# Polyomino that can tile itself

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.

• The tetramino $T$ also works, for $4$ copies of it tile a 4x4 square, so 4x4x4 copies tile $8T$. Jul 25 at 16:23
• But the T-tetromino doesn't work for n=2. The question is asking for tiles that work for all n. Jul 25 at 16:25
• 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. Jul 25 at 16:46
• This is a nice conjecture! Jul 26 at 5:13
• 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.
– PMar
Jul 26 at 13:28

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.