The hat looks more like a hat when you turn it 150° clockwise from this orientation, and is more often shown that way, but my priority here is to bring out how these shapes relate to each other by keeping them all in the same orientation, and I went with the orientation that is kinder to the turtle. It has been noted that in this orientation the hat looks more like a T-shirt, but I think Smith et al. were wise to avoid a hyphenated name, particularly one that — in mathematical contexts — looks like it begins with a variable.
For decades people had been searching for ever simpler aperiodic tiles, and up to this point, the smallest set of aperiodic tiles that anyone had managed to find was a set of two tiles, the best-known example of which is Roger Penrose's pair of rhombuses, one thick and one thin, with matching rules. The following examples from the Wikipedia article show both edge modifications and colored arcs which must meet the same color. Either one of these would be sufficient to enforce the matching rules.
Simplifying from two aperiodic tiles down to just one aperiodic tile was a major accomplishment. The search for such an aperiodic monotile had been known as the einstein problem. (Here, "einstein" isn't a reference to Albert Einstein, but a play on the German "ein Stein" meaning "one stone".)
The preceding brief summary of this year's achievements should probably prime you sufficiently for this forum post, but if you're new to this topic, or otherwise interested in some further background info, I suggest that you first check out the couple of Numberphile videos on the topic:
- A New Tile in Newtyle, featuring the excellent Ayliean MacDonald
- Discovery of the Aperiodic Monotile, an interview with Craig S. Kaplan, one of the members of the team who discovered these tiles
The reason I'm posting about these tiles on the Sagittal forum is that I found myself discussing these new tiles with Dave Keenan. We were asking each other about things like: in what ways exactly are each of these tiles special? How do they compare with each other in terms of their tiling properties, and their geometry? At first, we were driven mostly by curiosity, but later we found ourselves driven by a desire for clarity. Unsurprisingly (to us, anyway) we started to brainstorm names and terminology for things. After several weeks of this, we realized we'd produced enough interesting material that it would be worth gathering up and sharing. Perhaps many of y'all have found yourselves wondering about the same stuff as we've been wondering about, and I hope this will be a fun and illuminating read for you.
Our material sorted pretty nicely into two conceptual chunks, so I'll be posting the second half of it tomorrow.
Before I get into it, a brief word on my (Douglas Blumeyer's) background interest in geometric patterns: I was obsessed with houndstooth for a few years, and also took interest in traditional geometric Japanese patterns and quasicrystals.
The aperiodic monotile continuum
Our discussions began when Dave lamented that perhaps the most exciting of the newly discovered tiles hadn't been given a catchy name, aside from the prosaic "Tile(1,1)". This tile is neither the hat, the turtle, nor the spectre. But this naming vacuum has resulted in travesties where people online refer to it as the "straight spectre" or "polygonal spectre", when it isn't even a member of the class of tiles defined in the paper as spectres. And spectres in popular culture are noted for their wavy or poorly defined outlines, where Tile(1,1) has edges that are crisp and straight.
Actually, Smith et al. described more than just this handful of aperiodic monotiles, but an entire continuum of aperiodic monotiles. Each of their tiles can be named by "Tile(𝑎,𝑏)", where 𝑎 and 𝑏 are the lengths of its two different types of sides. The hat's name according to this scheme is Tile(1,√3), and the turtle is Tile(√3,1).
All that really matters is the proportion between 𝑎 and 𝑏, not their absolute sizes; for example, Tile(2,10) is essentially the same tile as Tile(1,5), just twice as big. That's why this is only a linear continuum of tiles — such that they could be plotted along a single line — rather than an entire two-dimensional space of possibilities, with one dimension for 𝑎 and one dimension for 𝑏.
In order to best understand where along this continuum a given tile is found, it's best to normalize a tile's name to where either one or the other of 𝑎 and 𝑏 is equal to 1. For instance, we'd prefer Tile(1,5) over Tile(2,10). This way, it's immediately apparent that this tile should more closely resemble Tile(1,√3) than it does Tile(√3,1).
