Aztec Diamond

Above, a random order 12 Aztec Diamond is being created using the Propp-Wilson Algorithm. What made the Aztec Diamond so popular was the circular arctic boundary that appeared in random samples. This boundary seperates the brickwork in the corners of the diamond from the variable center. Notice how in the tilings above, each started with perfect brickwork covering the region in two opposite phases. In this case, the four phases are characterized by their orientation (horizontal or vertical) and whether their top-left face covered a black or white face of the region. However, as rotations occur, brickwork forms in each corner. Why? Consider what happens if this were not true. If two vertical tiles were placed at the top of the diamond, on each side would be a square with only one neighbor. As a result, a tile is forced there. Extending this, it can be seen that this single change forces all the tiles along the top boundaries, creating in effect and Aztec Diamond of a smaller order. After further investigation, the arctic boundary was proved to be circular for arbitarily large regions. This insighted searches for the other regions with similar properties. One of these is the hexagon, the next example.

The other thing that makes Aztec Diamonds popular is the fact that the number of tilings equals exactly 2 ^ ( n(n+1)/2 ), where n represents the order of the diamond. Curiously the exponent equals the number of faces in one corner (or tiles in one half) of the region. The reason why this is so unusual is the fact that most regions, even common checkerboard shaped regions, have unusual numbers of tilings with seemingly random prime factors. Regions with "nice" numbers of possible tilings like this are also being searched for today.

Note: clicking the mouse within the applet will pause or resume the randomization process.

Send your questions and comments to: wooly@mit.edu

Copyright © 1997 Jason Woolever

Copyright © 1997 Massachusetts Institute of Technology

Last modified: 8/25/97