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| [[Image:Pavage domino.svg|thumb|right|Domino tiling of a square]]
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| A '''domino tiling''' of a region in the [[Euclidean plane]] is a [[tessellation]] of the region by [[domino (mathematics)|domino]]s, shapes formed by the union of two [[unit square]]s meeting edge-to-edge. Equivalently, it is a [[Matching (graph theory)|matching]] in the [[grid graph]] formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.
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| == Height functions ==
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| For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the nodes of the grid. For instance, draw a chessboard, fix a node <math>A_0</math> with height 0, then for any node there is a path from <math>A_0</math> to it. On this path define the height of each node <math>A_{n+1}</math> (i.e. corners of the squares) to be the height of the previous node <math>A_n</math> plus one if the square on the right of the path from <math>A_n</math> to <math>A_{n+1}</math> is black, and minus one otherwise.
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| More details can be found in {{harvtxt|Kenyon|Okounkov|2005}}.
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| ==Thurston's height condition==
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| [[William Thurston]] (1990) describes a test for determining whether a simply-connected region, formed as the union of unit squares in the plane, has a domino tiling. He forms an [[undirected graph]] that has as its vertices the points (''x'',''y'',''z'') in the three-dimensional [[integer lattice]], where each such point is connected to four neighbors: if ''x''+''y'' is even, then (''x'',''y'',''z'') is connected to (''x''+1,''y'',''z''+1), (''x''-1,''y'',''z''+1), (''x'',''y''+1,''z''-1), and (''x'',''y''-1,''z''-1), while if ''x''+''y'' is odd, then (''x'',''y'',''z'') is connected to (''x''+1,''y'',''z''-1), (''x''-1,''y'',''z''-1), (''x'',''y''+1,''z''+1), and (''x'',''y''-1,''z''+1). The boundary of the region, viewed as a sequence of integer points in the (''x'',''y'') plane, lifts uniquely (once a starting height is chosen) to a path in this [[three-dimensional graph]]. A necessary condition for this region to be tileable is that this path must close up to form a simple closed curve in three dimensions, however, this condition is not sufficient. Using more careful analysis of the boundary path, Thurston gave a criterion for tileability of a region that was sufficient as well as necessary.
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| ==Counting tilings of regions==
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| [[Image:Dominoes tiling 8x8.svg|thumb|right|Domino tiling of an 8×8 square using the minimum number of long-edge-to-long-edge pairs (1 pair in the center). This arrangement is also a valid [[Tatami]] tiling of an 8x8 square, with no four dominoes touching at an internal point.]]
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| The number of ways to cover an <math> m \times n </math> rectangle with <math> \frac{mn}{2} </math> dominoes, calculated independently by {{harvtxt|Temperley|Fisher|1961}} and {{harvtxt|Kasteleyn|1961}}, is given by
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| :<math> \prod_{j=1}^m \prod_{k=1}^n \left ( 4\cos^2 \frac{\pi j}{m + 1} + 4\cos^2 \frac{\pi k}{n + 1} \right )^\frac{1}{4},</math>
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| which is equivalent to
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| :<math> \prod_{j=1}^{\lceil\frac{m}{2}\rceil} \prod_{k=1}^{\lceil\frac{n}{2}\rceil} \left ( 4\cos^2 \frac{\pi j}{m + 1} + 4\cos^2 \frac{\pi k}{n + 1} \right ).</math>
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| A special case occurs when either ''m'' (or symmetrically ''n'') is set to 2: the sequence reduces to the [[Fibonacci sequence]] {{OEIS|id=A000045}} {{harv|Klarner|Pollack|1980}}.
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| Another special case happens for squares with ''m'' = ''n'' = 0, 2, 4, 6, 8, 10, 12, ... is
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| :1, 2, 36, 6728, 12988816, 258584046368, 53060477521960000, ... {{OEIS|id=A004003}}.
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| These numbers can be found by writing them as the [[Pfaffian]] of an <math>mn \times mn</math> [[skew-symmetric matrix]] whose [[eigenvalue]]s can be found explicitly. This technique may be applied in many mathematics-related subjects, for example, in the classical, 2-dimensional computation of the [[dimer-dimer correlator function]] in [[statistical mechanics]].
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| The number of tilings of a region is very sensitive to boundary conditions, and can change dramatically with apparently insignificant changes in the shape of the region. This is illustrated by
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| the number of tilings of an [[Aztec diamond]] of order ''n'', where the number of tilings is 2<sup>(''n'' + 1)''n''/2</sup>. If this is replaced by the "augmented Aztec diamond" of order ''n'' with 3 long rows in the middle rather than 2, the
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| number of tilings drops to the much smaller number D(''n'',''n''), a [[Delannoy number]], which has only exponential rather than [[super-exponential growth]] in ''n''. For the "reduced Aztec diamond" of order ''n'' with only one
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| long middle row, there is only one tiling.