So Tile(1,1) is found smack dab in the middle of this continuum. It's the only tile where 𝑎 = 𝑏 and therefore we have only one unique side length and we can call it equilateral. And because it still looks decidedly like a kind of turtle, we tried calling it the "equiturtle" for a while. We also considered "tortoise" because it looks more terrestrial compared to the original turtle's aquatic, but we concluded that most people don't know that a tortoise is a terrestrial turtle, a point that Craig S. Kaplan made when we suggested these over email.
The equilateral nature of Tile(1,1) also makes it the shape that is exactly halfway between a hat and a turtle, and that's why the name Dave and I eventually settled on is "hurtle". In case it's not obvious, "hurtle" is a portmanteau of "hat" and "turtle". And as you'll see below, this is consistent with a more general naming principle that we've adopted for the entire continuum. Readers familiar with Pokemon will also have a precedent in "squirtle" for treating such a portmanteau as a kind of turtle. Because I'll be referring to this tile quite often in this post, I'll use the name "hurtle" for it exclusively from this point onward.
We do recognize the irony that "hurtle" is exactly what land turtles don't do.
The basic properties that all the tiles along this continuum share are:
- They all have straight edges.
- They all have 14 total sides. (Two of them are colinear, though, so by some reckonings, one could say that they have only 13 sides; however, the 14-side understanding is more helpful when deconstructing how and why they work, so that's what we're using here.)
- Their 𝑎 and 𝑏 length sides all occur in the same sequence 𝑎, 𝑎, 𝑏, 𝑏, 𝑎, 𝑎, 𝑎, 𝑎, 𝑏, 𝑏, 𝑎, 𝑎, 𝑏, 𝑏.
- They all have the same sequence of angles: 120°, 90°, 120°, 270°, 120°, 180°, 120°, 90°, 240°, 90°, 240°, 90°, 120°, 270°.
- They all have area √3(2𝑎²+√3𝑎𝑏+𝑏²).
Toward one extreme of this continuum, the proportion between 𝑎 and 𝑏 gets extremely small; here we find tiles like Tile(1,99999) and Tile(1,99999999). Toward the other extreme, the proportion gets extremely huge; here we find tiles like Tile(99999,1) and Tile(99999999,1). And while the continuum is infinite in the sense that it contains an infinite count of aperiodic tiles, that does not mean that it continues on forever in either direction. Actually, there are two special tiles that cap off the continuum on either end. Here's one good way to think about it: once either 𝑎 or 𝑏 has gone infinite, it's clearer instead to say that the other side length has become 0. That is, Tile(1,∞) is better named as Tile(0,1), and Tile(∞,1) is better named as Tile(1,0).
The first of these two extreme tiles, where 𝑎 = 0, has 6 sides, one for each 𝑏 in that 𝑎, 𝑎, 𝑏, 𝑏, 𝑎, 𝑎, 𝑎, 𝑎, 𝑏, 𝑏, 𝑎, 𝑎, 𝑏, 𝑏 sequence. This shape is called the chevron. At the other extreme, we have the tile where 𝑏 = 0, which has 8 sides, one for each 𝑎 in the same continuum (note that 6 + 8 = 14, the total count of sides for any tile between the two extremes). This shape is called the comet.
A final thing worth noting is that while any tile along the continuum is aperiodic, neither the chevron nor the comet — these two extreme tiles at the ends — are aperiodic.
For convenience, Dave and I not only gave a name to the tile midway between the hat and the turtle, but also to the tiles between the chevron and the hat, and between the turtle and the comet. We named these in a similar way, that is, by blending the official names that had already been given to their neighboring shapes. Any tile between a chevron and a hat became a "chat", and any tile between a turtle and a comet became a "turret". We were fortunate enough that real English words like "hurtle", "chat", and "turret" made themselves available for each of these name-crossings. Fortunately "hurtles" and "chats" have no visual appearance that might conflict with the named shape, since one is an action and the other a sound or string of characters, and the "turret" can in fact be seen as a large shape having two small kite-shaped turrets projecting from it. (Another observation: the transformation from turret to turtle looks a bit like the turtle coming out of its shell.)
