# Definition:Polyomino

## Definition

A **polyomino** is a configuration of $n$ unit squares, for a given (strictly) positive integer $n$, which are placed side by side with vertices touching, to form a plane figure.

A **polyomino** of $n$ squares can be referred to as an **$n$-omino**.

### Free Polyomino

A **free polyomino** is a polyomino which is not distinguished from its image under a reflection.

That is, it is considered **free** of the plane in which it is embedded, and can be "lifted up and turned over".

### Fixed Polyomino

A **fixed polyomino** is a polyomino which *is* distinguished from its image under a reflection.

That is, it is considered **fixed** in the plane in which it is embedded, and (while it may be translated and rotated in the plane) *cannot* be "lifted up and turned over".

## Also known as

An **$n$-omino** can also be referred to by the more unwieldy name **$n$-polyomino**.

## Also see

For small $n$, **polyominoes** have their own names:

- $n = 1$: Monomino
- $n = 2$: Domino
- $n = 3$: Tromino
- $n = 4$: Tetromino
- $n = 5$: Pentomino
- $n = 6$: Hexomino
- $n = 7$: Heptomino
- $n = 8$: Octomino

- Results about
**polyominoes**can be found**here**.

## Linguistic Note

The word **polyomino** is derived by back-formation from the word **domino**, which can be defined as a polyomino formed from $2$ unit squares.

Thus a **polyomino** is a **domino** which is generalised for $n$ unit squares.

The plural of **polyomino** is **polyominoes**.

## Sources

- Weisstein, Eric W. "Polyomino." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Polyomino.html