The Sagittal forum

Earlier this year, as I'm sure many of you will already know, mathematics was blessed with the discovery of aperiodic monotiles — not just one of them, but many! In March, the team consisting of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss described the hat and the turtle, and then a couple of months later in May, the same team showed us the curvy-edged spectre.



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

You might also like to go straight to the team's papers themselves:

• An Aperiodic Monotile, from March 2023
• A Chiral Aperiodic Monotile, from May 2023

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).

So the only thing that differs between the tiles along this continuum is their proportion of length between 𝑎 and 𝑏.

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".



Chirality

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.



Periodicity

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.



Aperiodic class

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.



Penrose tiles

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!

Okay, here goes the second half of our material, focusing on how these new aperiodic tiles relate to each other in terms of kites.



Polykites 

As anyone who has ever tried to hand draw a Penrose tiling from a reference image could tell you, it ain't easy. And graph paper doesn't even help.

Remarkably, however, any tiling using only hats or turtles can be traced out using graph paper! At least, it could be, if you printed out a specific type of graph paper. Instead of the typical square grid, you would need to print out what is called a deltoidal trihexagonal grid. Its name makes it sound complicated, but it's not so bad. It's just an overlay of both a triangular grid and a hexagonal grid (the two other possible regular polygon grids, besides the square grid) in such a way as to create cells that are deltoids, more commonly known as kites:



And here's an example of a hat tiling traced onto such a grid:



A kite is a simple geometric shape, a type of quadrilateral — like a square, trapezoid, or rhombus — but defined by having symmetry across one of its diagonals. In this case, we're dealing with a particular type of kite, the one with angles 90°, 60°, 90°, and 120°. The two right angles make this a right kite.

So both the hat and turtle, then, can be called polykites. A polykite is a polygon that can be built up out of some number of kites. The hat is an 8-kite, that is, it is made up of 8 kites; you can confirm this by counting out the kites of any hat in the image above. The turtle is actually not an 8-kite. It's a 10-kite. So that makes the turtle slightly bigger than a hat. (You're welcome to work out the 10 kites of the turtle yourself, or you can just wait to see them in the next section.)



Transforming kites

Interestingly, however, the hurtle — the shape halfway between being a hat and being a turtle — is not a polykite; it can not be traced onto a grid of kites. So this got me wondering: what is the relationship between the 8 kites that make up the hat and the 10 kites that make up the turtle? Is there any way that we could describe a transformation between them, and describe the hurtle as being made up of some sort of in-between shape, or shapes? And whatever this transformation turns out to be, would it continue outward along the aperiodic tile continuum all the way to the two extreme tiles, the chevron and the comet?

Well, after quite a few false starts, Dave and I eventually arrived at a good description of such a transformation, from chevron through chat, hat, hurtle, turtle, turret, and comet. Actually, it turned out that there was not just one, but two different ways to describe this transformation, which were equally valid. The following diagram lays out both of these interpretations (click for full-size image):



The chevron, hat, hurtle, turtle, and comet are all tiles with specific 𝑎 and 𝑏 sizes, but we had a choice of which chat and turret to show. Smith et al. used a 1:4 ratio in their paper. But when I asked Dave if he had a good suggestion for a midpoint, he made the excellent observation that all of these 𝑎 and 𝑏 values can be described using the functions sine and cosine, and this approach points to clear midpoints between chevron and hat for our chat, and between turtle and comet for our turret. These are about 1:3.7. We've included the sin and cos angles in the tile descriptions above.

The first things to notice here are the two tiles we placed in boxes. Consider these to be the starting points for each of the two interpretation rows. In the first row, we start by fully coloring the turtle, grouping its 10 kites into 5 pairs, which we have colored red, green, yellow, blue, and purple. In the second row, we start by fully coloring the hat, grouping its 8 kites into 4 pairs, which we colored using the same set of colors, minus the purple.



