A polyominoMathworldPlanetmath consists of a number of identical connected squares placed in distinct locations in the plane so that at least one side of each square is adjacent to (i.e. completely coincides with the side of) another square (if the polyomino consists of at least two squares).

A polyomino with n squares is called an n-omino. For small n, polyominoes have special names. A 1-omino is called a monomino, a 2-omino a domino, a 3-omino a tromino or triomino, etc. The famous Tetris video game derives its name from the fact that the bricks are tetrominoes or 4-ominoes.

FigureĀ 1: All distinct 1-, 2-, 3-, 4-, and 5-ominoes. Pentominoes have been scaled in the figure to fit on the page.

Fixed polyominoes (which are also called lattice animals) are considered distinct if they cannot be translated into each other, while free polyominoes must also be distinct under rotationMathworldPlanetmath and reflection.

FigureĀ 2: All distinct, fixed dominoes and trominoes.

The topic of how many distinct (free or fixed) n-ominoes exist for a given n has been the subject of much research. It is known that the number of free n-ominoes An grows exponentially. More precisely, it can be proven that 3.72n<An<4.65n.

Polyominoes are special instances of polyforms.

Title polyomino
Canonical name Polyomino
Date of creation 2013-03-22 15:20:18
Last modified on 2013-03-22 15:20:18
Owner s0 (9826)
Last modified by s0 (9826)
Numerical id 10
Author s0 (9826)
Entry type Definition
Classification msc 05B50
Defines n-omino
Defines domino
Defines tromino
Defines tetromino
Defines fixed polyomino
Defines lattice animal