Polyomino


A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in Fairy Chess Review between the years 1937 to 1957, under the name of "dissection problems." The name polyomino was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 "Mathematical Games" column in Scientific American.
Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes.
In statistical physics, the study of polyominoes and their higher-dimensional analogs is applied to problems in physics and chemistry. Polyominoes have been used as models of branched polymers and of percolation clusters.
Like many puzzles in recreational mathematics, polyominoes raise many combinatorial problems. The most basic is enumerating polyominoes of a given size. No formula has been found except for special classes of polyominoes. A number of estimates are known, and there are algorithms for calculating them.
Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only simply connected polyominoes.

Enumeration of polyominoes

Free, one-sided, and fixed polyominoes

There are three common ways of distinguishing polyominoes for enumeration:
The following table shows the numbers of polyominoes of various types with n cells.
, Iwan Jensen has enumerated the fixed polyominoes up to n = 56; the number of fixed polyominoes with 56 cells is approximately 6.915. Free polyominoes have been enumerated up to n = 28 by Tomás Oliveira e Silva.

Symmetries of polyominoes

The dihedral group D4 is the group of symmetries of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the x-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of D4. However, those images are not necessarily distinct: the more symmetry a free polyomino has, the fewer distinct fixed counterparts it has. Therefore, a free polyomino that is invariant under some or all non-trivial symmetries of D4 may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group D4.
Polyominoes have the following possible symmetries; the least number of squares needed in a polyomino with that symmetry is given in each case:
In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows:
The following table shows the numbers of polyominoes with n squares, sorted by symmetry groups.
nnone mirror mirror C2 D2 D2 C4 D4
100000001
200001000
300101000
411011001
552211001
6206252000
7849743100
8316235184111
91,1963826194002
104,4619022738100
1116,750147917310200
1262,8783417927815333

Algorithms for enumeration of fixed polyominoes

Inductive algorithms

Each polyomino of order n+1 can be obtained by adding a square to a polyomino of order n. This leads to algorithms for generating polyominoes inductively.
Most simply, given a list of polyominoes of order n, squares may be added next to each polyomino in each possible position, and the resulting polyomino of order n+1 added to the list if not a duplicate of one already found; refinements in ordering the enumeration and marking adjacent squares that should not be considered reduce the number of cases that need to be checked for duplicates. This method may be used to enumerate either free or fixed polyominoes.
A more sophisticated method, described by Redelmeier, has been used by many authors as a way of not only counting polyominoes, but also proving upper bounds on their number. The basic idea is that we begin with a single square, and from there, recursively add squares. Depending on the details, it may count each n-omino n times, once from starting from each of its n squares, or may be arranged to count each once only.
The simplest implementation involves adding one square at a time. Beginning with an initial square, number the adjacent squares, clockwise from the top, 1, 2, 3, and 4. Now pick a number between 1 and 4, and add a square at that location. Number the unnumbered adjacent squares, starting with 5. Then, pick a number larger than the previously picked number, and add that square. Continue picking a number larger than the number of the current square, adding that square, and then numbering the new adjacent squares. When n squares have been created, an n-omino has been created.
This method ensures that each fixed polyomino is counted exactly n times, once for each starting square. It can be optimized so that it counts each polyomino only once, rather than n times. Starting with the initial square, declare it to be the lower-left square of the polyomino. Simply do not number any square that is on a lower row, or left of the square on the same row. This is the version described by Redelmeier.
If one wishes to count free polyominoes instead, then one may check for symmetries after creating each n-omino. However, it is faster to generate symmetric polyominoes separately and so determine the number of free polyominoes by Burnside's lemma.

Transfer-matrix method

The most modern algorithm for enumerating the fixed polyominoes was discovered by Iwan Jensen. An improvement on Andrew Conway's method, it is exponentially faster than the previous methods.
Both Conway's and Jensen's versions of the transfer-matrix method involve counting the number of polyominoes that have a certain width. Computing the number for all widths gives the total number of polyominoes. The basic idea behind the method is that possible beginning rows are considered, and then to determine the minimum number of squares needed to complete the polyomino of the given width. Combined with the use of generating functions, this technique is able to count many polyominoes at once, thus enabling it to run many times faster than methods that have to generate every polyomino.
Although it has excellent running time, the tradeoff is that this algorithm uses exponential amounts of memory, is much harder to program than the other methods, and can't currently be used to count free polyominoes.

Asymptotic growth of the number of polyominoes

Fixed polyominoes

Theoretical arguments and numerical calculations support the estimate for the number of fixed polyominoes of size n
where λ = 4.0626 and c = 0.3169. However, this result is not proven and the values of λ and c are only estimates.
The known theoretical results are not nearly as specific as this estimate. It has been proven that
exists. In other words, An grows exponentially. The best known lower bound for λ is 4.00253. The best known upper bound, not improved since the 1970s, is.
To establish a lower bound, a simple but highly effective method is concatenation of polyominoes. Define the upper-right square to be the rightmost square in the uppermost row of the polyomino. Define the bottom-left square similarly. Then, the upper-right square of any polyomino of size n can be attached to the bottom-left square of any polyomino of size m to produce a unique -omino. This proves. Using this equation, one can show for all n. Refinements of this procedure combined with data for An produce the lower bound given above.
The upper bound is attained by generalizing the inductive method of enumerating polyominoes. Instead of adding one square at a time, one adds a cluster of squares at a time. This is often described as adding twigs. By proving that every n-omino is a sequence of twigs, and by proving limits on the combinations of possible twigs, one obtains an upper bound on the number of n-ominoes. For example, in the algorithm outlined above, at each step we must choose a larger number, and at most three new numbers are added. This can be used to obtain an upper bound of 6.75. Using 2.8 million twigs, Klarner and Rivest obtained an upper bound of 4.65.

