Inspiring popular video games like Tetris while contributing to the study of combinatorial geometry and tiling theory, polyominoes have continued to spark interest ever since their inventor, Solomon Golomb, introduced them to puzzle enthusiasts several decades ago. In this fully revised and expanded edition of his landmark book, the author takes a new generation of readers on a mathematical journey into the world of the deceptively simple polyomino. Golomb incorporates important, recent developments, and poses problems, inviting the reader to play with and develop an understanding of the extraordinary properties of polyominoes. Polyominoes : Puzzles, Patterns, Problems, and Packings.
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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 , and the enumeration of pentominoes is dated to antiquity.
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 which are often referred to as lattice animals in this literature 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. There are three common ways of distinguishing polyominoes for enumeration: [6] [7]. The dihedral group D 4 is the group of symmetries symmetry group 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 D 4. 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 D 4 may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group D 4.
Polyominoes have the following possible symmetries; [10] the least number of squares needed in a polyomino with that symmetry is given in each case:. The following table shows the numbers of polyominoes with n squares, sorted by symmetry groups.
This leads to algorithms for generating polyominoes inductively. 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 [12] to generate symmetric polyominoes separately by a variation of this method [13] and so determine the number of free polyominoes by Burnside's lemma.
The most modern algorithm for enumerating the fixed polyominoes was discovered by Iwan Jensen. 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 many gigabytes of memory are needed for n above 50 , is much harder to program than the other methods, and can't currently be used to count free polyominoes.
Theoretical arguments and numerical calculations support the estimate for the number of fixed polyominoes of size n. The known theoretical results are not nearly as specific as this estimate. It has been proven that. In other words, A n grows exponentially. 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.
Refinements of this procedure combined with data for A n 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 since at most three unnumbered squares are adjacent to any numbered square.
This can be used to obtain an upper bound of 6. Using 2. 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 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. 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 in other words, each column has no holes. 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, [20] column or row convex polyominoes, [21] and convex polyominoes [22] 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 polyominoes with fewer than 15 squares, the perimeter always exceeds the area. In recreational mathematics , challenges are often posed for tiling a prescribed region, or the entire plane, with polyominoes, [24] and related problems are investigated in mathematics and computer science. 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. Another class of problems asks whether copies of a given polyomino can tile a rectangle , and if so, what rectangles they can tile. 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 for example, if a polyomino tiles the half plane then it tiles the whole plane and less so for example, if a polyomino tiles an enlarged copy of itself, then it tiles the quadrant. Polyominoes of orders up to 6 are characterized in this hierarchy with the status of one hexomino, later found to tile a rectangle, unresolved at that time. In 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 has also been much discussed. It was noted in that all polyominoes up to hexominoes [36] and all but four heptominoes tile the plane.
Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects orientations 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 n , it is possible to combine n 2 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. 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 s. Jorge Luis Mireles and Giovanni Resta have published websites of systematic results, [45] [46] and Livio Zucca shows results for some complicated cases like three different pentominoes. The first compatibility figure for the L and X pentominoes was published in and had 80 tiles of each kind.
No algorithm is known for deciding whether two arbitrary polyominoes are compatible. 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 spelled "Tetriminos" in the game , and the board game Blokus uses all of the free polyominoes up to pentominoes.
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Polyominoes : Puzzles, Patterns, Problems, and Packings - Revised and Expanded Second Edition
Solomon W. Many of our ebooks are available through library electronic resources including these platforms:. Inspiring popular video games like Tetris while contributing to the study of combinatorial geometry and tiling theory, polyominoes have continued to spark interest ever since their inventor, Solomon Golomb, introduced them to puzzle enthusiasts several decades ago. In this fully revised and expanded edition of his landmark book, the author takes a new generation of readers on a mathematical journey into the world of the deceptively simple polyomino. And if you haven't yet discovered the fascination of these tantalizing tiles, this is the book to introduce you to them. Polyominoes Solomon W.
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Solomon W. Golomb
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 , and the enumeration of pentominoes is dated to antiquity. 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 which are often referred to as lattice animals in this literature is applied to problems in physics and chemistry.
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