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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:

enter image description here

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:

enter image description here

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:

enter image description here

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.

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    $\begingroup$ I think you need to clarify your question. My set can be 3 squares of different size, and since all of them are similar, your criterion is satisfied. $\endgroup$
    – greenturtle3141
    Commented Jun 30, 2017 at 20:43
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    $\begingroup$ Can you use multiple copies of each polyomino in the construction? $\endgroup$
    – Deusovi ♦
    Commented Jun 30, 2017 at 20:44
  • $\begingroup$ @Deusovi No, you can't. $\endgroup$
    – Franklin Pezzuti Dyer
    Commented Jun 30, 2017 at 21:30
  • $\begingroup$ @greenturtle3141 No, you can't do that, because you can't assemble two or three distinct squares to get another square $\endgroup$
    – Franklin Pezzuti Dyer
    Commented Jun 30, 2017 at 21:30
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    $\begingroup$ It's not clear to me whether the similar polyomino must use only one copy each of the other polyominos. If so then the set must contain at least five polyominos for the simple reason that a linear factor of two corresponds to an areal factor of four. $\endgroup$
    – Peter Taylor
    Commented Jun 30, 2017 at 22:28

2 Answers 2

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Yes, it's possible.

Take three 12-ominos: a 1x12 rectangle (A), 2x6 rectangle (B) and 3x4 rectangle (C).

It's rather trivial to prove that it's possible to create enlarged copies of each polyonimo using only one of the others.

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A solution with total area 12:

L, square, and I tetrominoes. Each can easily combine with a copy of itself to make a 4x2 rectangle, and two of those can make any of those tetrominoes. Like the other answer, only one type of tetromino is required.

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