A finite connected union of unit squares on the square lattice, joined along whole sides. Counting conventions may identify shapes under translation, rotation, or reflection, so those conventions must be specified.
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.
A fixed polyomino is a polyomino considered up to translation only. Moving a finite side-connected square-lattice shape does not change it, but rotating or reflecting it may produce a different fixed shape.
There are two fixed dominoes, horizontal and vertical, although they form one free polyomino class. A fixed square tetromino remains the same under quarter-turn rotation because it already has that symmetry. Thus the word fixed records which transformations are identified, rather than prohibiting a shape from having symmetries.
One can represent a fixed polyomino by a finite set of lattice cells and translate it so the minimum horizontal and vertical coordinates are both zero. This removes positional redundancy for enumeration; connectivity along whole cell sides remains part of the polyomino definition.
A tromino is a polyomino made of three unit squares joined along whole sides. Up to translation, rotation and reflection, its two shapes are the straight three-square strip and the L shape. Their area is three, so a board tiled by trominoes must have area divisible by three; that condition alone need not ensure a board tiling. For example, a rectangle admits two straight trominoes.
