What does it mean for a tiling (in particular, one involving the recently discovered "Hat" monotile) to be "aperiodic"?

Question

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?

Answer 1

A periodic tiling of the plane is one where you can translate the entire pattern to another point and overlay it exactly with itself. For example, if you look at a hexagonal grid then if you move any one hexagon to overlap another, the entire pattern overlaps.

An aperiodic tiling, such as the one discovered using the Einstein tile, has the property that no translation of the pattern will overlap perfectly with itself - although you've found three sections that are the same, notice that if you overlay two of them together then the rest of the pattern doesn't match. Aperiodic tilings had previously been found using multiple shapes - the Penrose tiling uses two different tiles to cover the plane, and it also has rotational symmetry meaning that if you rotate the pattern 72 degrees you get a perfect overlap - but this is the first time that an aperiodic tiling has been found that uses only one shape.

Answer 2

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.