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 $n$)<br/>
I conjecture that there are only $4$ types of such polynomino. The trivial type is rectangle with integer side length, and three of the other types are polyominoes which similar to one of the three polyominoes below:<br/>
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/G3H3f.png
Is there any polyomino which not in the $4$ types above and satisfies that property? What if we replace polyomino by polyiamond, polyabolo, polycube,...? I have searched self-tiling, rep-tile and other similar key word, but I couldn't find this problem.<br/>
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.