Click for the full-size image:
Note that while we use the term "chat" for any shape between a chevron and a hat, and "turret" for any shape between a turtle and a comet, we only use "hurtle" for the single equilateral shape otherwise known as Tile(1,1). A shape between a hat and a hurtle might be called a "hattle", and between a hurtle and a turtle, a "hurturtle".
Now, you may have been wondering how the spectre fits into this situation. Well, as you can see from the diagram just above, the spectre is just a curvy-edged variant of the hurtle. So why is it, Dave and I wondered, that it's the spectre that got an official name and lots of attention, while the hurtle seems to linger in its shadow, despite it also qualifying as an aperiodic monotile, and having a simpler shape? Asked another way, what edge (har har) do the spectre's curvy edges give it over its straight-edged hurtle partner, that justifies the extra complexity of curviness? Who cares, and why?
Well, as it turns out, it has to do with one's definition of "tile". Or perhaps better said, it has to do with how one decides whether or not two tiles are the "same tile".
In particular, we need to ask ourselves about sameness with respect to a specific one of the fundamental geometric operations: reflection. One way to think about the reflection of a shape is how it would look in a mirror, with left becoming right, and right becoming left. Another way to think about the reflection of a shape is how it would look if you picked it up off the page and flipped it over onto its backside.
Some shapes — any regular polygon, like the square, for example — look the same as their own reflections. Put another way, we could cancel out any reflection with some combination of rotation and translation. Since rotation and translation are the two fundamental operations used when tiling a plane with copies of tiles, these are not considered to change a tile. A tile rotated or translated is still the same tile.
Other shapes, however, do not look the same as their reflections. The hat, turtle, hurtle, and spectre, in fact all of the newly discovered aperiodic tiles are like this. Once we reflect them, we can not get them back to their original shape by rotating and translating them. Many people would say, therefore, that such a mirror-imaged or flipped-over tile is a different tile. Accordingly, these people would say that you couldn't get away with referring to a set containing both a tile and its mirror image as a "monotile".
The concept we're dealing with here is chirality, sometimes also known as handedness. That's because our right and left hands are the most immediate examples of mirror-imaged objects. Also, the Greek root of the word "chiral" means "hand". So the technical term for shapes like the hat, turtle, and hurtle — those have both a left version and a right version — is chiral. These are all chiral shapes. Those that don't have a left and right version like this are called achiral, or not chiral.
Tilings can then be described as either heterochiral, for "different chiralities", or homochiral, for "same chirality". A heterochiral tiling not only uses a chiral tile, but it uses both of that tile's chiralities: its left, and its right. A homochiral tiling either uses only achiral tiles, or if it uses chiral tiles, then they all have the same chirality; in other words, it uses no mirror-imaged tiles.
Smith et al. have used different terminology than Dave and I used regarding chirality. Wherever we have "homochiral", they merely have "chiral", and wherever we have "heterochiral", they have nothing. We find this misleading and likely to lead to some confusion.
Now, if you are the type of person whose intuitions tell them that only homochiral tilings matter — and also you're a type of person who doesn't need to worry about what the other type of person thinks, such as for some professional reason! — then you can safely live your life considering the hurtle to be an aperiodic monotile. For example, if you're setting out to physically tile your kitchen or bathroom, then you're probably assuming homochirality. Your ceramic tiles have a definitive front side and a back side. You can't flip them over and still use them. The front side is the for-show side; it's glazed, or painted, or perhaps beveled, or whatever. The back side is the one to stick to the surface with adhesive. So if your goal is to tile your home with the simplest known aperiodic tile set, in that context, you should go with the hurtle.
