Seekers of the One-Stone

There are some who would begin the story this way:

Long before Earth was formed, before any planet or star existed, there was the One-Stone. Not an actual stone, of course – just an idealized shape that certain two-legged, one-headed inhabitants of Earth would later call “the hat”. It existed in the realm of Pure Form, awaiting instantiation and recognition. The waiting would take billions of years, but the One-Stone did not know impatience. None of its Kind could. It simply Was, and it waited.

The people who would start the story this way are called mathematical Platonists. Some of them talk about mathematical objects in reverent, mystical tones, imputing a timeless reality to them, though I’ve noticed that they don’t talk about art in the same way; they don’t say that Shakespeare’s Tempest predated Shakespeare just because the potential for arranging strokes to form letters and for letters to be arranged in that one particular Tempest-uous way has existed since the dawn of time.¹ It would be interesting to argue that that the difference between artistic creation and mathematical discovery is a difference not of kind but of degree, but I’m not going to try to make that argument today (partly because I’m not convinced it’s true). Instead, I’ll dive into one very recent example of mathematical creativity intertwined with esthetic choices.

THE WEIRD CORNER

Some shapes, such as the regular hexagon, like to tile the plane in a very simple way:

You’re supposed to imagine that the picture extends out to infinity. This is an example of a periodic tiling: if someone were to shift the infinite pattern up or over by a suitable displacement while your eyes were closed, you wouldn’t notice that anything had changed when you opened them. Regular hexagons can only tile the plane periodically. Other shapes, such as the domino (two squares stuck together), are more versatile and can tile the plane in periodic ways and also in nonperiodic ways.

(In the second picture, I divide the plane into 2-by-2 squares and then use infinitely many coin tosses to decide which squares get covered by two horizontal dominos and which squares get covered by two vertical dominos. We’re allowed to rotate our tiles, so a horizontal domino and a vertical domino count as the “same” tile.) Still other shapes can’t tile the plane at all. Such “nontilers” are the norm: if you draw a complicated shape at random, it’s almost certain to be a nontiler.

We can arrange these three possibilities in a chart:

The empty cell at the bottom right of this chart corresponds to a weird fourth possibility: a shape that can tile the plane, but can’t tile the plane periodically. You’re forgiven if an example of such a shape doesn’t leap into your mind, and you’re in good company if you’re inclined to suspect that no such shape exists; that’s exactly what the logician/mathematician Hao Wang conjectured back in 1961. In fact, he conjectured that the same was true for finite sets of shapes. That is, he surmised that if you have a finite bunch of (finite) shapes and there’s some way to tile the plane with them, then there’s got to be some way of tiling the plane with them that’s purely repetitive, consisting of some possibly large motif that just gets repeated over and over in every direction, without the slightest deviation from that pattern.

It was a surprise when Wang’s student Robert Berger showed in 1966 that in fact you could have a set of tiles that did tile the plane but didn’t tile the plane in a periodic fashion. Berger came up with a set of 20,426 nontilers that, taken together, could tile the plane but couldn’t do so in a periodic way. It’s not that he couldn’t find a way to tile the plane periodically with these tiles; he was able to prove that no such tiling existed.

You may be wondering “Even if Berger’s result were true, how could you ever prove it? If the tiling doesn’t repeat, it’s basically random, so you can’t say anything about it, so how can you be sure it doesn’t repeat?” But nonperiodic doesn’t mean random. For instance, consider the sequence of digits of the successive counting numbers: 1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,…It’s not hard to show that this sequence isn’t periodic, but it’s also far from random.

Similarly, fractals like the Koch snowflake curve shown above aren’t periodic, but they aren’t random; they show the same behavior at every scale. They aren’t repetitive as you pan across them, but they are repetitive as you zoom in. Berger’s tilings were analogous to fractals, but they were repetitive as you zoom outward rather than inward, with a hierarchy of structure at ever-larger scales governing the way patterns seen at smaller scales would break down.

Once Berger had shown that his particular set of 20,426 nontilers could tile the plane nonperiodically and only nonperiodically, it didn’t take long for others to find other, smaller sets of tiles that also belonged in the weird corner of the two-by-two table.