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| <gallery>
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| File:Diamant azteque.svg|An Aztec diamond of order 4, with 1024 domino tilings
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| File:Diamant azteque plein.svg|One possible tiling
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| </gallery>
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| {{clear}}
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| ==See also==
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| *[[Statistical mechanics]]
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| *[[Gaussian free field]], the scaling limit of the height function in the generic situation (e.g., inside the inscribed disk of a large aztec diamond)
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| *[[Mutilated chessboard problem]], a puzzle concerning domino tiling of a 62-square subset of the chessboard
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| *[[Tatami]], floor mats in the shape of a domino that are used to tile the floors of Japanese rooms, with certain rules about how they may be placed
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| == References ==
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| * {{Citation | last1=Bodini | first1=Olivier | last2=Latapy | first2=Matthieu | title=Generalized Tilings with Height Functions | url=http://www-rp.lip6.fr/%7Elatapy/Publis/morfismos03.pdf | year=2003 | journal=Morfismos | issn=1870-6525 | volume=7 | issue=1 | pages=47–68}}.
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| * {{cite journal
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| |first1=F. | last1=Faase
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| |journal=Ars Combin.
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| |title=On the number of specific spanning subgraphs of the graphs G X P_n
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| |volume=49 | mr=1633083 | pages=129-154 | year=1998 }}
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| * {{cite journal
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| |first1=J. L. | last1=Hock
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| |first2=R. B. | last2=McQuistan
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| |title=A note on the occupational degeneracy for dimers on a saturated two-dimenisonal lattice space
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| |journal=Discrete Appl. Math.
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| |volume=8 | pages=101-104 | year=1984
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| |doi = 10.1016/0166-218X(84)90083-0 |mr=0739603 }}
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| *{{citation
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| | title = The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice
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| | journal = [[Physica (journal)|Physica]]
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| | volume = 27 | issue = 12 | year = 1961 | pages = 1209–1225
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| | first = P. W. | last = Kasteleyn | bibcode=1961Phy....27.1209K
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| | doi = 10.1016/0031-8914(61)90063-5}}.
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| *{{Citation | last1=Kenyon | first1=Richard | title=Directions in mathematical quasicrystals | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=CRM Monogr. Ser. | mr=1798998 | year=2000 | volume=13 | chapter=The planar dimer model with boundary: a survey | pages=307–328}}
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| * {{Citation | last1=Kenyon | first1=Richard | last2=Okounkov | first2=Andrei | author2-link=Andrei Okounkov | title=What is … a dimer? | url=http://www.ams.org/notices/200503/what-is.pdf | year=2005 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=52 | issue=3 | pages=342–343}}.
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| *{{citation
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| | last1 = Klarner | first1 = David
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| | last2 = Pollack | first2 = Jordan
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| | doi = 10.1016/0012-365X(80)90098-9
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| | issue = 1
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | mr = 588907
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| | pages = 45–52
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| | title = Domino tilings of rectangles with fixed width
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| | volume = 32
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| | year = 1980}}.
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| *{{cite arxiv| first1=Richard J. | last1=Mathar | eprint=1311.6135
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| |year=2013 | title=Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings}}
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| *{{citation
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| | first=James | last=Propp | authorlink = Jim Propp
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| | title=Lambda-determinants and domino-tilings
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| | journal = [[Advances in Applied Mathematics]]
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| | volume = 34 | issue = 4 | year = 2005 | pages = 871–879
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| | doi = 10.1016/j.aam.2004.06.005
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| | arxiv=math.CO/0406301}}.
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| * {{cite journal | first1=Frank | last1=Ruskey
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| |first2=Jennifer | last2=Woodcock
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| |title = Counting fixed-height Tatami tilings
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| |journal=Electron. J. Combin | volume=16 | number=1
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| |page=R126 | mr=2558263 | year=2009 }}
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| *{{citation
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| | title = Domino tilings and products of Fibonacci and Pell numbers
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| | journal = Journal of Integer Sequences
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| | volume = 5
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| | year = 2002 | issue = Article 02.1.2
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| | url = http://www.emis.de/journals/JIS/VOL5/Sellers/sellers4.html
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| | first = James A. | last = Sellers}}.
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| * {{cite journal
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| |first1=Richard P. | last1=Stanley
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| |title=On dimer coverings of rectangles of fixed width
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| |journal=Discrete Appl. Math
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| |volume=12 | pages=81-87 | mr=0798013 | year=1985}}
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| *{{citation
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| | authorlink = William Thurston | last = Thurston | first = W. P.
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| | title = Conway's tiling groups
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| | journal = [[American Mathematical Monthly]]
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| | volume = 97
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| | issue = 8
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| | year = 1990
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| | pages = 757–773
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| | doi = 10.2307/2324578
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| | publisher = Mathematical Association of America
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| | jstor = 2324578}}.
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| *{{citation
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| | title = [[The Penguin Dictionary of Curious and Interesting Numbers]]
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| | edition = revised | year = 1997 | isbn = 0-14-026149-4 | page = 182
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| | last = Wells | first = David
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| | publisher = Penguin
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| | location = London}}.
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| [[Category:Statistical mechanics]]
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| [[Category:Lattice models]]
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| [[Category:Exactly solvable models]]
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| [[Category:Combinatorics]]
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| [[Category:Recreational mathematics]]
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| [[Category:Tiling puzzles]]
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| [[Category:Matching]]
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