Now I Get It: Why the Hat and Spectre have been making news

They can tile an indefinite plane without repeating a pattern. The search for such shapes began in the 1960s. Why does any of this matter? Find out.

In April, an unusual discovery made news. Researchers had identified a two-dimensional shape with 13 sides, not particularly pretty, which they described as the “Hat”. Copies of the Hat could be used repeatedly to tile an infinite plane, leaving no gaps or overlaps, and with no pattern repeating. This is called aperiodicity.

How much more is out there, that we don’t know about? What does it take to stumble upon new discoveries? How big a role does luck play? The Hat is, more than anything, a story of discovery. (University of Waterloo)

Within months, the same team discovered another shape that not only tiled an indefinite plane without repeating a pattern, but also did this without the use of its mirror image. The researchers named this one the “Spectre”.

The Hat is an “einstein”, or “one stone”, because the single shape (and its reflection) enables the infinite tiling. The Spectre is being called a “vampire einstein”, because vampires supposedly do not reflect themselves in the mirror.

All this does not, however, explain why this is such a big deal.

Well, here goes. The Hat and the Spectre are the first single shapes ever found to exhibit aperiodicity, in a search that has been underway since the 1960s.

These shapes may find practical use in the making of quasicrystals (structures that can fill all of an available space without repeating a pattern). Quasicrystals are used, for instance, as coatings on frying pans and surgical equipment, among other things.

What makes the new discoveries significant, however, is not the promise of practical application. In mathematics, as in any branch of science, the thrill of solving a difficult puzzle is in itself a reward. The significance of the two shapes boils down to a story of discovery.

The idea of aperiodicity has its genesis at least four centuries ago. In 1611, Johannes Kepler, the German astronomer and mathematician best known for his laws on planetary motion, published an article wondering why every snowflake shows a six-sided symmetry. Why not four or five sides?

Kepler assumed that the hexagonal shape allows for the tightest possible way of packing the snowflakes together. This is known as the Kepler Conjecture, formally proven only in 2011. Although Kepler did not provide definitive answers to the questions he laid on snow, his work is considered to have formed the foundation of modern crystallography (the scientific study of the structure and properties of crystals).

Now, as anyone who has tiled a floor knows, it is possible to pack a two-dimensional plane with a single shape such as a square, equilateral triangle, or regular hexagon. In all these tilings, however, various patterns will repeat themselves periodically.

One of the abiding questions in crystallography became: Could there be shapes that could pack a two-dimensional plane aperiodically (with no repetitions)?

In the 1960s, the Chinese-American logician Hao Wang articulated the idea of aperiodicity, but he dismissed it as an impossibility, says Craig S Kaplan, one of the researchers who described the two new shapes, and whose work at the University of Waterloo focuses particularly on interactions between mathematics and art.

Wang was wrong. In 1974, the British mathematician and physicist Roger Penrose — who would win the 2020 Physics Nobel — published a paper describing aperiodic tiling using a set of different shapes. There were no patterns repeated.

But a single aperiodic tile, or monotile, remained elusive.

“In truth, it’s not clear that there’s any particular reason why aperiodic monotiles were so hard to find. It really came down to luck,” Kaplan says. “The universe of shapes we might consider as candidates is infinitely vast, and there was no clear indication of where we might start looking for aperiodic monotiles.”

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The Hat eventually emerged last November, when David Smith, a hobbyist mathematician in Brighton, constructed it with software. Noticing that it could aperiodically fill a plane, he contacted Kaplan. Subsequently, Smith, Kaplan and two other mathematicians described it in a preprint paper in March, followed by another preprint paper on the Spectre in May.

“As is turns out, we spent 50 years looking in the wrong places,” Kaplan says. “Then David Smith just happened to be experimenting last year with polykites (shapes formed by gluing together a small number of kite shapes) and noticed that the Hat was behaving strangely, not settling into any obvious patterns. Fortunately, he knew enough about the history of aperiodicity to recognise the potential impact of this discovery.”

Kaplan is certain there are more einsteins and vampire einsteins out there.

“There’s nothing remarkable about the Hat as a shape. It’s just a boring, unassuming polygonal blob. There’s every reason to believe that we’ll be able to find many more aperiodic monotiles.” he says. Once again, though, “we just don’t know where to look for them!”