Here is a figure taken from page 307 of Martin Gardner’s book “Penrose Tiles to Trapdoor Ciphers … and the Return of Dr. Matrix” showing a set of three tiles devised by Robert Ammann (more about him below) which, taken together, can tile the plane but can’t tile the plane periodically.

Around 1976, Ammann also found a set of two tiles (independently discovered by physicist Roger Penrose at about the same time) that accomplishes the same feat. But it wasn’t known if a single tile, taken all by itself, could have this weird property, and that’s where things stood for decades.

There’s a terminological problem here, and a potential source of confusion between properties of tilings and properties of tiles and properties of sets of tiles. Some science journalists, in the interest of brevity, have recently written things like “For a long time, mathematicians didn’t know if it was possible to tile the plane nonperiodically.” But of course that’s not true; our random domino tiling is just such a tiling. Other journalists have written things like “For a long time, mathematicians didn’t know of an individual tile that could tile the plane nonperiodically.” This is wrong for the same reason, but you can make it true if you replace “tile” by “only tile” (so that the second half of the sentence expands to “can tile the plane, but can only tile the plane nonperiodically”). But most readers will skip over the word “only” or won’t grasp the work it’s trying to do in that sentence; I imagine that a busy copy-editor might simply cross out the word. Experts in the theory of tiling came up with nomenclature to clear up this potential confusion, by introducing a distinction between the words “nonperiodic” and “aperiodic”, saying that a set of tiles is aperiodic if all tilings that use the tiles in that set are nonperiodic; unfortunately the distinction never caught on.

For the past half-century experts and non-experts alike have been infatuated with the weirdest of the four corners of the two-by-two chart: tiles (or more generally sets of tiles) that can tile the plane but can’t tile the plane in a periodic fashion. Why? Because it’s the most mysterious corner, and mystery is catnip to mathematicians.

WAS IST UND WAS SOLL EIN “EINSTEIN”?

A hypothetical shape that tiles the plane but doesn’t tile the plane periodically was dubbed an “einstein”, or “one-stone”, by geometer Ludwig Danzer. (Albert Einstein was already dead and therefore not around to complain that people would mistakenly think he was somehow mixed up with this business.) Danzer never gave a precise definition of what an einstein is. In 2011, physicist Joshua Socolar wrote “Candidates with einstein–like features have been presented before, but there is no precise definition of the einstein problem, and several candidates that could be argued to qualify have not passed the consensus ‘I know it when I see it’ test.” (The title of this section, by the way, is a play on the title of Dedekind’s famous book on the nature of numbers, “Was ist und was sollen die Zahlen?”)

One issue that was never completely resolved was whether a proposed einstein’s mirror-image could be included for free, or whether it had to be considered a second tile. Certainly the playful coinage “einstein” smacked of recreational mathematics, and the dean of recreational mathematics, Martin Gardner, went on record with the pronouncement “Rotating and reflecting tiles are allowed.” But focussing on whether to allow or forbid reflection misses the point. Socolar got more to the heart of the matter when he wrote “We seek structures with long–range correlations that are not immediately evident from the examination of a single tile.” Going farther than “not immediately evident”, Socolar wrote (about one proposed three-dimensional einstein tile) “The nonperiodicity … does not seem mysterious enough to count.” A true einstein should have a good story to go with it.

Why the obsession with getting one tile to accomplish what mathematicians already knew how to do with two? The urge to push an already-impressive trick even further reminds me of the Dr. Seuss’s Cat in the Hat, each of whose stunts at the start of the book tops the stunt that came before:

“I can hold up the cup and the milk and the cake!
I can hold up these books and the fish on a rake!
I can hold up the ship and a little toy man!
And look! With my tail I can hold a red fan!
I can fan with the fan as I hop on the ball!
But that is not all. Oh, no. That is not all.”

But the mathematical version of “That is not all” is the opposite of the Cat’s. Mathematicians have a tendency toward minimalism, and often their way of challenging themselves is not to do more, but to make do with less.²

WHO TILES?

