A diamond of squares covered by dominoes, chosen fairly from every way to do it, drawn as the paths hidden in it. Outside a circle everything freezes into straight lines; inside it the paths wander like ridges.
Elkies, Kuperberg, Larsen & Propp, 1992 · Jockusch, Propp & Shor, 1995
Stack rows of squares two, four, six and on up to the middle, then back down, and you have the Aztec diamond. There are 2 to the power n(n + 1)/2 ways to cover one of order n with dominoes, a count found by Noam Elkies, Greg Kuperberg, Michael Larsen and James Propp in 1992, along with domino shuffling, a way to choose one of them perfectly fairly.
Shuffling grows the tiling one order at a time: dominoes about to collide are taken away, the rest each slide one square, and the holes left behind are filled with a pair at the toss of a coin. Large random tilings all look alike. Near the four corners every domino lies the same way, frozen into brickwork, and only inside the circle that touches the diamond's sides is there disorder; William Jockusch, James Propp and Peter Shor proved this arctic circle in 1995.
Drawn as paths, every tiling becomes a set of lines that never cross, one per row: level through some dominoes, climbing or falling through others, and missing from the rest. Kurt Johansson used these paths to study the edge of the circle. In the frozen corners they run straight, stacked or slanting or absent, and inside the circle they wander like ridges on a range of hills.
Drawn as tones, the dominoes are shaded by the way each one moved in the last shuffle, so the four frozen corners take four tones. In motion the diamond is shuffled up from nothing, one order at a time.
Behind the mathematics
Domino shuffling, 1992, and the arctic circle, 1995
American mathematician at the University of Massachusetts Lowell who with Noam Elkies, Greg Kuperberg and Michael Larsen counted the domino tilings of the Aztec diamond and found domino shuffling, and in 1995 proved its arctic circle with William Jockusch and Peter Shor.
Counting the tilings and domino shuffling, 1992
American mathematician at Harvard, where at 26 he became the youngest professor ever given tenure, whose 1992 paper with Kuperberg, Larsen and Propp counted the domino tilings of the Aztec diamond, 2 to the power n(n + 1)/2 of them, and gave domino shuffling to choose one fairly.
Counting the tilings and domino shuffling, 1992
Polish-born American mathematician at the University of California, Davis, and one of the four authors who in 1992 counted the Aztec diamond's domino tilings and introduced domino shuffling.
The tilings as paths that never cross, 2002
Swedish mathematician at KTH Stockholm who turned random tilings of the Aztec diamond into families of paths that never cross, and showed that the edge of the arctic circle fluctuates like the largest eigenvalue of a random matrix.
