polyomino

A polyomino 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.

Figure 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 rotation and reflection.

Figure 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 $A_n$ grows exponentially. More precisely, it can be proven that .

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