At first, let's just look at the hats and turtles; ignore the hurtles in between them for now. Notice how, in the first row, each of the 5 colored kite pairs in that turtle matches up with a single same-colored kite in the corresponding hat. With 8 total kites in this hat, that leaves it with 3 leftover kites, and we find these neatly grouped together as an equilateral triangle, which we've colored grey. Similarly, in the second row, each of the 4 colored kite pairs in that hat matches up with a single same-colored kite in the corresponding turtle. With 10 total kites in this turtle, that leaves it with 6 leftover kites, which we also find neatly grouped together, though this time as a parallelogram (or a double equilateral triangle), which we've also colored grey.

Don't worry about the equilateral triangle and parallelogram just yet. Focus on the matching colored single kites and kite pairs. Notice how each kite pair points in the opposite direction from the single kite. That's because the pointy end of each kite has an internal angle of 60°, and when two of those points come together as they do in our kite pairs, that internal angle sums to 120°, which is the same angle that we find on the opposite, stubbier end of the kite! So, one very simple way to look at how we get from a kite to a kite pair would be: we start shaving down the pointy end of the kite, and keep going until we make the shape which is the same as two kites of the same shape (but smaller). Or in the other direction, we un-shave down one side of the kite pair until they make a shape that's the same as one kite (but bigger).



And what does our shape look like when we're halfway through? Well, the second we started shaving down that point, we created a new edge, and so our 4-sided kite became a 5-sided pentagon. And our shape is still a pentagon once it reaches the same shape as two kites put together. So it's a pentagon the whole way along. The important thing that changes is the proportion of its sides to one another. The halfway point is found where four of the five sides all have the same length as each other. This corresponds with it being the tile where 𝑎 = 𝑏, i.e. Tile(1,1), AKA the hurtle. As for that fifth side, it's slightly shorter than the other four sides. And as for which side it is, it's the side where we've been shaving down what used to be the kite's pointy end.

Okay, now it's time for us to take a look at the hurtles.



In the first row, as you'll recall, the turtle has been completely colored in with our red, green, yellow, blue, and purple colors, and while the hat in this row does contain a bit of each of those colors, it also contains a largish grey area in the shape of an equilateral triangle. Well, the overall shape of the hurtle is in between that of a turtle and a hat, so, unsurprisingly, it's in between them in terms of these characteristics, too: it's mostly colored in with our red, green, yellow, blue, and purple, but not entirely, as is the turtle; and it has an equilateral triangle of grey area, but not nearly as big as the hat's. We could imagine that as the turtle transforms into the hat here — as its kite pairs un-shave themselves out into single kites — a grey triangle starts growing up out of a point between the red, green, and purple pentagons. We reach the hat as soon as this grey triangle is exactly the same size as three kites put together. And it's when we're exactly halfway there — which is also when the pentagons have four equal sides — that's where we have the hurtle.

(An alternative decomposition of the hurtle into an equilateral triangle and pentagons like these is possible, but it cannot be used to link kites in the hat and turtle as we've done here. It can be seen here: https://twitter.com/tessellationfan/sta ... 5742730240)



In the second row, we get it the other way around. The hat is completely colored in, and it's the turtle that has a large grey area. But the hurtle is still halfway between hat and turtle, so it's mostly colored in, with a smaller grey area. The grey area in this row is always a parallelogram, remember. So we could imagine that as the hat transforms into the turtle here — as its kite pairs un-shave themselves out into single kites — a grey parallelogram starts growing up out of a point between the colored shapes (actually, here it grows up out of a line, not a point, meaning that unlike the triangle it is not merely scaling but also changing proportions; but that won't be important until later). We reach the turtle as soon as this grey parallelogram is exactly the same size as six kites put together. And it's when we're exactly halfway there — which is also when the pentagons have four equal sides — that's where we have the hurtle.



Outer tile transformations

Finally, let's take a look at the outer tiles: the turrets and comets, and chats and chevrons. We've got four more sections of these transformations to detail: one for each of the two continuum extremes, and one of these continua per interpretation row. Some of these transformations are easy to visualize and understand, but others are pretty tricky, so we'll work our way up from the easiest one to the trickiest.



Probably the easiest transformation to understand is the top-right one, from turtle to comet in the first row. We've described how we get from a single kite to a pair of kites, by shaving down the pointy end of the single kite until our resultant pentagon is the same shape as two (smaller) kites. Well, we can keep shaving! If we shave down enough, we eventually completely wipe out two of the pentagon's sides, and are left with a quite stubby isosceles triangle. You can fit five such isosceles triangles in the first row's comet.

