. Kaplan: A computer scientist at the University of Waterloo.
- Joseph Samuel Myers: A software developer.
- Chaim Goodman-Strauss: A mathematician at the National Museum of Mathematics.
Together, they proved mathematically that “The Hat” was indeed an aperiodic monotile. It is technically a “polykite” formed by gluing eight kites together. When you arrange “Hats” on a flat surface, they fit together perfectly, but the pattern they create never repeats translationally. If you walked infinitely across a floor tiled with “Hats,” you would never see the exact same layout twice.
The Twist: "The Spectre"
When the team published their paper on “The Hat,” there was one small catch. To make “The Hat” tile the plane aperiodically, you had to flip some of the tiles over. Essentially, you needed the shape and its mirror image. While this still counted as a monotile solution in many mathematical definitions, purists wondered if a shape existed that worked without needing to be flipped.
David Smith went back to work. Astonishingly, just a few months later in May 2023, the same team announced a second shape. They modified the edges of the original shape to prevent it from fitting with its mirror image.
They called this new shape “The Spectre.” Because it does not require reflection (flipping) to tile the plane, the team jokingly refers to it as a “vampire” tile (since vampires have no reflection). The Spectre is considered a “chiral” aperiodic monotile. This shape is the strict, undeniable solution to the problem posed in the 1960s.
Why This Matters Outside of Math
While this might sound like an abstract puzzle, the discovery has concrete implications for science and material engineering.
Quasicrystals
The most direct application is in the study of quasicrystals. In 2011, Dan Shechtman won the Nobel Prize in Chemistry for discovering materials where atoms are arranged in orderly but non-repeating patterns. This structure makes materials more durable and changes how they conduct heat and electricity. Now that we know a single shape can create these structures, scientists have a new geometric model to help design synthetic materials.
Material Strength
Periodic structures (like standard crystal lattices) have specific lines of cleavage where they can break easily. Aperiodic structures, like those created by the Spectre tile, do not have these straight lines of weakness. This could lead to the development of materials that are resistant to stress and fracture in unique ways.
Art and Design
Beyond physics, the aesthetic potential is immediate. Since the pattern never repeats, “Einstein” tiles offer a unique visual texture. We are already seeing these shapes appear in architectural renderings, puzzle designs, and fabric prints.
Summary of the Breakthrough
- The Discoverer: David Smith, an amateur mathematician.
- The First Shape: “The Hat,” announced March 2023. It requires reflection (flipping) to work.
- The Final Shape: “The Spectre,” announced May 2023. It requires no flipping.
- The Definition: An aperiodic monotile is a single shape that tiles a plane but never repeats the pattern.
Frequently Asked Questions
Is the Einstein tile named after Albert Einstein? No. It comes from the German phrase ein stein, meaning “one stone.” It refers to the use of a single tile to cover a plane, rather than the two tiles required by Penrose tiling.
Can I 3D print these tiles? Yes. The geometric specifications for both “The Hat” and “The Spectre” are open source. Many enthusiasts have already uploaded files to 3D printing repositories like Thingiverse and Printables.
Why is this discovery considered a surprise? For decades, many experts believed an aperiodic monotile might not exist. The fact that the solution was found by a hobbyist using paper cutouts, rather than a supercomputer, made the discovery even more shocking to the scientific community.
What does the shape look like? “The Hat” resembles a fedora or a shirt. “The Spectre” looks similar but has more curved, interlocking edges that prevent it from being paired with its mirror image. Both are 13-sided polygons.