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The Einstein Problem



"It's a wild ride. It's completely unlikely and should not be taken as a model for how scientific discovery ever happens. It's completely ridiculous."

— Craig Kaplan, co-author of the recent paper on aperiodic monotiles

Sometimes, from the simplest of beginnings, emerge whole worlds of complexity. Einstein's general relativity can be expressed in a compact equivalence: Gμν= (8πG/c4)Tμν, while the psychedelic fractals of Benoit Mandelbrot's "set" emerge from this slim equation : Fc(z) = z² +c. And, as of a few months ago, mathematicians have been going gaga over not an equation, but a simple shape. It's the "hat" (or half tucked-in tee shirt) pink polygon shown above: the world's first aperiodic monotile.

Consider a tiled bathroom floor. Obviously square or rectangular tiles, all the same size, can fill the entire area of the bathroom without gaps or overlaps. Same with triangular or hexagonal tiles, including combinations. These are examples of periodic tilings: You could pick up a portion of whole tiling, slide it by some distance without rotating it, and superimpose it perfectly on top of a stationary copy. Then there are aperiodic tilings: No matter how you move a swatch of tiles around, you'll never be able to line it up on top of a similar-size swatch. Even with an infinite-size bathroom, the tiles will never match up because there will never be a repeating pattern.

In the early 1960s, mathematicians didn't know if aperiodic tilings were even possible, but apparently the problem caught the interest of some of them. In 1966, electrical engineer Robert Berger figured out how to do it with a set of 20,426 distinct tile shapes, which he later reduced to 104. Soon after, British polymath Roger Penrose discovered a set of just two different shapes — "kites" and "darts" — that he proved could only be laid in a non-repeating mosaic. Until last year, that's where the arcane world of aperiodic tiling stood, with a minimum of two shapes (including other pairs besides Penrose's original) needed to fill an infinite plane in never-repeating patterns. Thus was conceived the Holy Grail of Tiling — could two be reduced to one? It was dubbed "the einstein problem," not after Albert but from the German for one stone, ein stein). For nearly 60 years, the general belief was that if such a shape existed, it had to be wildly complicated — otherwise someone would have easily found it.

But it wasn't complicated at all; it was practically waiting in plain sight! The "hat" is merely eight right-angled "kites" drawn within a hexagonal grid. Self-styled "hobbyist," Yorkshireman Dave Smith, was playing around with polygons using Jaap Scherpuis's Polyform Puzzle Solver application when he stumbled on the shape last November. He emailed University of Waterloo math professor (and tiling enthusiast) Craig Kaplan, asking him to confirm that he'd solved the einstein problem. Kaplan checked it to the limits of his own computer software before bringing in two mathematical colleagues, Joseph Myers and Chaim Goodman-Strauss, to be doubly sure. In March, the four of them published an 89-page paper (arXiv:2303.10798) presenting the hat shape to the world, including not one but two proofs of its aperiodicy.

At which point, the dam burst. Within weeks, hobbyists and mathematicians discovered endless variations on the original hat, which requires flipping over about one in seven tiles (black in the photo). Soon after, the "spectre" was found, which eliminates the flipping. And the flood of discoveries is ongoing. If I were advising on The Graduate today, instead of telling Benjamin (Dustin Hoffman) that the future was plastics, I'd say, "Tiles."

Barry Evans (he/him, has found new meaning in life with his set of hat tiles from Etsy.

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