Image by Cmglee, CC BY-SA 4.0, via Wikimedia Commons
A square can tile a plane but can form a repeating pattern. Is there a single shape that can tile but never repeats? That’s what’s called the “einstein problem”.
In 2010, the first never-repeating tile was discovered: the Socolar-Taylor tile. But it’s a bit weird, having several separated, disconnected bits.
In 2022, “The Hat” (shown in pic) was discovered, and it’s a lot less weird. It only has 13 sides and nice angles that are multiples of 30°.
Is it just me or is the radial pattern not apparent to others? Starting with the red “hat” top center, work outward in a spiral. It’s not bilaterally symmetrical, but it appears to be chiral.
Going to read more about this.
Here’s the paper that proves the aperiodicity of the shape.
And here’s a site about this shape by Craig S. Kaplan, one of the mathematicians that contributed to the proof.
cool. a zoomed out version of the image is all I needed.