NON-TECHNICAL EXPLANATION OF THE 1998 TILINGS RESEARCH GROUP SHIRT

Both the front and the back of the shirt illustrate properties of RANDOM TILINGS.

The front of the shirt shows a region bounded by 200 (50+50+50+50) small quarter-circle arcs, which bend outward along the left and right side of the region and inward along the top and bottom. This region has been tiled with smaller shapes (2500 in all) called "gaskets" and "baskets". (To say that these shapes "tile" the large region is just to say that there are no gaps or overlaps.) The tiles in the corner are all baskets; it's only in the middle of the region that you see gaskets as well. There are two ways to orient a gasket and four ways to orient a basket. We've used six different colors to differentiate among these possibilities.

What's so special about this particular tiling of the big region? Nothing! It was chosen "at random", in the mathematical sense of the phrase. That is, the tiling was chosen using a random-generation process that gives equal likelihood to each of the (finitely many) possible tilings. This is not the same thing as the procedure of just starting somewhere, laying down some tiles, spreading outward, backtracking when you get stuck, etc., with no particular plan in mind --- such a (psychologically random) procedure would probably not give equal likelihood to all the different possibilities, but would favor some over others.

What's surprising is that, even though the tiling is random, it doesn't look random. Why do so many baskets segregate themselves in the corners? Did we "ask" them to do that? No, we didn't; but somehow they do it on their own. Explaining how this happens in detail, in this and other related tiling problems, is a major theme of my research.

In particular: is it mere coincidence that the region in which gaskets occur appears to be roughly circular? That's something I'd very much like to know!

Tilings like these were first studied in an equivalent form by physicists and chemists who were trying to understand ordinary ice, but as a pure mathematician I have very different goals from theirs. I pursue topics that illustrate order, symmetry, and surprise, regardless of how much or how little practical usefulness they have.

The back of the shirt shows a different view of the very same tiling that was shown on the front of the shirt. Where before we had gaskets and baskets, now we have black paths that join up points on the boundary to other points on the boundary, along with other black paths in the interior that join up with themselves to form closed loops. We've colored the regions between the black paths alternately blue and green, but the blue/green coloring is purely ornamental --- it's the wigglings of the black paths that really matter. The collection of black paths is mathematically equivalent to the tiling shown on the front of the shirt, even though the two sides look quite different. You can see this in a rough qualitative way if you squint; in both cases, all the real randomness seems to concentrated in a roughly circular central region.

Since this is supposed to be a non-technical explanation, I won't say more about this, except to say that mathematicians sometimes find new phenomena by replacing a picture by one that is mathematically equivalent but highlights different properties, and that this case is no exception --- looking at the black paths has uncovered new mysteries that still await explanation.

The members of the Tilings Research Group during 1997-1998 were: Jim Propp, David Wilson, Henry Cohn, Ben Raphael, Karen Acquista, Matthew Blum, Carl Bosley, Constantin Chiscanu, Edward Early, Nicholas Eriksson, David Farris, Lukas Fidkowski, Marisa Gioioso, Harald Helfgott, Eric Kuo, Yvonne Lai, Annie Oreskovich, Vis Taraz, Ben Wieland, Lauren Williams, Jason Woolever, and Juwell Wu.