Polyominoes
Polyominoes are plane geometric figures formed by joining one or more equal squares edge to edge. They are a type of polyform, a geometric figure formed by joining together identical basic shapes. Polyominoes have a rich mathematical structure and numerous applications in tiling puzzles, recreational mathematics, and even computer science.
Polyominoes are classified based on the number of squares they contain. Below is a table summarizing the different types of polyominoes, including the number of distinct shapes for each type.
| n | name | free | one-sided | fixed | total | with holes | without holes |
|---|---|---|---|---|---|---|---|
| 1 | monomino | 1 | 0 | 1 | 1 | 1 | 1 |
| 2 | domino | 1 | 0 | 1 | 1 | 1 | 2 |
| 3 | tromino | 2 | 0 | 2 | 2 | 2 | 6 |
| 4 | tetromino | 5 | 0 | 5 | 7 | 7 | 19 |
| 5 | pentomino | 12 | 0 | 12 | 18 | 18 | 63 |
| 6 | hexomino | 35 | 0 | 35 | 60 | 60 | 216 |
| 7 | heptomino | 108 | 1 | 107 | 196 | 196 | 760 |
| 8 | octomino | 369 | 6 | 363 | 704 | 704 | 2,725 |
| 9 | nonomino | 1,285 | 37 | 1,248 | 2,500 | 2,500 | 9,910 |
| 10 | decomino | 4,655 | 195 | 4,460 | 9,189 | 9,189 | 36,446 |
| 11 | undecomino | 17,073 | 979 | 16,094 | 33,896 | 33,896 | 135,268 |
| 12 | dodecomino | 63,600 | 4,663 | 58,937 | 126,759 | 126,759 | 505,861 |
A monomino is the simplest polyomino, consisting of just one square.
A domino consists of two squares joined edge to edge. It is the second simplest polyomino and is often used in tiling puzzles.
A tromino is made up of three squares. There are two distinct types of trominoes:
- Straight tromino: A straight line of three squares.
- L-shaped tromino: Three squares forming an "L" shape.
Tetrominoes are made up of four squares. The five free tetrominoes include:
- I-tetromino: A straight line of four squares.
- O-tetromino: A 2x2 block of four squares.
- T-tetromino: A T-shaped figure.
- S-tetromino: A zigzag shape.
- Z-tetromino: Another zigzag shape, mirrored from the S-tetromino.
Pentominoes consist of five squares. There are 12 distinct free pentominoes, each resembling a different letter or shape, such as:
- F, I, L, P, T, U, V, W, X, Y, Z, and N.
As the number of squares increases, the complexity and the number of distinct shapes increase exponentially. Hexominoes (6 squares), heptominoes (7 squares), and so on, follow similar naming conventions.
Tiling, or tessellation, is the process of covering a plane using one or more geometric shapes with no overlaps and no gaps. Tiling with polyominoes involves arranging these shapes in a way that completely covers a region, often a rectangle or other regular shapes.
- Rectangular Tiling: Covering a rectangular region using a given set of polyominoes.
- Checkerboard Tiling: Using dominoes to tile a checkerboard pattern.
- Puzzle Solving: Solving puzzles where specific shapes must fit into a defined area without overlapping.
Polyominoes have a wide range of applications, including:
- Puzzle games: Such as Tetris, which uses tetrominoes.
- Mathematical research: Studying the properties and behaviors of polyominoes.
- Computer algorithms: Developing algorithms for tiling and pattern recognition.
Polyominoes offer a fascinating glimpse into the world of geometric shapes and their applications. Whether used for recreational puzzles or serious mathematical research, they continue to captivate and challenge enthusiasts and professionals alike.