Some areas of math are forbidding to outsiders (here I’m thinking of subjects like K-theory, which fewer than one percent of mathematicians could even define). Others pose accessible problems but require a certain amount of mathematical sophistication if one wants to avoid reinventing centuries-old blunders (here I’m thinking of number theory, and of the many amateurs whose misunderstandings have led them to mistakenly think they’ve solved longstanding problems – to think that they’re scaling a peak when they’ve walked off the edge of a cliff and just don’t know it). Tiling theory is accessible like number theory but without pitfalls to mislead the unwary, so it’s a perfect topic for exploration by mathematical enthusiasts who lack training in the daunting armamentarium of modern mathematics. One such explorer was tilings-hobbyist Marjorie Rice, who found several new ways to tile the plane with congruent pentagons. In some ways, being an amateur may have helped her; not knowing that a problem is regarded as hard by experts can unlock creativity that might otherwise be hindered by lack of confidence.

The same might be said of Robert Ammann, a self-described “amateur doodler with a math background”, to whom tiling expert Branko Grunbaum once wrote “Without exaggeration, I am convinced that you have shown more inventiveness than the whole rest of us taken together.” Although Ammann was a computer programmer when he began his correspondence with Martin Gardner, he later switched to sorting mail for the post office; it was steady work, and it allowed him to let his mind wander. And did it ever wander! I’ve already mentioned that he discovered Penrose tilings independently of Penrose, and shown you his aperiodic set of three shapes made of squares. You can read more about Ammann (pronounced “AM-man”) and his work in Marjorie Senechal’s lovely article “The Mysterious Mr. Ammann”.

Shortly after Ammann died in 1994, mathematician Petra Gummelt found something like an einstein, but in the domain of coverings rather than tilings (so that overlaps are permitted). Gummelt found a way to decorate a regular decagon so that copies of the decorated decagon can cover the whole plane (with identical decorations on overlaps) but not in a periodic way. Gummelt’s “stone” permitted many ways to cover the plane, but none of them were periodic.

A notable near-miss in the hunt for an einstein was the tile found by Joan Taylor and Joshua Socolar. Taylor is an amateur mathematician; Socolar is a physics professor. Combining their strengths, they came up with several near-einsteins. Taylor’s first attempt was a marked hexagon; if one tries to tile the plane with copies of her hexagon, subject to the constraint that the markings of nearby tiles must match up properly, then one can do it, but not in a periodic way. (The image below is taken from their joint article, listed in the References.)

Socolar modified her idea to get rid of the markings, but at a cost: the resulting tile is disconnected. In the physical world, objects that aren’t attached to each other mostly move independently, but there are charming exceptions; my favorites are magnets with holes in the middle. If you mount two of them on a spindle of some kind (a pencil will do) arranged so as to repel each other, one will levitate above the other, and slowly raising the lower one will cause the upper one to move upward in tandem by the same amount, as if they were one rigid object. In our universe there’s no way to make two flat jigsaw puzzle pieces move in tandem as if they were a single rigid object (is there?), but we can still imagine a mathematical universe in which this is possible, and that’s where the Taylor-Socolar tile lives.

There are many others who looked for an aperiodic tiling set consisting of just a single tile (an “aperiodic monotile”); as the problem’s fame grew, the hunt for a monotile became a natural target for professional and amateur alike. Computer scientists Joseph Myers and Craig S. Kaplan were two of many who got bitten by the bug. Myers did extensive work on shapes made up of squares, regular hexagons, and equilateral triangles, known as polyominoes, polyhexes, and polyiamonds. Myers’ webpage Polyform Tiling documented the fruits of computer exploration of trillions of shapes, every single of one of which (prior to 2023) failed to be an einstein.

Building on Myers’ work, Kaplan studied a relative of the einstein tile problem called the Heesch problem. (See the video Heesch Numbers and Tilings.) Heinrich Heesch, who died in 1995, came up with a way to measure how high a nontiler could raise your hopes before dashing them. A nontiler has Heesch number n if you can surround the shape with n layers of copies of itself (but no more than n). Using Myers’ code to filter out “non-nontilers”, Kaplan determined the Heesch numbers of all the nontiling polygons that can be formed by sticking together up to 19 squares, up to 17 regular hexagons, or up to 24 equilateral triangles. Kaplan’s survey uncovered nontilers with Heesch number 1, 2, 3, and 4. But this didn’t set a new record; tiling-fan David Smith, working by hand, had already come up with shapes with Heesch number 5 back in 2019, and tiling-fan Bojan Bašić had topped this in 2020 by finding a shape with Heesch number 6 (still the reigning champion in April of 2023).

