Editing Polyomino (section)

{{Short description|Geometric shapes formed from squares}}
{{Redirect|Polyominoes|the book by Solomon Golomb|Polyominoes: Puzzles, Patterns, Problems, and Packings}}
[[Image:All 18 Pentominoes.svg|thumb|The 18 one-sided [[pentomino]]es, including 6 mirrored pairs.]]

A '''polyomino''' is a [[Shape|plane geometric figure]] formed by joining one or more equal [[square]]s 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 [[puzzle]]s since at least 1907, and the enumeration of [[pentomino]]es is dated to antiquity.<ref>Golomb (''Polyominoes'', Preface to the First Edition) writes "the observation that there are twelve distinctive patterns (the pentominoes) that can be formed by five connected stones on a [[Go (game)|Go]] board ... is attributed to an ancient master of that game".</ref> Many results with the pieces of 1 to 6 squares were first published in ''[[Fairy Chess Review]]'' between the years 1937 and 1957, under the name of "dissection problems." The name ''polyomino'' was invented by [[Solomon W. Golomb]] in 1953,<ref>{{cite book |last=Golomb |first=Solomon W. |author-link=Solomon W. Golomb |title=Polyominoes |title-link= Polyominoes: Puzzles, Patterns, Problems, and Packings |year=1994 |publisher=Princeton University Press |location=Princeton, New Jersey |isbn=978-0-691-02444-8 |edition=2nd}}</ref> and it was popularized by [[Martin Gardner]] in a November 1960 "[[Mathematical Games (column)|Mathematical Games]]" column in ''[[Scientific American]]''.<ref>{{cite journal |last1=Gardner |first1=M. |title=More about the shapes that can be made with complex dominoes (Mathematical Games) |journal=Scientific American |date=November 1960 |volume=203 |issue=5 |pages=186–201 |jstor=24940703 |doi=10.1038/scientificamerican1160-186 }}</ref>

Related to polyominoes are [[polyiamond]]s, formed from [[equilateral triangle]]s; [[polyhex (mathematics)|polyhexes]], formed from regular [[hexagon]]s; and other plane [[polyform]]s. Polyominoes have been generalized to higher [[dimension]]s by joining [[cube (geometry)|cubes]] to form [[polycube]]s, or [[hypercube]]s 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 [[Branching (polymer chemistry)|branched polymers]] and of [[percolation]] clusters.<ref>{{cite book |last1=Whittington |first1=S. G. |last2=Soteros| first2=C. E.|author2-link=Chris Soteros |editor-last1=Grimmett|editor-first1=G.|editor-last2=Welsh|editor-first2=D.|title=Disorder in Physical Systems|chapter=Lattice Animals: Rigorous Results and Wild Guesses |year=1990 |publisher=Oxford University Press }}</ref>

Like many puzzles in [[recreational mathematics]], polyominoes raise many [[combinatorial]] problems. The most basic is [[enumeration|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 [[algorithm]]s 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.<ref>{{cite book |last=Grünbaum |first=Branko |author-link=Branko Grünbaum |author2=Shephard, G.C. |title=Tilings and Patterns |location=New York |publisher=W.H. Freeman and Company |year=1987 |isbn=978-0-7167-1193-3 |url-access=registration |url=https://archive.org/details/isbn_0716711931 }}</ref>
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