The remaining space is taken up by a grey equilateral triangle. Notice, however, that this triangle is oriented differently from the one found in this row's hat. You can imagine that the hat's grey triangle shrinks, smaller through the hurtle, and once at the turtle, it reaches zero size, with all sides shrinking down to and converging upon a point. Then, past the turtle into the turret, the sides just keep going in the direction they'd been going, but now they're increasing in size again, and on the opposite side of that point they'd converged on. Cool!



The next transformation we'll look at is the top-left one, from hat to chevron in the first row. This is the first transformation that requires subtractive area to explain. In other words, this transformation involves shapes that partially overlap each other, one with normal (positive) area, and the other with subtractive (negative) area, causing them to cancel each other out.

We can see that the grey triangle in the hat continues to grow until it reaches the size of a triangle within which the chevron is completely inscribed. The chevron can be broken down into four equilateral-triangle-shaped chunks, and the grey triangle — if it were fully intact here — would be the size of nine such triangular chunks. Each one of those five missing chunks corresponds to one of this row's five differently-colored kite pairs.

So: we've already described how we can un-shave a kite pair out into a big single kite. But we can keep going this way too! When we do, we technically have a pentagon shape again, but it's a weird one: a pentagon whose sides cross each other. It looks like a kite that is growing a triangle off of its pointy end, but what's really happening is that those sides aren't coming to a point; rather, those two sides are shooting right past each other, and so the part of the pentagon's area that is bounded by them and the new side that connects them is inside-out. Being inside-out is what makes this area negative, or subtractive.

Eventually, the positive part of the shape gets so tiny that it's essentially zero-sized compared with the negative part of the shape; in other words, the negative triangle gets so big relative to the positive kite that it's growing off of, that the kite can be treated as nothing but a point. So we're left with just a negative triangle.

So when the red pentagon's negative triangle cancels out the top tip of the big grey triangle, the green pentagon's negative triangle cancels out the bottom-left tip, and the yellow, blue, and purple pentagons' negative triangles come together to cancel out the bottom-right tip and most of the rest of the bottom, this leaves us precisely with a chevron shape. It's somewhat tricky, but not too hard to see, and no less cool than the transformation from turtle to comet!



The third transformation is on the same level of trickiness as the second one, pretty much. This is the one from turtle to comet, but in the second row, so in the bottom-right. It involves the same essential idea of the colored pentagons turning partially inside-out and subtracting chunks from the ever-growing grey area. But in this case, we have an ever-growing grey parallelogram instead of a triangle.

Like the chevron in the first row, the comet can be built up out of equilateral-triangle-shaped chunks, specifically, eight of them (note that these ones are 30° in orientation from the first row's). And the grey parallelogram that it's inscribed within would be as big as twelve of these chunks, were it intact. But here, the red pentagon's inside-out part has negated the left tip of the parallelogram, the blue pentagon's inside-out part has negated the right tip, and the green and yellow pentagons' inside-out parts have come together to negate the bottom tip, thus leaving us precisely with a comet shape. Nice.



Now, the fourth and final of these outer transformations is the real doozy: the one in the bottom-left, from the fully-colored hat to the chevron.

It's easy to see, at least, that if each of the four differently-colored kite pairs in this hat transforms into a stubby equilateral triangle (as we saw happen in the top-left case) then these four stubby triangles would be perfect for assembling the chevron. They're even all found in the proper orientation to make this happen.

Where things get much less obvious, however, is how each colored shape gets from its point A to its point B. Well, the green and yellow shapes are fairly straightforward, I suppose. Perhaps it's really just the red shape that has a ways to go to reach its final position.

Another thing that might throw us off is the lack of a grey parallelogram in this chevron's composition. If we were expecting complete visual parallelism with what happens in the top-right case, we would expect some sort of parallelogram to appear here, as an equilateral triangle had made an appearance there.

Well, fret not. We can find a satisfying explanation for all of these mysteries, but it's quite a bit trickier to get one's head around.