After a half century of search for their holy grail, Kaplan, Myers, and scores of other combinatorial geometers had failed to find an einstein. They had to occasionally wonder: Could it be possible that there aren’t any? The quick reduction of twenty-thousand-plus tiles in 1966 (Berger) down to just two tiles in 1976 (Ammann and Penrose, independently), followed by a long hiatus with no progress at all, suggested some sort of invisible but impermeable barrier to further progress. Kaplan had no strong intuitions one way or the other. Socolar had written a few years earlier: “It is not yet clear whether computer search will beat human creativity to finding the elusive unmarked, simply connected, two-dimensional einstein — if such a thing exists at all.”

It seemed that there were three possible endings to the saga: an einstein might be found; or a proof that no einstein exists might be found; or humanity might simply never know, either because the proposition is independent of the standard axioms of mathematics or because the smallest einstein is so big that we could never find it by computer search and so subtle that we could never design it through merely human cleverness.

“NOW WOULDN’T THAT BE A THING?”

In November of 2022, the aforementioned David Smith, retired print technician and self-described “shape hobbyist”, came up with an asymmetric shape that puzzled him more than any of the hundreds of shapes he’d played with before. There was no obvious way to tile the whole plane periodically with the shape and its mirror image, but there was no obvious way to show that it couldn’t be done. The new shape wasn’t made of squares, regular hexagons, or equilateral triangles like the ones Myers and Kaplan had studied; instead it was made of triangles with angles of 30, 60, and 90 degrees (sometimes called “drafters” in the recreational math literature, on account of their resemblance to what used to be called a draftsman’s triangle).

These triangles were paired into quadrilaterals often called “kites”.

Smith’s new shape was made of eight such kites.

Smith cut out dozens of copies of the shape and found that it and its mirror image seemed to be able to tile large patches in the plane, and perhaps even all of it, but he couldn’t coax it into tiling the plane periodically.

Smith knew of Kaplan’s program for computing Heesch numbers of tiles, and thought that the raw power of the program, trying out combinations billions of times faster than his brain and fingers could, might hit on a pattern. On November 20th, Smith sent a picture of the tile to Kaplan by email, writing: “Can the ‘CryptoMiniSat’ program deal with drafters or kites (from regular hexagons)? Below is one such shape composed of eight kites. It has a Heesch number of at least three, if it’s a non-tiler (I couldn’t get it to tile periodically).” Given Smith’s brilliant earlier work on Heesch numbers, Craig knew that if Dave thought a tile was interesting, it was worth looking into. Kaplan’s code had already been enhanced by undergraduate Ava Pun to allow the study of polydrafters (regions made up of drafters joined together), so he was able to study the new shape immediately. The computer went to work and methodically set about trying to surround Smith’s shape with copies of itself and its mirror image. Two layers, three layers, four, five, six, . . . It wasn’t till it got to sixteen layers that Kaplan interrupted it. Either it was a nontiler with a Heesch number that blew the previous record-holder out of the water, or it was something even better. On November 24th, Smith wrote “Could this shape be an answer to the so called ‘einstein problem’ – now wouldn’t that be a thing?”

The pictures that Kaplan’s program generated contained intriguing hints of order. Strikingly, even though Smith permitted both the left-handed and right-handed versions of the tile to occur wherever they could fit, a kind of spontaneous symmetry-breaking led to one version of the tile dominating over the other. Even more strikingly, occurrences of the rarer tile formed a kind of grid of their own: not an exact grid, but a grid with displacements reminiscent of what researchers had seen with other aperiodic tile sets such as those of Ammann and Penrose. (The following two images are stills from the video “A Hat for Einstein” listed in the References.)

Encouraged, and drawing inspiration from that earlier work, Smith and Kaplan sought motifs in the tiling the computer had drawn, in the hopes that these motifs would join to form super-motifs, and those super-motifs would join in a similar fashion to form super-duper-motifs, and so on, ad infinitum.

