Find all polyomino $P$ such that we can tile $nP$ with $n^2$ copies of $P$ for all $n\in \mathbb{N}$. ($nP$ is a polynomino similar to $P$ with scale factor $n$)
I conjecture that there are only $4$ types of such polynomino:
-The first trivial type is rectangle with integer side length.
-The second type is created by $3$ rectangles with integer side as follow:
The last two type is polyomino similar to one of two polyominoes below:
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? And as Timothy suggest, what if we allow all but finitely many $n$?
Here some link relate to this problem, but none has any attempt to solve it:
-More rep-tile polyominoes
-The same question but with polygonal
I think for too complex polyomino, combine copies of it would make either more complex polyomino or too simple polyomino, so it can't tile itself. So I guess we need some invariants which measure the complexity of polyomino. And $n=2$ may be the most important case.
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2$\begingroup$ The tetramino $T$ also works, for $4$ copies of it tile a 4x4 square, so 4x4x4 copies tile $8T$. $\endgroup$– Pietro MajerCommented Jul 25, 2022 at 16:23
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4$\begingroup$ But the T-tetromino doesn't work for n=2. The question is asking for tiles that work for all n. $\endgroup$– Matt FayersCommented Jul 25, 2022 at 16:25
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9$\begingroup$ You may want to additionally search for "rep-tile", the term for self-tiling shapes, and see "Tiling with polyominoes" by Solomon W. Golomb. By Theorem 4, a necessary condition is that the shape can cover at least one corner of its rectangular hull, which eliminates shapes like (e.g.) the x-pentomino. $\endgroup$– John S. AdairCommented Jul 25, 2022 at 16:46
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2$\begingroup$ This is a nice conjecture! $\endgroup$– Per AlexanderssonCommented Jul 26, 2022 at 5:13
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2$\begingroup$ I know that the so-called 'sphynx' polygon (formed from 6 equilateral triangles, hence not a polyomino) is rep-$k^2$ for all k. If that helps any. $\endgroup$– PMarCommented Jul 26, 2022 at 13:28
1 Answer
As John S. Adair commented, the relevant keyword is rep-tile. Wikipedia provides a partial answer to your second question (shapes other than polyominoes); it cites a paper by Viorel Niţică, "Rep-tiles revisited," on pages 205–217 of MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics (AMS, 2003), which is devoted precisely to the question,
Which polygonal rep-4 tiles are also rep-$k^2$ tiles for any $k\ge 2$?
However, Niţică does not show that there are no other examples. Many "near-misses" (which are rep-$k^2$ for all but finitely many $k$) may be found on Andrew Clarke's webpage (click on the picture at the top to view pentominoes, hexominoes, heptominoes, and higher-order polyominoes).
As far as I can tell, your conjecture about polyominoes that are rep-$k^2$ for all $k$ is open. I could not find any mention in either of the standard books on polyominoes (by Golomb and Martin). Even a complete investigation of hexominoes is laborious, as one can see from the paper Solving Rep-tile by Computers: Performance of Solvers and Analyses of Solutions, by Banbara et al.
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5$\begingroup$ I wanted to ask Viorel Niţică if he had any more recent information, but then I learned that he recently passed away. $\endgroup$ Commented Jul 26, 2022 at 12:22