Aztec diamond

In combinatorial mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy |x| + |y| ≤ n. Here n is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both x and y are half-integers.

The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order n is 2^n(n+1)/2. The arctic circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle.

• Diamant azteque.svg

  An Aztec diamond of order 4, with 1024 domino tilings

• Diamant azteque plein.svg

  One possible tiling.

• Arctic Circle.svg

  A tiling of a hexagon chosen uniformly at random, with the "frozen" tiles being depicted in white. (Arctic Circle theorem.)

It is common to color the tiles in the following fashion. First consider a checkerboard coloring of the diamond. Each tile will cover exactly one black square. Vertical tiles where the top square covers a black square, is colored in one color, and the other vertical tiles in a second. Similarly for horizontal tiles.

External links

• Weisstein, Eric W., "Aztec Diamond", MathWorld.