Eventually they had an idea for a specific hierarchy of shapes, inspired by the output of Kaplan’s program. They could show that the proposed hierarchical structure could be used to create an infinite tiling of the plane. Moreover, the nature of the hierarchy automatically implied that the tiling would have to be nonperiodic.³ But that’s only half of the job that Smith and Kaplan had to carry out. It wasn’t enough to show that Smith’s new tile could participate in a nonperiodic tiling; they had to show that every tiling of the plane by the new tile was nonperiodic. The task was daunting.

To make the story even more complicated, in December Smith found another tile made of kites that appeared to be an aperiodic monotile. He announced it to Kaplan almost sheepishly: “Probably not the right time but I came across another polykite of interest (a modified hat)”. (The duo had considered a number of names for Smith’s original tile, and “hat” was the one they eventually went with.) This second tile got named the turtle.

Kaplan felt increasingly hopeful that Smith’s artistic sensibility had led him to discover not one but two aperiodic monotiles. But he didn’t want the kind of attention that would come if he made a premature announcement. On a few occasions he told acquaintances that he was working on “new research in aperiodic tilings”, but didn’t say more. Meanwhile, he was beginning to feel that he and Smith needed to bring new people into the project.

In January of 2023, Smith and Kaplan reached out to two other researchers: Chaim Goodman-Strauss and the already-mentioned Joseph Myers. Myers stunned Kaplan and Smith by finding the desired proof in just eight days. Myers’ computer-assisted proof required separate consideration of the 188 different ways a hat can be surrounded by two layers of other hats and showing in each case that if the local patch were to have any chance of being extended to a full tiling of the plane, the patch would have to accord with the proposed hierarchical structure. Kaplan wrote his own code to corroborate what Myers had found, and the two results agreed. The team had found a proof that the hat was an aperiodic monotile. The innocent, almost mundane shape, when forced to tile the entire plane, self-organizes into a communication network that disrupts periodic order at all scales.

But Myers found something else: an explanation of why Smith, by playing around with polykites, had found two aperiodic monotiles rather than just one.

What Myers noticed is that even though the interiors of the hat and the turtle look different (one is made of eight kites and the other is made of ten), their boundaries are similar: not only are both of them 13-sided, but in each case, one of the boundary edges actually comes from two different kites, and so might be viewed as two consecutive edges that happen to be parallel. Viewing each of the shapes as a 14-gon rather than a 13-gon, you’ll find that the boundary of the hat consists of 8 short edges and 6 long edges, while the boundary of the turtle consists of 6 short edges and 8 long edges. If you lengthen the short edges (shown in blue) and shorten the long edges (shown in red), the hat turns into a (rotated) turtle and vice versa!

The understanding that ultimately emerged from Myers’ brainstorm was that the hat story and the turtle story were the same story. The two shapes belonged to a continuum of tiles, obtained by choosing the lengths of short and long edges independently, and with just three exceptions the resulting tile would be an aperiodic monotile. Now there were not just two known aperiodic monotiles, but infinitely many. Kaplan has created a video showing how the different tiles, and the tilings they belong to, morph into one another as the lengths of the sides are adjusted. What’s more, the fact that the hat and the turtle have similar boundaries (combinatorially speaking) but different areas is a key to one of the proofs of aperiodicity of each of those two tiles.

Nothing like this had been seen before in the world of aperiodic tile sets.

On March 20, the four authors released a preprint entitled “An aperiodic monotile”, and the world took notice. Major newspapers reported the discovery, and even the researchers’ own children were impressed; one said she would like to get a tattoo done with a cluster of hats, and another was surprised to see that their father had ascended to the firmament of TikTok fame. Kaplan was especially pleased when friends (online and in person) told him how they’d run across an article about the hat and immediately thought “I bet Craig would be interested in this,” only to discover that Craig was already extremely aware of it.

