Imagine a big square. Choose a tiling with dominos (1-by-2 and 2-by-1 rectangles) uniformly at random out of all the possibilities. What would such a tiling look like? One would expect it to look completely disordered, and in fact it does. Here is an example:
In the figure, the colors are determined by the positions of the dominos. They distinguish horizontal and vertical dominos, and within these two classes the colors tell how the dominos are located relative to a checkerboard coloring of the underlying square grid.
One might futhermore expect that random domino tilings would look equally disordered for pretty much all regions, but that's not the case. "Aztec diamonds" show surprising inhomogeneity:
The boundaries of an Aztec diamond clearly put more constraints on a tiling than the boundaries of a square do. What's surprising is that the boundary effects don't die out rapidly as one moves into the diamond. In a large Aztec diamond, a key role is played by the inscribed circle, called the arctic circle. In the polar regions outside the arctic circle, the dominos will almost always be frozen into place, while in the temperate zone inside, they will almost always be disordered. Crossing the arctic circle amounts to a phase transition between a solid and a liquid state (and this analogy is mathematically deeper than it may sound). Even inside the temperate zone, the disorder shows spatial inhomogeneity that does not occur in squares.
Phenomena of this sort are not limited to domino tilings. For example, here is a picture of a random lozenge tiling of a regular hexagon:
In some of the papers on my research publications page, my coauthors (Noam Elkies, Rick Kenyon, Michael Larsen, Jim Propp) and I analyze Aztec diamonds and hexagons in detail and set up a general framework for analyzing any sort of region.
The pictures on this page were created by Jim Propp's Tilings Research Group (particularly Matt Blum and Jason Woolever).