Socolar-Taylor Tiles.
These groundbreaking tiles are the first known solution of “the einstein problem.” No, it isn’t a physics problem--just a clever German play on words (ein Stein is german for “one tile”).
Anyways, the problem asks for a single tile capable of tessellating space aperiodically. Let’s unpack that in two separate parts.
Simply put, a tessellation is a tiling of the plane without any gaps or overlaps. For a given set of tiles (prototiles) to tile the plane, we arrange them to cover a flat surface (or fill space in higher dimensions). If the tiling can be continued indefinitely, the prototiles are said to tile the plane. A tiling can be described mathematically by a sequence of reflections, rotations, and translations.
Loosely, periodicity means something repeats in regular intervals or periods. An aperiodic tiling, then, does not repeat at scale. This means we cannot find arbitrarily large patches of tile with any repeating pattern. Another way to think of this is that the “pattern” is always changing.
The first image is a 2-dimensional monotile. The black markings on the tile are used as guides for matching rules, which restrict how the tiles are arranged.
The second image is an patch of 25 tiles. Notice how the markings generate an infinite hierarchy of successively larger triangles. This concept, called limit-periodicity, is one of the main structures by which aperiodic tilings are constructed.
Observe that the patch cannot tile the plane without leaving holes. Thus, the 2D monotile does not tessellate.
However, in the third image, we see a 3D monotile (here the guide markings are in red). This is, indeed, an einstein--it aperiodically tessellates 3D space. It should be noted that this tile does admit another tiling that is periodic. Tiles having this property are called “weakly aperiodic.”
The fourth image depicts a 3D patch. The limit-periodic structure is exhibited by the Sierpiński Triangle-looking design on its surface.
The final image removes a tile to show the internal structure of the patch.
One last, fascinating tidbit: don’t go and buy a bunch of 3D-printed copies hoping to fill space. The aperiodic tiling can only be constructed by allowing reflections. So, unless you have access to 4D space, you are out of luck.
Interactive 3D tile here. Rotate it! It’s like a weird lego, I guess.
Mathematics is beautiful. <3