FUN WITH HATS AND TURTLES

Smith, Myers, Kaplan, and Goodman-Strauss have opted not to try to copyright or patent their tiles, preferring instead to make a free gift of them to the world. Teachers everywhere can dive right in and encourage their students to have fun exploring the new tilings. But how can one do this? As Seuss’s Cat says, “It is fun to have fun but you have to know how.” The first thing you need is a good supply of tiles! See Christian Lawson-Perfect’s webpage of printable and cuttable hats and turtles (listed in the References).

If you’re going to try to make a large tiling using hats, it’s helpful to know that the flipped hats want to spread apart evenly from one another in an approximate triangular grid. Even so, if you don’t have the hierarchical plan in mind, you’re probably going to end up having to backtrack a fair bit as you try to extend your tiling to ever-larger patches of the plane. That’s because of a theorem of Stephen Dworkin and Jiunn-I Shieh that says that for any aperiodic tile set in which the tiles appear in only finitely many orientations (which is certainly true for the hat), there must exist “deceptions of arbitrarily large resolution”. What this means is that there’s a tiled patch that doesn’t extend to a tiling of the entire plane even though every sub-patch of size one million is fine (that is, does extend to a tiling of the entire plane), where the number “one million” can be replaced by any large number. If you use an algorithm that only uses information coming from a portion of the tiling of bounded size, you’ll inevitably make mistakes.

Rather than try to tile the infinite plane (which you could never finish doing anyway), it’s fun to create and solve finite jigsaw puzzles based on the hat. To create such a puzzle, put a bunch of hats together and trace their outline. Have a friend do the same. Then swap outlines and try to fill them in.⁴

Another way to have fun with hats is to decorate them in various ways; a few possibilities are shown near the end of the Museum of Mathematics video A Hat for Einstein. For instance, you might try to color the tiles using four colors, one color for the flipped tiles and three colors for the non-flipped tiles, in such a way that no two adjacent tiles are the same color. It’s believed that you can always do this in a tiling of the infinite plane by hats, and moreover that you can essentially do it in only one way, but this has not been proved. Specifically, this would require a proof that the process of converting hat-tilings into hexagon tilings shown in Figure 2.2 from the article of Smith et al. actually works for infinite tilings, which I believe has not yet been done. For more on the coloring problem, see the MathOverflow discussion listed in the References.

Not to cast any aspersions, but hat-tilings don’t have the same immediate visual pizazz as Penrose tilings. Penrose rhombs induce a pleasing Cubist vertigo, while kites and darts, with hints of 5-fold symmetry that get broken but then reassert themselves at larger scales, are suggestive of infinite hierarchy. Not so with hat tilings. Smith would like to find a way to mark tiles so as to visually bring out the latent hierarchical structure. He is also interested in tilings that combine the hat with other shapes. In particular, if you allow hexagons as well as hats you can come up with periodic tilings of the plane. In a recreational vein, it would be fun to come up with more such patterns; in a research vein, it would be challenging to try to classify them.

As far as I know, nobody has tried replacing the hat by a non-planar polygon, but I think it’s worth doing. As long as you have three red directions oriented at 120 degrees relative to each other, and you have three blue directions oriented at 120 degrees relative to each other, the 14 segments will close up to form a polygon (though it’s no longer guaranteed that the original tile and its mirror image, once modified in this manner, will still be mirror images of one another; in fact there might be twelve different nonplanar polygons). I imagine that this can be done with Zome tools; the result will be an approximately flat network of non-planar polygons that fit together in the same fashion (combinatorially speaking) as the hat and its mirror image. If any of you create a structure like this, please send me photos!

One reader asks: “How would an interested student begin to design a set of shapes that tile the plane aperiodically?” I have no good answers for that. Designing aperiodic sets of tiles is still very much a black art. My advice would be, start from some sort of hierarchical design for covering the plane with shapes of ever-larger size, then come up with marked tiles that enforce that hierarchy, and then find ways to replace the markings with notches of various kinds; this is what Taylor and Socolar did, starting from a hierarchical way of covering the plane by triangles (see Figure 1(c) in their article).

WHAT’S NEXT?

