Domino (mathematics)
{{Short description|Geometric shape formed from two squares}}
{{About|the mathematical polygon|the game|dominoes}}
In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge.{{cite book |last=Golomb |first=Solomon W. |authorlink=Solomon W. Golomb |title=Polyominoes |title-link= Polyominoes: Puzzles, Patterns, Problems, and Packings |year=1994 |publisher=Princeton University Press |location=Princeton, New Jersey |isbn=0-691-02444-8 |edition=2nd }} When rotations and reflections are not considered to be distinct shapes, there is only one free domino.
Since it has reflection symmetry, it is also the only one-sided domino (with reflections considered distinct). When rotations are also considered distinct, there are two fixed dominoes: The second one can be created by rotating the one above by 90°.{{cite web| last=Weisstein |first= Eric W |url=http://mathworld.wolfram.com/Domino.html |title=Domino |publisher=From MathWorld – A Wolfram Web Resource |accessdate=2009-12-05}}{{cite journal |last=Redelmeier |first=D. Hugh |year=1981 |title=Counting polyominoes: yet another attack |journal=Discrete Mathematics |volume=36 |issue=2 |pages=191–203 |doi=10.1016/0012-365X(81)90237-5|doi-access=free }}
In a wider sense, the term domino is sometimes understood to mean a tile of any shape.{{cite journal | last = Berger|first = Robert|title = The undecidability of the Domino Problem| journal = Memoirs Am. Math. Soc. | volume = 66 | year = 1966}}
Packing and tiling
{{Main|Domino tiling}}
Dominos can tile the plane in a countably infinite number of ways. The number of tilings of a 2×n rectangle with dominoes is , the nth Fibonacci number.[http://www-cs-faculty.stanford.edu/~knuth/gkp.html Concrete Mathematics] {{Webarchive|url=https://web.archive.org/web/20201106232418/http://www-cs-faculty.stanford.edu/~knuth/gkp.html |date=2020-11-06 }} by Graham, Knuth and Patashnik, Addison-Wesley, 1994, p. 320, {{ISBN|0-201-55802-5}}
Domino tilings figure in several celebrated problems, including the Aztec diamond problem in which large diamond-shaped regions have a number of tilings equal to a power of two,{{Citation | last1=Elkies | first1=Noam | author1-link = Noam Elkies | last2=Kuperberg | first2=Greg | author2-link = Greg Kuperberg | last3=Larsen | first3=Michael | author3-link = Michael J. Larsen | last4=Propp | first4=James | author4-link = Jim Propp | title=Alternating-sign matrices and domino tilings. I | doi=10.1023/A:1022420103267 | mr=1226347 | year=1992 | journal=Journal of Algebraic Combinatorics | volume=1 | issue=2 | pages=111–132| doi-access=free }} with most tilings appearing random within a central circular region and having a more regular structure outside of this "arctic circle", and the mutilated chessboard problem, in which removing two opposite corners from a chessboard makes it impossible to tile with dominoes.{{citation|doi=10.2307/4146865|title=Tiling with dominoes|first=N. S.|last=Mendelsohn|journal=The College Mathematics Journal|volume=35|issue=2|year=2004|pages=115–120|publisher=Mathematical Association of America|jstor=4146865}}.