The Einstein Hat Monotile Explained

Tiling the plane

A tiling is an endless cover of the plane by copies of shapes—like infinitely extending floor tiles—with no gaps and no overlaps. We can tile using a set of different shapes or using a single shape (a monotile). If the tiling pattern repeats (you can slide the whole picture by some fixed distance and it matches perfectly), the tiling is periodic. If no such "slide" exists, it’s aperiodic.

Which regular polygons tessellate?

For edge ‑ to ‑ edge tilings made from a single regular polygon, the angles meeting at each corner must add to a full turn (360°). This simple fact rules out most polygons (for example, regular pentagons have 108° corners, and 360° ÷ 108° is not an integer). Only three regular polygons tile on their own: equilateral triangles, squares and hexagons. Many other tessellations exist using multiple shapes (including highly symmetric “semi‑regular” tilings) or

Aperiodic tiling and the "one-tile" surprise

It’s very easy to see that tilings using regular polygons are repeating (periodic). Mathematicians have long been able to define sets of multiple tile shapes that tile the plane aperiodically, but for decades it remained unknown whether a single shape could force a non‑repeating pattern. In 2023 researchers discovered the first such example: the so‑called Hat monotile. Its geometry forces copies of itself to combine into ever‑larger “super‑tiles,” locking the pattern into a non‑repeating hierarchy. Below is an illustration of a patch of Hat tiling.