A free polyomino is a polyomino considered up to translations, rotations, and reflections of the square lattice. Thus two finite side-connected unions of unit squares represent the same free shape whenever a lattice symmetry carries one to the other. Its absolute position and orientation are discarded.
For example, a two-square horizontal domino and a two-square vertical domino are the same free polyomino. A reflected L-shaped polyomino is also identified with its original. Under the fixed polyomino convention, translation is still ignored but rotation and reflection are not, so more shapes are distinguished. A shape with its own symmetries need not have eight distinct orientations: the orbit-stabilizer theorem accounts for the repetitions.
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