Free polyominoes

Approximations for the number of fixed polyominoes and free polyominoes are related in a simple way. A free polyomino with no symmetries corresponds to 8 distinct fixed polyominoes, and for large n, most n-ominoes have no symmetries. Therefore, the number of fixed n-ominoes is approximately 8 times the number of free n-ominoes. Moreover, this approximation is exponentially more accurate as n increases.

Special classes of polyominoes

Exact formulas are known for enumerating polyominoes of special classes, such as the class of convex polyominoes and the class of directed polyominoes.
The definition of a convex polyomino is different from the usual definition of convexity, but is similar to the definition used for the orthogonal convex hull. A polyomino is said to be vertically or column convex if its intersection with any vertical line is convex. Similarly, a polyomino is said to be horizontally or row convex if its intersection with any horizontal line is convex. A polyomino is said to be convex if it is row and column convex.
A polyomino is said to be directed if it contains a square, known as the root, such that every other square can be reached by movements of up or right one square, without leaving the polyomino.
Directed polyominoes, column convex polyominoes, and convex polyominoes have been effectively enumerated by area n, as well as by some other parameters such as perimeter, using generating functions.
A polyomino is equable if its area equals its perimeter. An equable polyomino must be made from an even number of squares; every even number greater than 15 is possible. For instance, the 16-omino in the form of a 4 × 4 square and the 18-omino in the form of a 3 × 6 rectangle are both equable. For polyominoes with fewer than 15 squares, the perimeter always exceeds the area.

Tiling with polyominoes

In recreational mathematics, challenges are often posed for tiling a prescribed region, or the entire plane, with polyominoes, and related problems are investigated in mathematics and computer science.

Tiling regions with sets of polyominoes

Puzzles commonly ask for tiling a given region with a given set of polyominoes, such as the 12 pentominoes. Golomb's and Gardner's books have many examples. A typical puzzle is to tile a 6×10 rectangle with the twelve pentominoes; the 2339 solutions to this were found in 1960. Where multiple copies of the polyominoes in the set are allowed, Golomb defines a hierarchy of different regions that a set may be able to tile, such as rectangles, strips, and the whole plane, and shows that whether polyominoes from a given set can tile the plane is undecidable, by mapping sets of Wang tiles to sets of polyominoes.

Tiling regions with copies of a single polyomino

Another class of problems asks whether copies of a given polyomino can tile a rectangle, and if so, what rectangles they can tile. These problems have been extensively studied for particular polyominoes, and tables of results for individual polyominoes are available. Klarner and Göbel showed that for any polyomino there is a finite set of prime rectangles it tiles, such that all other rectangles it tiles can be tiled by those prime rectangles. Kamenetsky and Cooke showed how various disjoint polyominoes can tile rectangles.
Beyond rectangles, Golomb gave his hierarchy for single polyominoes: a polyomino may tile a rectangle, a half strip, a bent strip, an enlarged copy of itself, a quadrant, a strip, a half plane, the whole plane, certain combinations, or none of these. There are certain implications among these, both obvious and less so. Polyominoes of orders up to 6 are characterized in this hierarchy.
In 2001 Cristopher Moore and John Michael Robson showed that the problem of tiling one polyomino with copies of another is NP-complete.

Tiling the plane with copies of a single polyomino

Tiling the plane with copies of a single polyomino has also been much discussed. It was noted in 1965 that all polyominoes up to hexominoes and all but four heptominoes tile the plane. It was then established by David Bird that all but 26 octominoes tile the plane. Rawsthorne found that all but 235 polyominoes of order 9 tile, and such results have been extended to higher orders by Rhoads and others. Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects in which the tiles appear in them.
The study of which polyominoes can tile the plane has been facilitated using the Conway criterion: except for two nonominoes, all tiling polyominoes up to order 9 form a patch of at least one tile satisfying it, with higher-order exceptions more frequent.
Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a rep-tile tiling of the plane. For instance, for every positive integer, it is possible to combine copies of the L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed.

Tiling a common figure with various polyominoes

The compatibility problem is to take two or more polyominoes and find a figure that can be tiled with each. Polyomino compatibility has been widely studied since the 1990s. Jorge Luis Mireles and Giovanni Resta have published websites of systematic results, and Livio Zucca shows results for some complicated cases like three different pentominoes. The general problem can be hard. The first compatibility figure for the L and X pentominoes was published in 2005 and had 80 tiles of each kind. Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible.

Polyominoes in puzzles and games

In addition to the tiling problems described above, there are recreational mathematics puzzles that require folding a polyomino to create other shapes. Gardner proposed several simple games with a set of free pentominoes and a chessboard. Some variants of the Sudoku puzzle use nonomino-shaped regions on the grid. The video game Tetris is based on the seven one-sided tetrominoes, and the board game Blokus uses all of the free polyominoes up to pentominoes.

Etymology

The word polyomino and the names of the various orders of polyomino are all back-formations from the word domino, a common game piece consisting of two squares, with the first letter d- fancifully interpreted as a version of the prefix di- meaning "two." The name domino for the game piece is believed to come from the spotted masquerade garment domino, from Latin dominus.
Most of the numerical prefixes are Greek. Polyominoes of order 9 and 11 more often take the Latin prefixes nona- and undeca- than the Greek prefixes ennea- and hendeca-.

Online polyomino solvers

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