If you wanted to lay tiles on a flat surface, but never have their basic geometric pattern repeat, could you do it? If you could, is there a single uniform tile geometry you could use?
The first question's answer of "yes" has been known for some time, but finding the answer to the second question has perplexed mathematicians for decades. Its answer has only just become understood after decades of research. Veritasium describes the progress they made in whittling down the number of differently shaped tiles you would need to do the job from 20,426, to 96, to 6, then 2, which was the state of the art until 2023.
That brings us to now, as the shape of what might be called "an aperiodic monotile" capable of completely filling a planar surface using a single uniformly shaped tile with no gaps has gotten closer to reality thanks to the work of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. The following illustration from their preprint paper shows the basic 13-sided "hat-shaped" polykite tile geometry they've identified:
The hexagonal pattern underlying the basic polykite shape helps explain why this particular geometry can succeed in tiling a plane into infinity, but their solution is not limited to this particular geometry. The following video shows how it can be morphed into other viable variations.
If you look closely at the pattern however, you'll see a number of these tiles represent a "mirrored" or "reflected" variation of the basic polykite geometry. On Twitter, Robert Fathauer took a creative approach to illustrating that property:
The new aperiodic monotile discovered by Dave Smith, Joseph Myers, Craig Kaplan, and Chaim Goodman-Strauss, rendered as shirts and hats. The hat tiles are mirrored relative to the shirt tiles. pic.twitter.com/BwuLUPVT5a
— Robert Fathauer (@RobFathauerArt) March 21, 2023
That small difference raises the challenge of whether a two-dimensional plane could be fully covered with a non-reflected single geometric tile, which has become the final frontier for fully resolving the mathematical challenge.
Given the relative simplicity of the newly defined geometry, the first applications to take advantage of the discovery will almost certainly artistic in nature, which already includes its fast incorporation in video games like HyperRogue. Beyond that, aperiodic tilings involving multiple tile geometries have applications in materials science, where they help understand a unique class of materials called quasicrystals.
Building on that base, aperiodic tiling patterns are also being investigated for application in lightweight structures that are subjected to high vibration environments. For example, the new 13-sided aperiodic monotile would be a candidate to replace traditional isogrid-based geometry in machined structures on aerospace vehicles.