9464

Random Domino Tilings

Consider an rectangle and rectangular tiles of size 1×2 (dominoes). A domino tiling of the rectangle is a placement of dominoes that covers the rectangle completely without overlaps. A tiling exists if and only if and are not both odd, implying is even. One tiling can readily be found: suppose is even, place dominoes vertically in the first column and repeat for the next columns. This Demonstration generates random tilings of rectangles of chosen sizes and computes the total number of tilings possible.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

If is the number of tilings of an rectangle, it is easy to see that . As and , is a Fibonacci number. The original motivation for tackling the tiling problem by dominoes (or dimers) was to provide a simple model to describe the thermodynamic behavior of fluids. The amazing formula for was first obtained by Kasteleyn in 1961: , which is the formula used in this Demonstration.
References
[1] P. Kasteleyn, "The Statistics of Dimers on a Lattice I. The Number of Dimer Arrangements in a Quadratic Lattice," Physica, 27, 1961 pp. 1209–1225.
[2] R. Kenyon and A. Okounkov, "What Is...a Dimer?" Notices of the American Mathematical Society, 52(3), 2005 pp. 342–343.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+