The Deep Mathematics of Tiling

By Jordana Cepelewicz

Each week Quanta Magazine explains one of the most important ideas driving modern research. This week, senior math writer Jordana Cepelewicz describes the mathematical utility of studying how shapes “tile” the plane.

Tilings are all around us. These geometric patterns — arrangements of interlocking shapes that cover a flat surface — appear in ancient mosaics and M.C. Escher drawings, in decorative wallpaper and bathroom floor designs. They also crop up in nature, in the structure of honeycombs, of snakes’ skin, of rock formations .

Mathematicians have been captivated by such patterns, which they also call tessellations, for millennia. The central question they want to answer is a simple one: Can a given set of shapes “tile” the infinite 2D plane, filling the entire space without gaps or overlaps? Squares can; regular pentagons cannot. It turns out that the general version of this problem is maddeningly difficult — what theoretical computer scientists call “undecidable .” It’s impossible to write an algorithm to determine the answer for all possible sets of tiles.

And so rather than trying to study tiling problems this way, mathematicians have set their sights on individual patterns, testing — often through trial and error — what might happen when they play with different shapes and different rules. To do so, they’ve had to wander through the mathematical kingdom, from geometry and topology to algebra and number theory. They’ve had to reckon with the infinite, and with surprising kinds of symmetry . And they’ve had to push up against the limits of human intuition. (As a mathematician once remarked to me, “You want to understand the structure of tilings. How crazy can they get?”)

These are only some of the reasons mathematicians find tilings so compelling. Here’s another: It’s one of those rare areas of mathematical study where there is room for amateurs and hobbyists. Many nonprofessionals have made major advances, including Robert Ammann, a mail sorter; David Smith, a retired print technician; and Marjorie Rice , a California housewife. In a field known for its technical jargon and high barrier to entry, the study of tilings is an important exception, reminiscent of an older era of mathematics.

Which is not to say that it’s easy.

Recently, though, there’s been a flurry of exciting progress by mathematicians and nonmathematicians alike — culminating in the unexpected discovery of a very special tile.

What’s New and Noteworthy

Mathematicians have sought answers to several questions about tilings. First: What’s possible, and what’s not? Some shapes simply cannot tile the plane, no matter how hard you try to wedge them together. But stretch or squish them a little, making the shape’s outline just a bit more irregular, and you might succeed .

Similarly, there are lots of different ways you might try to tile the plane using a given set of shapes. You can cover a surface with squares, for instance, by simply placing copies of a square tile next to one another in rows and columns. But if you’re using triangles, this won’t work; you’ll also have to rotate some of the tiles.

Both squares and triangles can form periodic tilings — tessellations built from a repeating pattern. But mathematicians are also interested in aperiodic tilings , which lack a globally repeating pattern (and can never be rearranged into something periodic). Those are much harder to find. The first aperiodic tiling, discovered in the 1960s, relied on an arrangement of a whopping 20,426 different tiles. Mathematicians immediately wanted to knock that number down. In the mid-1970s, Roger Penrose found two tiles that create an aperiodic pattern. (His aperiodic tilings have figured in research on naturally occurring “quasicrystal” structures, and in recent work by physicists on quantum error-correcting codes.)

But for decades, mathematicians couldn’t find a single-tile solution.

They found some workarounds. In 2010, Joshua Socolar, a physics professor at Duke University, and Joan Taylor, an amateur mathematician in Tasmania, designed a single disconnected tile that could be shifted, rotated and flipped to cover the 2D plane aperiodically. In 2022, I wrote about the similar discovery of an immensely complicated tile that tiles higher-dimensional space aperiodically without having to be rotated or reflected.

Then, last year, an eclectic team — Smith, the retired print technician, along with a mathematician, a computer scientist and a software engineer — announced that they had finally found an “einstein,” a single connected shape capable of aperiodically tiling the 2D plane. (In German, ein stein means “one brick” or “block.”) The group discovered a whole family of such tiles. Where once mathematicians wondered if an einstein could exist at all, now they know of infinitely many such shapes.

This finding will not put an end to the search for novel tiles and tilings. There are still new patterns to uncover, new questions to answer — whether there are broader properties that all einstein tiles share, for instance — and new geometric boundaries to probe.

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Around the Web:

·? Scientific American published a first-person account of the discovery of the einstein tile earlier this year.

·??The Guardian ran a puzzle column that featured tiling-related problems in February 2019.

·? The Numberphile channel on YouTube detailed the importance of Penrose tilings in a 2012 video .