Domino (mathematics)

This article is about the mathematical polygon. For the game, see 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.^[1] When rotations and reflections are not considered to be distinct shapes, there is only one free domino.

The single 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

Main article: 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 ${\displaystyle F_{n}}$, 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

• Dominoes, a set of domino-shaped gaming pieces
• Tatami, Japanese domino-shaped floor mats

References

1. Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8.
2. Weisstein, Eric W. "Domino". From MathWorld – A Wolfram Web Resource. Retrieved 2009-12-05.
3. Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics. 36 (2): 191–203. doi:10.1016/0012-365X(81)90237-5.
4. Berger, Robert (1966). "The undecidability of the Domino Problem". Memoirs Am. Math. Soc. 66.
5. Concrete Mathematics Archived 2020-11-06 at the Wayback Machine by Graham, Knuth and Patashnik, Addison-Wesley, 1994, p. 320, ISBN 0-201-55802-5
6. 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
7. 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.