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Domino (mathematics)

Geometric shape formed from two squares

Domino (mathematics)
Summary

Geometric shape formed from two squares

the mathematical polygon

The single free domino

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. 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°.

In a wider sense, the term domino is sometimes understood to mean a tile of any shape.

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 F_n, the nth Fibonacci number.

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, 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.

References

References

  1. Golomb, Solomon W.. (1994). "Polyominoes". Princeton University Press.
  2. Weisstein, Eric W. "Domino". From MathWorld – A Wolfram Web Resource.
  3. Redelmeier, D. Hugh. (1981). "Counting polyominoes: yet another attack". Discrete Mathematics.
  4. Berger, Robert. (1966). "The undecidability of the Domino Problem". Memoirs Am. Math. Soc..
  5. ''[http://www-cs-faculty.stanford.edu/~knuth/gkp.html Concrete Mathematics] {{Webarchive. link. (2020-11-06 '' by Graham, Knuth and Patashnik, Addison-Wesley, 1994, p. 320, {{ISBN). 0-201-55802-5
  6. (1992). "Alternating-sign matrices and domino tilings. I". Journal of Algebraic Combinatorics.
  7. Mendelsohn, N. S.. (2004). "Tiling with dominoes". Mathematical Association of America.
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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