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An animal in the plane is a finite set of grid-aligned unit squares in $\mathbb{R}^2$. (The definition is the same as a polyomino, but where we relax the connectivity requirement.)

In this investigation of tilings of rectangles by 1D animals, it is mentioned offhandedly that Coppersmith showed every 4-celled animal tiles the plane in 1985 (and can do so without reflections); I was able to locate the paper here (link to PDF). 6-celled animals do not, in general, since they can have a hole:

enter image description here

Has any progress been made on the 5-cell case since 1985 (either with or without reflections permitted)? I'd be interested in computational results as well, e.g. "every size-5 subset of a $7\times 7$ grid tiles the plane".

Edit: Since I realize the answer to this question may well be "no further progress has been made and the problem is difficult", I'm interested in any statements that can be made about restrictions of this problem to natural subcases. For instance:

  • Can every size-5 subset of $\mathbb{Z}$ tile the plane?

  • I didn't specify in the original question, but one can consider whether reflections are allowed in a tiling or not. If so, a proof may be easier (it seems Coppersmith restricted himself to the rotation case), and if not, a counterexample might be more tractable.

Edit 2021-05-04: I contacted Dr. Coppersmith about the problem, and he wasn't aware of any further research in this direction.

RavenclawPrefect
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    A very minor result, but through a combination of tiling software and manual checking I've confirmed that every size-5 subset of a $3\times 3$ square tiles the plane without reflections (that is, only using rotations and translations of the original tile), and that every size-5 subset of a $4\times4$ square at least can cover a $10\times 10$ grid without holes (again, using only rotations and translations). – RavenclawPrefect Dec 28 '20 at 22:50
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    Something I noticed in the first link you posted; the 1D-animal made by XXXOOXOX (where X is animal, O is nothing) can apparently not tile a rectangle. This is probably not enough to also imply it can't tile the plane, but might be a starting point for a counterexample. There are also other examples there, such as XXOXOOOXX. – Xander L Dec 28 '20 at 23:24
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    Update: I have crossposted this question to MathOverflow here. – RavenclawPrefect Jan 05 '21 at 22:00

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