If on the other hand (har har), you want or need to stay open-minded about mirror-imaged tiles being considered the same tile, then that's where you'll need the curved edges of the spectre in order to ensure aperiodicity. Many mathematicians are perfectly happy to consider a reflected tile as being the same tile, since reflection is as much of a length-and-angle-preserving operation as translation and rotation, and "congruence" has been defined to allow reflection since Euclid.
In order to fully understand this distinction between the nature of aperiodicity of the spectre and the nature of the aperiodicity of the hurtle, we need to be careful about the definition of "aperiodic". There are some differences between "aperiodic" and "nonperiodic" which I suspect many readers are not aware of, as neither Dave nor I were sensitive to them before our discussions.
Between the two, the simpler property is nonperiodicity, so we'll unpack that one first. Nonperiodicity is a property of a tiling, not a tile or a set of tile shapes. Ignoring some complications designed to eliminate trivial solutions, it's the property where the tiling has no translational symmetry, though it may exhibit some rotational symmetry. In other words, we might be able to rotate it about a central point and overlay it onto itself as an exact copy (that's rotational symmetry), however, we could never lift it up and slide it over somewhere else and overlay it onto itself as an exact copy (that's translational symmetry). Most simple geometric tilings exhibit translational symmetry and are thus periodic.
So now, let's look at aperiodicity. This is not a property of a tiling; it's a property of a tile, or rather a set of tiles. A set of tiles, which possibly contains only a single tile, is considered aperiodic not simply whenever it admits at least one nonperiodic tiling. A tile set can only qualify as aperiodic if it admits only nonperiodic tilings, or in other words, we can make nonperiodic tilings with it, but there's no possible way to make any periodic tilings with it.
So why do we care about the difference? Nonperiodic tilings are all pretty cool if you ask me. Well, aperiodic tiles are even more rare and special. It turns out to be much trickier to find a simple nonperiodic tiling whose tiles cannot also form some periodic tiling. More on this later.
When we put together what we've just learned about the distinction between aperiodic and nonperiodic together with what we learned about chirality in the section before that, then we can finally make complete sense of what exactly makes the hat, turtle, hurtle, and spectre special (or not so special).
The spectre can only produce nonperiodic tilings, no matter what. So it is aperiodic, full stop.
The hurtle can produce both nonperiodic tilings and periodic tilings. Our knee-jerk reaction, then, might be: so it's not aperiodic! However, it's important to note that the only periodic tilings that the hurtle can produce are heterochiral, that is, they're tilings that use both the hurtle and its mirror image. So in this way, the hurtle is aperiodic, with an asterisk: we can say that it's aperiodic as long as we enforce homochirality, such as by keeping the glazed side up.
Chirality is also involved in explaining why many people weren't quite satisfied with the hat or turtle, the tiles that were published with the first paper, in March. For clarity, I will speak only of the hat for the rest of this paragraph, but everything said here goes for the turtle as well. The hat can only produce nonperiodic tilings. So our knee-jerk reaction now might be: well, it's aperiodic, then! However, we have another important thing to note here, and that's that the only tilings the hat can produce are heterochiral, that is, they include hats of both chiralities, a left hat and a right hat. So many people considered this to be sort of cheating, or that it was indeed tantamount to using a two-tile set, not a monotile. It wasn't until the second paper, focusing on the spectre and hurtle, that Smith et al. gave these detractors what they were hoping for.
As for the reason why Smith et al. dismissed the hurtle from consideration as an aperiodic monotile in the first paper, that was because their intuition was to treat reflection as a sameness-preserving operation, just like translation and rotation. And as we've noted, when reflections are permitted, the hurtle (like the extreme ends of the continuum, the chevron and comet) admits periodic tilings and is thus not aperiodic.
