My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied.

Does there exist a tile such that when you put a bunch of copies of it on a table and push from all sides, they always form a tiling?

My friend illustrated with physical (uniform density) lozenge tiles that they do not have this property, by throwing some on the table, and pushing them together. More specifically this suggests the stronger property that a typical initial configuration will get stuck. The tiles this was demonstrated on had positive friction.

The informal question as stated is a bit ambiguous. I am not going to try to formalize the physics of the problem, but I’ll at least try to specify how the force is applied in a hopefully unambiguous (but somewhat arbitrary) way. You can suggest a better variant in the same spirit if e.g. it’s easier to solve or mine misses the point for a “stupid” reason.

Let’s say a **tile** is a nice enough subset $P subset mathbb{R}^2$, you can pick what that means. E.g. if going for a negative answer, you can choose something like “simply connected convex polygon”. If going for a positive answer, I could imagine something like piecewise smooth being helpful. (For physics considerations it’s a zero friction rigid body, and let’s say of uniform density.)

Let $G = mathbb{R}^2 rtimes S^1$ be the rototranslation group (so no flips), which acts on $mathbb{R}^2$ from the left. A **partial tiling** is a subset of $T subset G$ such that the interiors of $t cdot P$ for distinct $t in T$ are disjoint. We say a partial tiling $T$ **fills** $C subset mathbb{R}^2$ if $T cdot P supset C$.

A **jam** is a finite partial tiling $T subset G$ such that, assuming the tiles have zero friction and behave according to physics, if you stretch a rubber band around the convex hull of $T cdot P = bigcup_{t in T} {t cdot P}$, the tiles will not budge. Intuitively, jams always exist aplenty, just put some tiles on the table, stretch the band around them and let go (if there’s a third dimension available there’s a problem with that strategy, but you see what I mean).

Definition. A tile $P$ is a

rubber band monotileif all $r > 0$, there exists $R > 0$ such that every jam whose convex hull contains the ball of radius $R$ fills a ball of radius $r$?

Observe that any rubber band monotile admits a partial tiling that fills the entire plane. In usual terminology, $P$ tiles the plane under rototranslations, and such $P$ is sometimes called a monotile.

In case this question is non-trivial, here’s some starters:

Is the equilateral (or any) triangle a rubber band monotile? Is the square (or any other rhombus, e.g. the lozenge)? Is the hexagon? Any of the pentagon monotiles?

I’m also interested in higher dimensions of course (my friend may or may not be). In one dimension I was able to solve the problem myself.