Home > Polyomino
A polyomino is a polyform with the square as its base form. It is constructed by placing a number of identical squares in distinct locations on the plane in such a way that at least one edge of each square coincides with an edge of one of the remaining squares, and all squares are simply connected. If the corner of any square touches the edge of another square at any place other than a corner, the result is not a polyomino. For each positive natural number, n, there are a finite number of distinct such arrangements of the n squares. There are two common ways of defining distinct polyominoes: free polyominoes must be different under translation, rotation, and reflection, while fixed polyominoes must be different only under translation. Therefore, there are always fewer free polyominoes than fixed polyominoes.In some contexts the definition of a polyomino is relaxed. Sometimes polyominoes are generalised to three or more dimensions by aggregating cubes or hypercubes. For some applications mirror image forms are treated as distinct.
The numbers of configurations for free polyominoes are listed below.
- 1 square: monomino: 1 configuration
- 2 squares: domino: 1
- 3 squares: triomino or tromino: 2
- 4 squares: tetromino: 5
- 5 squares: pentomino: 12
- 6 squares: hexominoA hexomino is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. As with other polyominoes, rotations and reflections of a hexomino are not considered to be distinct shapes and with this conventio: 35
- 7 squares: heptomino : 108 (including 1 configuration with a hole)
- 8 squares: octomino: 369 (including 6 with holes)
- 9 squares: nonomino: 1285 (37 with holes)
- 10 squares: decomino: 4655 (195 with holes)
- 11 squares: undecomino: 17073 (979 with holes)
- 12 squares: dodecomino: 63600 (4663 with holes)
These numbers are SloaneNeil James Alexander Sloane is a US-American mathematician. He received his Ph. from Cornell University in 1967 and joined AT&T Bell Labs in 1968. He became an AT&T Fellow in 1998. His major contributions are in the fields of sphere packing, error-correct's A000105 (for all configurations) and A000104 (for configurations without holes).
No formula for finding the exact number of polyominoes with n squares has been found. However, a number of estimates are known, and there are algorithmFlowcharts were often used to represent algorithms. An algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will result in a corresponding recognisable end-state (contrast with heuristic). Algors for counting them; see below for more information.
Polyominoes composed of seven or more squares may contain holes (that is, regions which are not tiled with squares but which are unconnected to the exterior of the polyomino).
Polyominoes have been used in popular puzzles since the late 19th century, but were first studied systematically by Solomon W. Golomb and were popularized by Martin GardnerMartin Gardner (born October 21, 1914) is an American recreational mathematician and author of the long-running but now discontinued Mathematical Games' column in Scientific American. Interests and writings Born in Tulsa, Oklahoma, Martin Gardner more or. Related to polyominoes are polyiamondA polyiamond is a counterpart to a polyomino where the polygon used as the building block is an equilateral triangle rather than a square. The word polyiamond is a back-formation from diamond motivated by the fact that this word is often used to describes (formed from equilateral triangles) and polyhexIn recreational mathematics, a polyhex is a polyform with a regular hexagon (or 'hex' for short) as the base form. The number of polyhexes made from a given number of hexagons: 1. 1 2. 1 3. 3 4. 7 5. 22 6. 81 7. 331 8. 1,435 9. 6,505 10. 30,086 11. 141,22es (formed from regular hexagon s). The game TetrisNintendo Game Boy Tetris is a video game invented by Alexey Pajitnov whilst he was working for the Academy of Sciences in Moscow, inspired by a pentominoes game he had purchased earlier. The game Tetrominoes or tetrads, shapes composed of 4 blocks each, a is based tetrominoes.
