Aperiodic Textures

These are two examples of patterns created with textured Penrose tiles. The texture image is a piece of a photograph of an underwater coral reef.

To create the original tiles I manipulated the coral texture in Photoshop so it would match up smoothly as the tiles were assembled. Then I handed the images to a custom program I wrote in C# that “grows” a Penrose pattern. When that was done, I brought the image back into Photoshop to add the drop shadow.

Note that these tilings can be extended infinitely. I just chose a nice fragment to illustrate the pattern (I call this the “running Nixon” shape). The cool thing about aperiodic tilings like these is that even if they continue out to infinity, they will never form a single chunk that repeats over and over. Though sections of the pattern will look like other sections, as a whole the pattern will never repeat.

Since I did this work in 1998, there have been two amazing discoveries in aperiodic tiling, made in quick succession (I had nothing to do with them; I was just a cheering fan like everyone else!).

The first happened in 2023. It was the “hat,” a single tile (rather than Penrose’s two) that could tile the plane aperiodically (a popular account is here, and the technical paper is here).

The hat was an amazing discovery, and a lot of people thought it was a once-in-a-lifetime kind of event. There was just one small issue that held this back from being an ultimate kind of event, and that was that to tile the plane aperiodically, you and to use both the original hat and and its flipped-over version. It felt like it was really two tiles, but related by mirror symmetry.

So close!

Then, in a move that shocked everyone, only a few months later members of that group announced a single tile that does the job and doesn’t require being flipped over. They called it an “einstein”.

I’ve always thought of a tile like this as the holy grail of aperiodic tiling. It’s the perfect thing. I don’t know if I really expected to see one in my lifetime, and I’m glad I did!