As you most likely already know, a polyomino is a polygon that you get by joining unit squares together in such a way that each edge of a unit square touching another unit square touches exactly one other unit square. Here are some pictures of polyominoes:
Those are all hexominoes. And before you ask, yes, that picture is from Wikipedia.
The concept that I am interested in is the construction of similar polyominoes. One would say that two polyominoes are similar if one could get from one to the other by replacing all unit squares in one with chunks of $4$ unit squares arranged in a $2$ by $2$ square. Here are two similar polyominoes:
Let us call a polyomino self-similar if it can be assembled with other copies of itself to form a larger similar copy of itself. My example was self-similar, and here is the construction to prove it:
My question is this: can anybody find a set of $3$ or more distinct polyominoes so that, for each polyomino in the set, a polyomino similar to it can be constructed using the other polyominoes in the set?
P.S. That last criterion is not essential - if you like, you could find a set in which, for each polyomino, a similar one can be constructed using all polyominoes in the set, including itself.
P.P.S. I have no idea if it's possible. If you suspect it isn't and you would like to shatter my beautiful problem by disproving the existence of such a set, go for it.