polyominos

Polyominoes are plane figures formed by joining one or more identical squares edge to edge. An n-omino is a polyomino composed of n squares. The term, coined by Solomon W. Golomb in the 1950s, brings together the study of these shapes in combinatorics and recreational mathematics.

Shapes are classified according to how they are considered equivalent. Fixed polyominoes regard two shapes as the same only if they differ by a translation; one-sided polyominoes treat rotations as allowing equivalence but reflections as distinct; free polyominoes identify shapes that can be rotated or reflected to match.

Most familiar examples include dominoes (n=2), trominoes (n=3), and tetrominoes (n=4). In popular culture, tetrominoes are well known from the game Tetris, which uses seven distinct shapes (I, O, T, S, Z, J, L); under symmetry there are five free tetromino types.

A central area of polyomino study is tiling and packing. A region is tiled by copies of a given polyomino if it can be covered without gaps or overlap. Classic results appear in tromino tilings of certain boards and pentomino tilings of rectangles such as 6×10 or 5×12. More generally, researchers investigate exact cover problems and algorithmic methods for constructing tilings with various polyominoes.

Enumeration and growth are also key topics. For each n, there are finitely many n-ominoes up to the chosen equivalence, and their numbers grow rapidly with n. Counts are known for small n and are tabulated in mathematical literature, with ongoing work in generating functions, tiling capabilities, and computational enumeration.