The key insight here is that a parallelogram does appear here, but only in the chat; the parallelogram is zero-sized in the hat, and it has returned to zero size again by the time it reaches the chevron. Moreover, unlike the turret's triangle in the top-right, the chat's parallelogram here in the bottom-left is subtractive! To understand why, I'll remind the reader of a parenthetical point made earlier: that in the hat, the parallelogram has been collapsed to zero area, but not at a point, but rather, along a line. In other words, each of the parallelogram's two pairs of parallel sides reaches zero length in some tile, but not the same tile; one pair of sides reaches zero length in the hat, and the other reaches it in the chevron. If either pair of sides are zero length, then the parallelogram is zero area. As we transform past the hat toward chevron, the first pair of sides to reach zero length continue to shrink in length, so that they now have negative length, while the other pair of sides have not reached zero length yet; they're still positive, and still shrinking. (Another way to think about why this parallelogram is negative while the reoriented triangle in the first row was not: vertices need to change order from clockwise to counterclockwise for a shape to be turned inside-out. This happens for the parallelogram, but not for the triangle.)

During this curious chat stage, not only is the grey parallelogram subtractive and canceling out with the colored shapes, but we get another strange occurrence that we've not witnessed elsewhere yet: the colored shapes are also overlapping with each other! The red pentagon slides across one edge of the green pentagon, crossing over to overlap with the yellow and blue pentagons. We've colored in these overlapping regions with orange (red + yellow) and purple (red + blue), respectively.

Now, were this to be the only thing happening, then these overlapping regions would in some sense be doubly-positive in area, and that's a concept that we don't really want to have to deal with in a tile. But as it works out, these overlapping regions also overlap with our subtractive grey parallelogram. And so these doubly-positive areas cancel out with the negative area, which sends them not all the way down to empty space, but just to a normal positive area.

There is one corner of the negative grey parallelogram, however, that does cancel out all the way down to empty space. This is the corner where it overlaps with only one colored pentagon, the blue one.

So that's about all I can say about this transformation. When I first drew this one out on paper, I found it so ridiculous that I almost couldn't believe it. But Dave reassured me that it was correct.



Each colored shape's individual transformation

At this juncture, I'd like to take a closer look at how we get from each colored pair of kites to a single kite. So far I've been describing the transformation as the shaving down of a single kite's pointy end until we find a shape that's the same shape as two (smaller) kites put together, pointing in the opposite direction. That description is nice and simple, and it worked just fine for explaining the transformations across the aperiodic monotile continuum in the previous sections. But there's an even stronger way to describe this transformation that Dave figured out. The only issue is that it's slightly trickier to understand. It's not that the simple way is disingenuous or anything; I just didn't want to open the can of worms that is this trickier way until a bit later on.

Without further ado, here it is (click for full-size image):


We can visualize every form that this shape takes on, within the confines of a 120° sector of a circle. Each shape can be formed by placing five vertices: one vertex at the center of the circle, two of them along the arc (which we'll call the "arc vertices"), and one each on the radii (which we'll call the "radius vertices"). Sometimes none of these vertices coincide, and that's where we get pentagonal shapes. Other times one or more of these vertices coincide, and that's where we get kites or triangles.

This arc is 120°, and we mark every 15° interval along this arc, and label them from 0° to 90° from one end, and then 0° to 90° again but from the other end, so that the labeling overlaps in the middle stretch. In the first shape, the arc vertices start at the two 0° positions. In each next shape, they move by these 15° increments, meeting in the middle, and eventually crossing each other. The two radius vertices also begin at these 0° positions (they're where the radii meet the arc), and in each next shape, they move a bit further along those radii toward the circle's center point, where they ultimately converge.

So beginning with the stubby isosceles triangle, as it is found in the first row's comet and second row's chevron, we have its arc vertices and radius vertices coinciding at the 0° starting points of the arc.

In the next shape, the short isosceles pentagon, the arc vertices have gone to the 15° marks on the arc, while the radius vertices inched ever-so-slightly toward the center.

Then we get the (isosceles pentagon that is shaped like a) kite pair. The arc vertices are now at the 30° marks, and the radius vertices are a smidge closer to the circle center.

