Aztec diamond

[share]

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.[1]

The Aztec diamond theorem states that the number of domino tilings of the
Aztec diamond of order n is 2n(n+1)/2.[2]
The Arctic Circle theorem says that a random tiling of a large Aztec diamond
tends to be frozen outside a certain circle.[3]

[share]

https://mathworld.wolfram.com/AztecDiamond.html

An Aztec diamond of order n is the region obtained from four staircase shapes of
height n by gluing them together along the straight edges.

It can therefore be defined as the union of unit squares in the plane whose edges lie
on the lines of a square grid and whose centers (x,y) satisfy

|x-1/2|+|y-1/2|<=n.

[share]

Random domino tilings the Aztec diamonds, along with random lozenge tilings of a hexagon,
is one of the most studied models of statistical physics.

It was first introduced by Elkies-Kuperberg-Larsen-Propp in [3],
and we refer to a recent survey by Johansson [4] and
references therein for detailed information about it.

http://math.mit.edu/~borodin/aztec.html

[share]

A New Simple Proof of the Aztec Diamond Theorem

The Aztec diamond of order n is the union of lattice squares
in the plane intersecting the square |x|+|y|<n.

The Aztec diamond theorem states that the number of domino tilings
of this shape is 2n(n+1)/2.
It was first proved by Elkies et al. (J. Algebraic Comb. 1(2):111–132, 1992).
We give a new simple proof of this theorem.

https://link.springer.com/article/10...373-015-1663-x