As I understand it: they found a single shape. Where existing Penrose tilings were composed of two (or more) shapes.

But in one sense this is actually 2 shapes that are mirror images. It's still really cool, but I don't think it is ultimately what we've been looking for. As proof that it's not, we all know that a paper presenting one that doesn't need its mirror image to tile the plane aperiodically would still be a big dea.

It's not really a single shape since the tiling contains the shape's reflection which is normally considered a different shape since its handedness changes.

Rigid transformations of a single shape https://en.wikipedia.org/wiki/Rigid_transformation

That's covered in the article.

It has a paywall.

They didn't indicate they couldn't read the article due to a paywall. They just asked for an ELI5 on something that's covered in the 4th paragraph of the article. My reaction would be different if you said you couldn't read the article and asked the same question.

But almost every HN post to pay-walled content includes an archive.is link to bypass the paywall.

It is just a single tile type, not two

The two Penrose tiles are also an affine transformation of each other.

Whereas in this new tiling there's a single shape and its rigid transformations.

I don't think they can be if you want to enforce the matching rules using the tile's shape.

> matching rules using the tile's shape.

That excludes affine transformation.