Next, the tall isosceles pentagon. The arc vertices are now at the 45° marks, and the radius vertices are about a third of the way to the circle center.

Now, the single kite. The arc vertices are now at the 60° marks, and thus coinciding. The radius vertices are a bit more than halfway to the center.

Almost there. We've arrived at the self-intersecting pentagon. Here, the arc vertices have reached the 75° marks, and for the first time are on opposite sides of each other, causing two of the edges of the pentagon to cross, and thus part of its area to go negative (that part shown in cyan). The radius vertices are now almost all the way to the center.

And the final shape: an equilateral triangle, whose area is completely negative. Here, the arc vertices have reached the 90° marks. And more interestingly, the radius vertices have finally converged on the circle center.

Note that these 15° increments correspond exactly to the angles used for the sinusoidal functions that give the sizes of 𝑎 and 𝑏 in the tiles.



A countably infinite family of aperiodic polykites

For one last point of interest, Dave highlighted something from the first Smith et al. paper, where they explain that not only are the hat and turtle polykites, but there are also an infinite number of polykite chats and polykite turrets. He wanted to see what they looked like. Well, here we are (click for full-size image):



The polykite tiles are all the ones with Tile(1,𝑘√3) or Tile(𝑘√3,1), for some odd positive integer 𝑘. In the above diagram, it is clear to see how this works out this way. The long side of each kite is √3 as long as the short side of each kite. As a chat gets closer and closer to being a chevron, the subtractive regions of the colored pentagons that are shaped like equilateral triangles get bigger and bigger, relatively speaking. Eventually, they get big enough that they're the same size as three kites. The side length of such a triangle is 2√3, since it's the same length as two long sides of kites. In the context of the chat shape, this results in the Tile(1,3√3) chat. The next polykite chat is found when each of the subtractive colored triangles can be subdivided into 4 smaller triangles, each of which is the size of three kites, and has a side length of 4√3, leading to the Tile(1,5√3) chat. We continue on like this, and anytime our subtractive triangle is just the right size such that it breaks down into some number of smaller triangles which are the size of 3 kites together, we get a new polykite chat. The same process works for the polykite turrets. In this case the turrets look like turkeys, getting fatter and fatter.

Note that the chat we chose as the midpoint between the chevron and hat, for purposes of our transformation visualization in the earlier sections, did not happen to be one of these polykite chats. The same goes for the turret we chose as the midpoint between the turtle and comet. Those midpoint tiles are Tile(sin 15°, cos 15°) = Tile(1,2+√3), and Tile(sin 75°, cos 75°) = Tile(2+√3,1), respectively. So these do not satisfy Tile(1,𝑘√3) or Tile(𝑘√3,1). The angles corresponding to these polykites are arctan(𝑘√3) and arctan(1/(𝑘√3)) = 90°−atan(𝑘√3). The pair for 𝑘=3 are at approximately 79.1° and 10.9°, not rational fractions of a circle.

It's interesting to consider the kites outside the large triangle and parallelogram as equivalent to modified edges that prevent the chevrons or comets from tiling periodically (as well as homochirally).

The kite counts of these tiles are related to the area formula given in the previous post. Simply drop the outer factor of √3 to find 2𝑎²+√3𝑎𝑏+𝑏². The polykite chats' kite counts are found by this formula with 𝑎 = 1 and 𝑏 = 𝑘√3:

2𝑎²+√3𝑎𝑏+𝑏²
2(1)²+√3(1)(𝑘√3)+(𝑘√3)²
2 + 3𝑘 + 3𝑘²
3𝑘² + 3𝑘 + 2

And the polykite turrets' kite counts are found by this formula with 𝑎 and 𝑏 assigned the other way around:

2(𝑘√3)²+√3(𝑘√3)(1)+(1)²
6𝑘² + 3𝑘 + 1

Alright, well, if you've made it this far, thanks for sticking around to the end! That's I'll that I've got on the topic for now. Please don't hesitate to let me know if I got anything wrong and I'll see what I can do to patch it. I'm sure this is only the beginning of fun observations that we can make on these new tiles. I encourage anyone reading to share any cool posts they've found elsewhere online, and of course to share any new ways that you yourself may have found to look at these incredible objects.