Here's a table we assembled that helped us organize this information for ourselves (click for full-size image):
What we've done here is categorize all of the aperiodic tile sets of interest by row according to their cardinality — are they monotiles, or pairs of tiles — and by column according to what we're calling their aperiodic class. We've defined three aperiodic classes, and included subtables that break down which types of tilings that tile sets of that aperiodicity class admit, according to similarity (homochiral or heterochiral) and periodicity (nonperiodic, or periodic).
To be clear, every tile set in this table may be considered aperiodic by some definition. It's up to you to decide if some definitions are not compelling.
We note some further differences in terminology between this post and the papers by Smith-Myers-Kaplan-(Goodman-Strauss). Wherever we say "conditionally homochiral", they say "weakly chiral", and our "(unconditionally) homochiral" is their "strongly chiral."
Note that the hat is used as our representative example of a heterochirally aperiodic monotile, but the turtle is in the same category. Also note that while the hat and turtle individually qualify as a heterochirally aperiodic monotile, when taken together in a two-tile set, this set is conditionally homochirally aperiodic, just like the hurtle, though not a monotile.
You may also be interested in this table, which shows example tilings for each of the three possible types of hurtle tiling. Tiles of the other chirality are colored grey (please forgive the unintentional differences in outline boldness):
Any hat & turtle (two-tile set) tiling can be found by taking one of these hurtle tilings and changing side lengths in a way that changes some tiles into hats and some into turtles. For the heterochiral tilings, the hurtles of one chirality will become hats (of that chirality), and the hurtles of the other chirality become turtles (of that chirality). For the homochiral tiling, hurtles whose colinear edges are oriented to odd multiples of 30° become hats and even multiples become turtles, but they'll be hats and turtles with the same chirality as each other. You can watch this transformation animated for the homochiral nonperiodic case on Craig S. Kaplan's YouTube channel here: Chiral aperiodic monotile animation.
And if we tried to change these hurtle tilings into spectre tilings, we'd find that the bottom two — the heterochiral ones — no longer worked. Only the homochiral one can be curved up without breaking, and that's the only possible tiling the spectre admits.
By the way, if you want to see the heterochiral nonperiodic tilings for either the hat or turtle, you can see them by pausing the following animation at the proper point. This is also from Craig S. Kaplan's YouTube channel: Aperiodic monotile animation.
We decided to include Penrose's tiles in our diagram, to help complete our picture of the situation. We chose the thick & thin rhombs as our representative example of a homochirally aperiodic pair of tiles, but any of his other pairs, such as his kite and dart, would do the trick too. Do note that his tiles require matching rules — or equivalently, modified edges — in order to qualify as aperiodic. Otherwise, they admit periodic tilings, such as this:
Dave and I had both been long-time fans of the Penrose tiles, so we were surprised to find ourselves both led to a feeling of disappointment with them, once we'd learned what we needed to learn about aperiodicity in order to prepare that big table in the previous section. We had been aware to some extent of the matching rules for Penrose tiles, but the full implications of their importance hadn't clicked in the right way for us until now.
To explain our disappointment, we can compare the Penrose tile situation with that of the spectre and hurtle. When we take away the spectre's curved edges, we get the hurtle, which remains aperiodic (assuming homochirality, anyway). However, when we take away the curved edges from the Penrose thick & thin rhombs, we no longer have an aperiodic tile set, and so they are of little interest.
Looking at this another way, if we were to forbid matching rules — considering nothing but tiles' shapes — then in order to achieve aperiodicity with Penrose's tiles, we need to modify their edges, and in doing so, we no longer find ourselves tiling with a simple and elegant pair of rhombuses.
Perhaps, however, we should avoid thinking any less of Penrose's remarkable accomplishments, and instead think even more highly of what Smith et al. discovered this year.
Alright, so that wraps up the first half of our material. I'll post the rest tomorrow, where we'll be focusing on how these new aperiodic tiles relate to each other in terms of kites. Thanks for your attention thus far!