Domino (mathematics)
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 nth 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, 35 (2), Mathematical Association of America: 115–120, doi:10.2307/4146865, JSTOR 4146865.
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