In these articles "Mathematicians Excited About New 13-Sided Shape Called 'the Hat'" (Gizmodo), "An 'einstein' tile? Mathematicians discover pattern that never repeats" (Interesting Engineering), from the paper An aperiodic monotile (arXiv) from Smith, et al., they make a claim
Researchers identified a shape that was previously only theoretical: a 13-sided configuration called “the hat” that can tile a surface without repeating."
If we look at the image, we can see at least three places where the pattern repeats
What do they mean exactly when they say that? Is it that they can repeat but not touch or something else?
What exactly are they proving to show this given we can see repeating patterns?
Best Answer
A tiling is aperiodic if it does not have any (global) translational symmetry. However, in many cases, every finite portion of an aperiodic tiling will repeat infinitely many times (as an example, you can see the same property in an irrational number's decimal expansion). This property is called repetitivity.
The novelty of the newly discovered "Hat" monotile is not only that it is possible to construct aperiodic tilings with it (it is also possible with 1-2 right triangles as in the pinwheel tiling), but also that it is impossible to construct a periodic tiling out of it.