Surely you didn’t expect news about aperiodic tilings to appear at regular intervals? You know how it is – you wait ages for a new aperiodic monotile discovery to come along, then two come in quick succession.
In March, we covered the discovery of an aperiodic monotile. The team of authors behind that discovery have been continuing their work and this week have an even bigger announcement.
The only slightly dissatisfying aspect of the previous discovery, as you may recall, was that in order to tile the plane, the Hat (and Turtle) tile each needed roughly one in every six tiles to be mirrored. This raised the question of whether a tiling using a single aperiodic tile would be possible without reflections.
It turns out the authors had kind of already found one – a tile they referred to in their previous paper as “Tile(1,1)” was the basis for a chiral aperiodic tile – one which doesn’t need mirroring in order to fully tile.
It’s actually the midpoint of the continuum of possible tile shapes that contains the hat and turtle, but had been previously considered uninteresting, since it can tile periodically if used along with its reflection. However, if you don’t use reflection, it gets more interesting.
Tile(1,1) has now been shown to be weakly chiral – meaning that if you use it without reflections, it must tile non-periodically.
This means we immediately have an infinite family of tiles with the property we’re looking for – by replacing each of the straight edges with anything that’s asymmetrical and oriented consistently, we can force the tiles to be oriented all the same way, producing an aperiodic monotile in the strong sense. (This is similar to what Edmund Harris shared in our math-off as a way to force Penrose tiles to be aperiodic.) The authors have used a simple curved line, creating a vaguely ghost-like shape they’ve christened The Spectre (because it doesn’t have a reflection, lol)
If you’d like more context and an idea of how the proof has been achieved you can check out Craig Kaplan’s webpage and his excellent Mastodon thread about it.