There’s work to be done on the structure of the set of all hat tilings. (Here I’ll focus on just the ones in which unreflected hats dominate, in contrast to their mirror images, in which reflected hats dominate.) One reader asked me: If I were to randomly land in a patch of a tiling of the plane by hats, would I be able to figure out where I was by looking around? This brings up an issue that I’ve ducked till now: are all hat tilings basically the same as each other? The answer to that question appears to be, “mostly”. Leaving aside some possible anomalous tilings that have not yet been fully classified, all non-anomalous tilings are similar to one another in the sense that any patch that occurs in one of them occurs in all the others. Indeed, any patch that occurs in one of them occurs infinitely often in all of them. It follows that no matter how widely you range and how much of the tiling you observe, there are still infinitely many places you could be in that particular tiling. In fact, no amount of finite observation can tell you with certainty which tiling you landed on! But in another sense, you can learn a lot by looking around. If you survey a big enough patch, you might be able to infer where many of the tiles are – perhaps even where most of the tiles are – all the way out to infinity. It would be good to have a more detailed understanding of what you can infer about distant parts of a tiling without going there. You can use the algorithms devised by Simon Tatham to generate random patches of hat-tilings (see the webpage listed in the References).

Right now, the continuum of tiles containing the hat and turtle tiles is magnificent but isolated. It’s natural to suspect that there must be other simple aperiodic tile sets that we haven’t discovered yet because we haven’t been looking in the right places. David Smith’s esthetic led us to the hat and the turtle; can it provide us with a clue about where we should look next? The hunt is on, and given how quickly things progressed from November 20, 2022 to April 20, 2023, I expect that the coming year will bring a bumper crop of new (and pleasingly small) aperiodic tile sets. Indeed, Kaplan and Myers are looking at extending their brute force approach to tilings to more complicated assemblages of pieces. Recall that their pre-2023 work looked only at shapes formed by sticking together regular polygons with 3, 4, or 6 sides. Smith’s extremely fruitful foray into the world of polykites suggests that by looking at other polyforms built in other Laves tilings we might find more aperiodic monotiles.

One natural question that hearkens back to our discussion of “What is an einstein anyway?” is, what if we don’t allow tiles to be flipped? Can we still find an “einstein” under this more restrictive definition? I don’t expect this to be solved in the positive or the negative anytime soon, but it’s certainly something we should strive to figure out.

Two broader problems about two-dimensional tiling are the Heesch problem (mentioned above) and tiling decision problem for monotiles. The latter problem asks for a criterion or algorithm for determining when a shape tiles the plane. The problems are related. If there were an upper bound (say one million, for the sake of definiteness) on how large the Heesch number of a nontiler can be, then there would be an inefficient but definitive way to decide whether a given shape tiles the plane: use brute force to establish whether or not one can arrange a million-and-one layers of tiles surrounding a central tile. If you can’t, then the region certainly doesn’t tile the plane; but if you can, then the (hypothetical) upper bound on Heesch numbers would imply the existence of a tiling. Right now we know very little about the asymptotics of Heesch numbers; for all we know, Heesch numbers are unbounded, but as I mentioned above, the largest Heesch number we know is 6.

Some people have wondered: “Will there be hat-tiling toilet paper?” Although hat tilings of the plane don’t contain strips that are tiled in a periodic fashion, there’s nothing to prevent hats from tiling a strip that’s as wide as you like (though your singly-periodic tiling of the strip won’t extend to a doubly-periodic tiling of the plane).

I’ve also been asked: “Does this work have any applications?” (Toilet paper for nerds doesn’t count.) Physicists may try to build materials whose microscopic structure embodies that of hat tilings. The work of Socolar suggests that if one can build such a material, its X-ray diffraction pattern will be unique in many respects. Will this translate into useful properties for propagation of particles and quasiparticles through the lattice? You can be sure that someone’s looking into it.

But since I’ve been mentioning possible applications, I can’t stress enough that applications are not what motivates pure mathematicians to work on problems like this. We tackle them because they’re hard but not impossible, and because we get satisfaction from working through the difficulties and emerging on the other side with a wriggling newborn theorem in our hands.

Thanks to Richard Amster, Chaim Goodman-Strauss, Sandi Gubin, Brian Hayes, Craig Kaplan, Joseph Myers, Tzula Propp, and David Smith.

