It's also funny that while the title uses the baity word "discover", the very first paragraph merely claims the mathematicians "described" the shapes.

I know that in newspapers and magazines, editors write headlines rather than authors to get clicks, regardless of accuracy. I would have thought Nature would try to be better though...

It's pretty egregious clickbait for Nature -- more along the lines of what I'd expect from Forbes or a similar outfit.

I mean, the title is saying that they "discovered" the "new class of shape" featured in this old kitchen tile: https://www.contemporist.com/reasons-why-you-should-get-crea...

Come on, now. The Egyptians, Greeks, and Romans were surely aware of it, and used similar pointed/curved and lenticular shapes in art and design.

The actual discovery seems to be buried in the midsection

>…suspected that the actual 3D chamber had no corners at all. “That sounded unbelievable,” says Domokos. “But later we found that she was right.”

Fwiw its also not obvious from the main paper, you have to look at fig 7 d-e for an idea

So in this case, i’d place some of the blame on the mathematicians themselves for failure to properly follow up on the bait. (But nature shall not be absolved from holding them to a higher standard)

A grid is a set of points, described by a basis. A tiling is like puzzle pieces, but with a fixed number of piece "shapes". A packing is a way to stuff a set of things into a space. Tilings and packings are related, but the subfields are asking different questions.

Some of these terms are pretty general and their usage will depend on the user and context. I'll try to define what I think are the most appropriate and common usages of each.

Grid - usually a regular D-dimensional boxes that are packed, axis aligned. Sometimes used synonymously with a set of points that are also regularly placed and axis aligned. I've used this to describe a (finite) rectangular cuboid (in 3D) but could just as easily be used to describe an infinite set of boxes. As in "Label each cell in the grid an alternating color of red or blue".

Tiling - A covering of some D-dimensional space from a (finite) set of smaller tiles, with no overlap and no gaps. I've used this to describe higher dimensional spaces but is often used for 2D. As in "A set of Penrose tiles can be used in a plane tiling".

Packing - Placing a (finite) set of smaller geometric elements into a large area such that the geometry doesn't overlap but gaps are allow. The larger area that can be be finite or infinite. The dimension can be arbitrary. This is often used in context of trying to minimize the gaps within the area being packed. As in "Randomely placing 3D oblong spheroids (aka 'M&Ms') in a box of side length L will yield a sub-optimal packing. Introducing gravity, friction and 'shaking' the box for some amount of time will yield a better packing"

Tesselation - A synonym for tiling.

A grid is a tiling. For example a 2d grid is a tiling/tesselation of the plane by boxes.

You may also like: lattice.

Tilings cover an entire plane with no gaps or overlaps. Opposed to packings which may leave gaps.

I may be wrong but I think 'packing' may allow the shapes to vary in size.

In 3D printing you need pieces that can fit together into each other, not merely tile together. It should however be quite interesting to extend the soft cell shapes to also fit together while preserving softness. Perhaps it is possible that the shown saddle-like shape in Fig 6, Panel 4 of the PNAS Nexus article can serve this purpose, but it is not clear how.