Definite shape of polyominos

Question

I'm currently exploring polyominoes and there's this question I'm pondering on. It said, "how can you convince yourself that a new shape is really 'different' from those already obtained?". I know that if two shapes can fit on top of one another, either by rotation or reflection, they are not unique. My question is, how can I better explain this reasoning? How can it be explained mathematically to be more convincing? (can it?)

Thank you so much for any help you can offer.

Answer 1

The short answer is that you just define the objects you're counting as equivalence classes under rotation and reflection.

The longer answer is that polyominos are an example of a combinatorial object whose identity can be defined in more than one way. (Other examples would be tilings, trees, colourings, ...). In particular, polyominos can have symmetry and can be related to each other by symmetries. With these kind of objects, it's typical that people should want to count

• The number of distinct objects ignoring all symmetries (e.g. for polyominos this is the OEIS' A000105)
• The number of distinct objects having a given symmetry (e.g. polyominos with no symmetry, polyominos with every symmetry possible)
• The number of distinct objects when you identify an object with those it relates to by a symmetry (e.g. for polyominos this is A001168)

Since polyominos are built up from squares, the symmetries of a polyomino must be a subset of the symmetries of a square. The symmetry group of a square has eight elements: the easy way to argue this is that any of the four corners can be mapped to the top-left, and then there are two choices for the corner which is mapped to the top-right. It turns out to have 10 subgroups (including itself and the trivial group), so there are 10 types of symmetry which a polyomino can have. (None, only vertical, only horizontal, only main-diagonal, only anti-diagonal, rotation by 180, rotation by 90, vertical + horizontal + rotation 180, main-diagonal + anti-diagonal + rotation 180, all four reflections + rotation by 90). Each of them has a symmetry group whose size is a factor of 8, and the number of distinct objects in the equivalence class is 8 divided by that size. E.g. the Z tetromino

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has a symmetry group of size 2 (the "do nothing" operation, and a rotation by 180 degrees), so there are $\frac{8}{2} = 4$ objects in its equivalence class:

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