Domino (mathematics)
Geometric shape formed from two squares
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Domino_green.svg/75px-Domino_green.svg.png)
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. [1] 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°. [2] [3]
In a wider sense, the term domino is sometimes understood to mean a tile of any shape. [4]
Packing and 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
n
th
Fibonacci number
.
[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 , [6] 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. [7]
See also
References
- ↑ Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8 .
- ↑ Weisstein, Eric W. "Domino" . From MathWorld – A Wolfram Web Resource . Retrieved 2009-12-05 .
- ↑ Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack" . Discrete Mathematics . 36 (2): 191–203. doi : 10.1016/0012-365X(81)90237-5 .
- ↑ Berger, Robert (1966). "The undecidability of the Domino Problem". Memoirs Am. Math. Soc . 66 .
- ↑ Concrete Mathematics by Graham, Knuth and Patashnik, Addison-Wesley, 1994, p. 320, ISBN 0-201-55802-5
- ↑ Elkies, Noam ; Kuperberg, Greg ; Larsen, Michael ; Propp, James (1992), "Alternating-sign matrices and domino tilings. I", Journal of Algebraic Combinatorics , 1 (2): 111–132, doi : 10.1023/A:1022420103267 , MR 1226347
- ↑ Mendelsohn, N. S. (2004), "Tiling with dominoes", The College Mathematics Journal , Mathematical Association of America, 35 (2): 115–120, doi : 10.2307/4146865 , JSTOR 4146865 .
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