The hat, by contrast, has no symmetry and is “almost mundane in its simplicity,” the authors wrote. What its tilings do have is a deep relationship with a particular periodic tiling: the honeycomb lattice of hexagons. To get a hat tiling from a hexagonal tiling, first connect the midpoints of the opposite sides of the hexagons. This divides every hexagon into six “kites.” Each hat is made of eight adjacent kites, combined from neighboring hexagons. With just a little work, anyone with a magic marker and a hexagonally tiled bathroom floor can trace out a hat tiling.

The hat tile, Senechal said, shows that periodic and aperiodic tiles are more closely linked than mathematicians had realized.

In the days since the announcement, mathematicians and tiling hobbyists have rushed to get their hands on the new tiles, making paper cutouts, 3D-printing them, and making hat quilts and cookies. The excitement the tiles have generated has felt “a bit surreal,” said Smith, who lives in the coastal town of Bridlington in northern England. “I’m not used to this kind of thing.”

But this is far from the first time a hobbyist has made a serious breakthrough in tiling geometry. Robert Ammann, who worked as a mail sorter, discovered one set of Penrose’s tiles independently in the 1970s. Marjorie Rice, a California housewife, found a new family of pentagonal tilings in 1975. And then there was Joan Taylor’s discovery of the Socolar-Taylor tile. Perhaps hobbyists, unlike mathematicians, are “not burdened with knowing how hard this is,” Senechal said.

Shapes Within Shapes

It’s easy to make tilings that aren’t periodic from tiles that also form periodic tilings. You can, for instance, use a couple of vertical dominoes while otherwise filling the plane with horizontal dominoes. “The real art is finding a shape that will allow you to tile the whole plane, but won’t let you do it in a periodic way,” Socolar said.

It’s impossible to create an algorithm that can determine, for every possible collection of tiles, whether they tile the plane (let alone whether they are aperiodic). So after Smith told Kaplan about the hat tile, Kaplan turned to a program he’d written that simply places copies of a tile around an initial seed tile in ever-growing rings. Apart from tiles that create repeating patterns, which have infinitely many rings, no one had ever found a tile that could keep going for more than six rings. This time, the program kept going and going. It filled 16 rings with hats before Kaplan told it to stop, figuring they had enough data to work with.

Meanwhile, to Kaplan’s shock, Smith made another discovery: a second tile, shaped like a turtle, that also appeared to be aperiodic. “The idea of identifying two einsteins back-to-back seemed too good to be true,” the researchers wrote.

By mid-January, Smith and Kaplan had enlisted two more researchers: Chaim Goodman-Strauss, a mathematician at the National Museum of Mathematics and the University of Arkansas, and Joseph Samuel Myers, a software engineer in Cambridge, England, with a doctorate in combinatorics. Myers started devoting all his spare time to the hat tile, and in just over a week, he had proved that it is aperiodic. “We were all pretty blown away by how quickly he nailed this all down,” Kaplan said.

The proof adapted an approach originated by Berger in the 1960s, which involves piecing together tiles into larger versions of themselves, creating a hierarchical structure. Myers began by identifying four intermediate shapes built of hats, which he called H, T, P and F. An H, for example, is made of four hats, joined together to form a shape roughly like a triangle with its tips chopped off. Myers showed that you can put together combinations of the four shapes to make bigger ones. You can make a bigger H, for instance, by surrounding one T with three H’s, then surrounding this object with a combination of P’s and F’s.

This provided a way to make bigger and bigger hat tilings. You can start with an H, increase its size, then fill it with the above combination of the four shapes. Next, you can inflate this whole assembly and fill all the shapes inside the (now huge) H with assortments of H, T, P and F shapes. You can repeat these steps indefinitely, building an increasingly large hierarchy of shapes within shapes. At the bottom level of the hierarchy is the hat.

The researchers proved that the tilings created by these hierarchical constructions are never periodic. They also proved that hierarchical constructions are the only way to make hat tilings. Therefore, a hat tiling can never be periodic. “It’s a very cool result,” Socolar said.

Resizing the Hat

That left the other tile Smith had found: the turtle. Was it just an amazing coincidence that he had come up with two different aperiodic tiles, when no one had found any for 50 years? The hat and turtle tilings looked strikingly similar, making the researchers suspect that the turtle was also aperiodic. But their suspicions were not a proof.

Then Myers made a discovery that the researchers describe in their paper as “both a relief and a revelation.” The hat and the turtle, he realized, belong to an infinite family of tiles that all tile the plane in the same way.

Each hat has 13 sides: six long ones and six short ones that correspond to kite edges, plus one more that’s made of two short kite edges. By tweaking the lengths of these sides, you can create a continuum of new shapes. Imagine a slider bar: As you move it left, the short sides get shorter (as does the solitary double-short side); as you move it right, the long sides get shorter. The turtle is somewhere off to the right of the hat, but there are also infinitely many other shapes.

If you push the slider all the way to the left, the short sides of the hat disappear, leaving a six-sided chevron shape; if you push it all the way to the right, the long sides disappear, leaving a seven-sided shape the researchers called a comet. Unlike the hat, the chevron and comet can tile the plane periodically. So can the shape at the center of the slider bar, where the long and short sides are equal.