Parameterizations

The scheme we use above, where we morph smoothly from the chevron to the comet, by deriving 𝑎 and 𝑏 as the sine and cosine of a single parameter θ, is not the only way to do this, although it's hard to imagine a simpler way. On the diagram below:



we can see that it keeps constant the length of the hypotenuse of a right triangle whose other sides are a and b, as this is the line from the center vertex to an arc vertex. So we can call this the constant-hypotenuse parameterization. It keeps √(𝑎²+𝑏²) constant.

Another parameterization of interest is the constant-area parameterization. This ensures that when morphing not just a single tile but a whole tiling, the tiling does not appear to zoom in or out. You can see such zooming in the second of these wonderful videos by Craig Kaplan, but not in the first one. So, clearly Craig has already figured out what I’m about to describe here. These are the same animations we linked above.

Aperiodic monotile animation
Chiral aperiodic monotile animation

Smith et al. told us, that the area of a tile in this continuum is given by √3(2𝑎²+√3𝑎𝑏+𝑏²). Here are the formulas I found for 𝑎 and 𝑏 that allow us to morph using a single parameter θ, while keeping the areas of the tiles constant.

𝑎 = √3sin(𝑥)−√5cos(𝑥)
𝑏 = sin(𝑥)+√15cos(𝑥)
where 𝑥 = 180°−2arctan( (2√6√(2tan²(θ)+√3tan(θ)+1)−√3tan(θ)+3) / (3√5tan(θ)+√15) )

The crazy-looking formula for 𝑥 in terms of θ (thanks Wolfram Alpha) lets us retain θ = arctan(𝑎/𝑏) as was the case for the sin/cos parameterization used in the above posts. This keeps the hurtle at the midpoint between the hat and turtle, and it keeps the hat and turtle at the ⅓ and ⅔ points between the chevron and comet.



An alternative notation

The fact that these relationships can be preserved for two very different parameterizations suggests the following alternative to the Tile(𝑎, 𝑏) notation:

chevron = Tile( 0°)
hat     = Tile(30°)
hurtle  = Tile(45°)
turtle  = Tile(60°)
comet   = Tile(90°)

I discovered that the following homochirally aperiodic tiling of "pears and leaves" (or lemons and leaves), is mutually locally derivable (MLD) from the homochirally aperiodic hurtle tiling. The set consists of two convex polyhedra: a chiral irregular 7-gon and a 45° right triangle, with matching rules enforced by requiring continuity of the white curves decorating the faces of the tiles. The 7-gon has six sides of length 1 and one side of length √2. Clockwise from the side of length √2, it has interior angles: 75°, 150°, 150°, 120°, 120°, 120°, 165°. The triangle has two sides of length 1 and one side of length √2, with angles 45°, 90°, 45°.

The hurtle tiling can be seen as the result of adding specific edge modifications to these 7-gons and triangles to enforce the matching rules, instead of relying on the continuity of the white curves. The edge modification consists of replacing every edge of length 1 with two edges of length √2⁄2 at right angles, with the right-angle pointing in to the polyhedron when the white curve is in the clockwise half of the edge, and pointing out of the polyhedron when the white curve is in the anticlockwise half of the edge. When this is done, the triangles vanish and we are left with only hurtles. The long side of the 7-gon is unaltered and becomes the pair of colinear edges of the hurtle.

Of course the heptagons and triangles were actually obtained from the hurtles, by starting from one end of the pair of colinear edges and skipping every other vertex. The non-convex 14-sided hurtle thereby becomes a convex 7-sided "pear" and some space is left over in the form of 45° right triangles and some squares which can be filled with two of the right triangles.

This process can also be done by starting in the middle of the colinear edges and skipping every other vertex. In this case the chiral irregular heptagons are not convex, but they are equilateral. I call them "oilcans" — they look like the antique kind with an angled spout and a diaphragm in the bottom that goes "pukka pukka". In this case the left-over triangles are equilateral. The reverse process of edge modification in this case requires adding 120° angles, 3 of which completely eliminate each equilateral triangle.