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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, keeping the shape connected, and in such a way that at least one edge of each square coincides with an edge of one of the other squares. Polyominoes with from 1 to 6 squares are called respectively monominoes, dominoes, trominoes (or triominoes), tetrominoes, pentominoes and hexominoes. Related to polyominoes are polyiamonds (formed from equilateral triangles), polyhexes (formed from regular hexagons), and other polyforms.
In some contexts, the definition of a polyomino is relaxed or refined. Sometimes it is requested that polyominoes are simply connected, while on other occasions may have holes (in other words, regions which are not tiled with squares but which are unconnected to the exterior of the polyomino). Sometimes polyominoes are generalised to three or more dimensions by aggregating cubes or hypercubes.
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 Gardner. The game Tetris is based on tetrominoes.
Polyominoes are a source of combinatorial problems, the first being to enumerate polyominoes for various sizes. No formula has been found except for special classes of polyominoes. However, a number of estimates are known, and there are algorithms for counting them.
Free and fixed polyominoes
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.
Let D8 be 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 fixed polyomino corresponds to at most 8 free polyominoes, which are its symmetric images under the symmetries of D8. However, those images are not necessarily distinct. Therefore, a fixed polyomino which is invariant under some or all non-trivial symmetries of D8 may correspond to only 4, 2 or 1 free polyominoes. Mathematically, fixed polyominoes are equivalence classes of free polyominoes under the group D8.
Number of polyominoes
We call n the number of squares, and An the number of fixed polyominoes with n squares (possibly with holes). An enumeration gives the following table:
|n||name||number of free polyominoes||number of free polyominoes with holes||An = number of fixed polyominoes|
|3||tromino or triomino||2||0||6|
As of 2004, Iwan Jensen has enumerated the fixed polyominoes up to n=56: A56 is approximately 6.915×1031. Free polyominoes have been enumerated up to n=28. See the external links for tables containing the known results.
Algorithms for enumeration of fixed polyominoes
Inductive exhaustive search
The most obvious method of enumerating the polyominoes, and also one of the slowest, is inductive exhaustive search. Given a list of polyominoes of area n, take each polyomino in turn, embed it in an n×n square, surround that square with a collar of cells to create an (n+2)×(n+2) square. For each vacant cell in that square that is adjacent to at least one occupied cell, fill the cell and strike out a bounding row of vacant cells and a bounding column of vacant cells. The resulting (n+1)×(n+1) square contains a candidate polyomino of area n+1. If this configuration has not been encountered before, it is added to the list of polyominoes of area n+1. Comparison with the polyominoes of area n+1 already seen must take account of position and symmetry, depending on whether fixed or free polyominoes are being counted. Position can be accounted for by translating the candidate polyomino to the top left corner of the (n+1)×(n+1) square. In order to compute the number of fixed polyominoes, rotations and reflections must also be accounted for.
This procedure can be applied repeatedly starting from the monomino to reach any desired area of polyomino, but this becomes computationally expensive for large areas. For example finding all the dodecominoes using this algorithm consumes nearly 90 seconds of CPU time on a 1 GHz Pentium.
This method 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. It will count each n-omino n times, once from starting from each of its n squares.
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. If one wishes to count free polyominoes instead, then one must check for symmetries after creating each n-omino. This algorithm can enumerate the dodecominoes in about 20 seconds on a 1 GHz Pentium. The running time scales proportionally to the number of polyominoes.
This method 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 which is on a lower row, or left of the square on the same row. With this improvement, the running time is divided by n, so it only takes about 1 second to enumerate the dodecominoes.
Conway's method and Jensen's 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 (however its running time is still exponential on n).
Both Conway's and Jensen's 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 (many gigabytes of memory are needed for n above 50), is much harder to program than the other methods, and cannot currently be used to count free polyominoes.
Asymptotic growth of the number of polyominoes
Theoretical arguments and numerical calculations support the estimate
where λ = 4.0626 and c = 0.3024. However, it should be emphasized that 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 3.90. The best known upper bound, not improved since the 1970s, is λ < 4.65.
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 (n+m)-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 Growth Method of enumerating polyominos. 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 (since at most three unnumbered squares are adjacent to any numbered square). 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. Undoubtedly, modern computers could carry the calculation to larger twigs, but no better results have been published.
Approximations for the number of fixed polyominoes and free polyominoes are related in a simple way. A free polyomino with no symmetries (rotation or reflection) corresponds to 8 distinct fixed polyominoes, and, for large n, most polyominoes have no symmetries. Therefore, the number of fixed polyominoes is approximately 8 times the number of free polyominoes. 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. A polyomino is said to be column convex if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be 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 polymino 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 (or row) convex polyominos, and convex polyominoes have been effectively enumerated for all n using generating functions.
- Karl Dahlke's polyomino finite-rectangle tilings
- Enumeration of polyominoes
- Enumeration of fixed polyominoes up to n=56
- An implementation and description of Jensen's method
- Klarner and Rivest's paper (PDF)
- A paper describing modern estimates (PS)
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