Topping the Hat

Siobhan Roberts is a Canadian science journalist, biographer, and historian of mathematics. She has an article that appeared in print in yesterday’s New York Times. It is on a second breakthrough by a team of mathematicians, improving their solution to a famous problem on tiling as we covered last March.

Yesterday, while reading the Times, I must admit that I was surprised. I was reading Section A, with tons of stuff on the war in Ukraine and the debt ceiling deal and on the Republican primaries for president. I was not expecting to see a math theorem given such prominence.

I knew Roberts’s previous work on math such as a terrific book a while ago on John Horton Conway. He discovered the Conway groups in mathematical symmetry and invented the aptly named surreal numbers as well as the cult classic Game of Life. Moving to Princeton in 1987, he deployed cards, ropes, dice, coat hangers, and even the odd Slinky as props to improve his lectures; see our memorial for more.

The Problem

Roberts begins by recapping the original solution:

In March, a team of mathematical tilers announced their solution to a storied problem: They had discovered an elusive “Einstein”— a single shape that tiles a plane, or an infinite two-dimensional flat surface, but only in a non-repeating pattern. “I’ve always wanted to make a discovery,” David Smith, the shape hobbyist whose original find spurred the research, said at the time.

The other members of the team are Craig Kaplan and Chaim Goodman-Strauss (to the left of Smith below) and Joseph Myers:

Composite crop of several sources

The rub with the tile that the collaborators named “the hat” is that it needed to be flipped into its mirror image to create a set that can tile the plane. This is evident in the following grid—note that the blue hats have their upper notch on the left not right:

The New Solution

To remove this hitch and obtain the strongest possible result, the team needed to achieve two objectives:

1. Create a single shape that can tile the plane without being flipped, but only aperiodically.

2. Show that the shape plus its flip cannot tile the plane periodically.

Note that the second clause makes the “but only aperiodically” part of the first clause redundant. The key with the second clause is that the team could begin with work already done for their original result, which included the flip. Indeed, they had created a whole continuum of tiles with the same properties as the original “hat.”

There were two other dimensions of freedom for the team to explore:

• Modify the edges of the tiles in ways other than the continuum.

• Combine basic tiles into “supertiles” . One possible idea is that would be OK for to include copies of both and flips so long as can tile the plane without itself being flipped.

It appears from the new paper that the second dimension is not used—or rather, ideas like it are used in the proofs but not in the definition of the new einstein tiles. The former dimension, however, made profit out of a step that at first seemed to undo the work they had done before. They had already observed the key point in their original March paper, from which we reproduce this snippet:

That is to say, “” defines the dimension of their original continuum which changes the relative lengths of edges of the polygon. The case is an isolated point at which aperiodicity breaks down—because it makes equal-length edges that permit sudden new ways tiles can fit. This is exploited by the simple columns of what they call “Tile(1,1)” and its flip in the figure. The key property of Tile(1,1) proved in the new paper is:

Tile(1,1) can tile the plane without being flipped—but only aperiodically.

Further and most important, the new paper shows that by altering the edges of Tile(1,1) in any of a whole spectrum of ways represented in the figure below (which is rotated from ones in the paper and NYT article), one can rule out the flips and preserve the aperiodic tilings without flips:

These tiles, which they call “Spectres,” are the new objection-free einsteins. They prove properties of a whole space of these tiles, fed by two main lines of hierarchical supertile constructions. Thus the case is like a Grand Central Station for connecting to the other dimensions.

Open Problems

What could be the payoff of this wonderful discovery? Perhaps a new drug? A new solution to a potential Nobel level problem? Or something else?

Dan Shechtman was awarded the 2011 Nobel Prize in Chemistry for the discovery of natural quasicrystals, making him one of six Israelis who have won the Nobel Prize in Chemistry.

Meeting at NIST in 1985 where Shechtman (on left) explains the atomic structure of quasicrystals: NIST source

Quasicrystals are defined by being aperiodic—the connection to the tiling problem is express in the Wikipedia page linked above. Will the new tiling results reported this spring lead to an insight worthy of a major award outside mathematics—a Nobel perhaps?