Researchers found a 14th family of tiling pentagons in 1985, and three decades later, another team found a 15th family using a computer search. No one knew if this discovery completed the list, or if there were more families still hiding. That question was answered in 2017 when Michaël Rao proved that all convex tiling pentagons — and with them, all convex tiling polygons — had been found.

All these tilings repeat. That is, they have a periodic symmetry, which basically means that if we were to trace the tiling on a piece of paper and slide that paper in certain directions, it would line up exactly with the tiling again.

Other kinds of symmetries are also possible. For example, a mirror symmetry implies that our patterns will line up if we flip our tracing paper upside down about a fixed line. Rotational symmetry means that they’ll line up if we rotate our paper. And we can combine actions to obtain a glide reflection symmetry, which is like sliding the paper and then flipping it over.

In 1891, the Russian crystallographer Evgraf Fedorov proved that there are only 17 ways that these symmetries can be combined. Since this restriction applies to all periodic decorations of the plane, these are widely referred to as the 17 “wallpaper groups.”

Once one is familiar with this classification of symmetry patterns, it is nearly impossible to see a periodic design, however intricate, and not view it as a puzzle to decode: Where and how, exactly, does it repeat? Where are those symmetries?

Of course, not every tiling design is periodic. It is possible, and often easy, to place tiles in the plane so that the resulting design never repeats. In our example with hexagons, squares and triangles, you can do this by simply rotating a single hexagon and the polygons surrounding it by 30 degrees. The resulting tiling no longer has translational symmetries.