ENDNOTES

#1. Since writing that sentence, I’ve been informed that some Platonists do believe in a (rather crowded) realm of Pure Form in which all possible works of art already exist.

#2. More like:

“But I can get by with much less!” said the Cat
As he winked and he took off his odd-looking hat,
Revealing another hat, just like the first,
And beneath that another, but mirror-reversed!
“And NOW,” said the cat with a dazzling smile,
“I’ll do it again but with only one tile!
You may want to sit; this could take me a while. . . ”

#3. If the tiling consisted of repetitions of a fundamental motif, say a big patch consisting of a ordinary tiles and b mirror-reversed tiles, then the ratio of ordinary tiles to reversed tiles in the tiling as a whole would have to be a : b. That is, if you drew a disk of radius R in the plane and computed the ratio of ordinary tiles to reversed tiles within the disk, that ratio would have to converge to a : b as R goes to infinity. But the specific hierarchical structure of the tiling can be shown to imply that as R goes to infinity, that ratio of ordinary to reversed tiles must converge to φ⁴ : 1, where φ is the golden ratio (1 + √5)/2 (see Adam Goucher’s proof of this, listed in the references). And that’s a contradiction, because φ⁴ is irrational. (This isn’t the proof of nonperiodicity that the team found, but it’s the proof that simplest to describe.) An analogous proof of the nonperiodicity of Penrose tilings is sketched in a very nice video from the MinutePhysics YouTube channel listed in the References.

#4. I believe that if you have a hat tiling of the entire plane, then it is rigid in the sense that if any finite number of tiles are removed, the only way to put them back into the frame is to put them in exactly where they were before, possibly permuted but not otherwise altered. Assuming this is true, any finite jigsaw puzzle based on a finite tiling of a patch that extends to an infinite tiling of the plane has exactly one solution. This is not necessarily the case for finite hat jigsaw puzzles, though, since most of them don’t extend to the whole plane. Can any reader find a hat jigsaw puzzle with more than one solution?

REFERENCES

Stephen Dworkin and Jiunn-I Shieh, Deceptions in quasicrystal growth, Comm. Math. Phys. 168(2): 337-352 (1995)

Martin Gardner, Tiling with Convex Polygons, chapter 13 in “Time Travel and Other Mathematical Bewilderments”

Martin Gardner, Tiling with Polyominoes, Polyiamonds, and Polyhexes, chapter 14 in “Time Travel and Other Mathematical Bewilderments”

Martin Gardner, Penrose Tiling, chapter 1 in “Penrose Tiles to Trapdoor Ciphers . . . and the Return of Dr. Matrix”

Adam Goucher, Aperiodic monotile

Craig Kaplan, Heesch Numbers of Unmarked Polyforms

Craig Kaplan, An aperiodic monotile

Craig Kaplan, Aperiodic monotile animation

Erica Klarreich, Hobbyist Finds Math’s Elusive ‘Einstein’ Tile, Quanta Magazine

Christian Lawson-Perfect, The aperiodic monotile in a variety of formats

Christian Lawson-Perfect, Katie Steckles and Peter Rowlett, An aperiodic monotile exists!, The Aperiodical

Robert Lipton, A new tiling

Casey Mann, Heesch’s Tiling Problem, The American Mathematical Monthly, Volume 111, 2004 – Issue 6, pp. 509–517

Ayliean MacDonald, The Shiny New Shape That Aperiodically Tessellates!

MathOverflow, Can we three-color a tiling of the plane with Smith, Meyers, Kaplan, and Goodman-Strauss’s einstein?

MinutePhysics, Why Penrose Tiles Never Repeat

Marjorie Senechal, The Mysterious Mr. Ammann, The Mathematical Intelligencer volume 26, pp. 10–21 (2004)

David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, An aperiodic monotile

Joshua E. S. Socolar and Joan M. Taylor, Forcing nonperiodicity with a single tile, The Mathematical Intelligencer, volume 34, pp. 18–28 (2012)

Siobhan Roberts, Elusive ‘einstein’ solves a longstanding math problem, New York Times, March 28, 2023

Simon Tatham, Two algorithms for randomly generating aperiodic tilings

Joan Taylor, Joan Taylor’s Tilings