Polyominoes

Polyominoes are plane geometric figures formed by joining one or more equal squares edge to edge on the square lattice. Each square shares a full edge with at least one other square, so the figure is connected by edge-adjacency. The term was coined by Solomon W. Golomb in 1953, from poly- meaning many and -omino from domino. Common families include monomino (1 square), domino (2), tromino (3), tetromino (4), and pentomino (5).

Classification: Polyominoes can be classified according to how symmetries are treated. Free polyominoes identify shapes up to translations, rotations, and reflections; one-sided polyominoes identify shapes up to translations and rotations but not reflections; fixed polyominoes are distinguished only by translation (no movement allowed).

Research and applications: Polyominoes are studied in combinatorics and recreational mathematics. Topics include enumeration (counting distinct shapes for a given number of squares), tiling problems (covering a region with copies of a polyomino or with a set of polyominoes), and efficient generation algorithms. The classic puzzle of tiling a rectangle with all twelve pentominoes is a well-known example; many rectangle sizes admit pentomino tilings, found by computational searches. In theory, tiling relates to exact cover problems and computational complexity.

Extensions and related ideas: The study also encompasses lattice animals, polyomino graphs, and related tilings in higher dimensions.