**WARNING: ****It seems as if there may be a problem with the rendering engine that will cause severe flickering on some devices. If you have a seizure disorder or are otherwise sensitive to flickering, please don’t start the game. Sorry about that.**

If you’d actually like to play it, I hope the controls are simple enough:

- Left and right arrow keys to move the current block around the well.
- Up and down arrow keys to rotate the current block.
- Space bar to drop.
- P to pause.

You might need to click in the game window in order to get it to recognize your keystrokes.

Because of the extreme perspective, I expect the game is largely unplayable (though playing Tetris without being able to read the well is an interesting experience, akin to blindfold chess). If you really wish to get through it (the perspective zoom at the end of the level, inspired by Tempest, is pretty badass), press the D key to activate a more conventional Tetris visualization in an overlay.

There’s not much more to say about this experiment. The whole thing is written in P5.js, with the vector display simulation operating as WebGL shaders. The game doesn’t add much to the Tetris (or Tempest) genre, though I haven’t seen a Tetris variant in which the well can be a cylinder, and that turns out to be somewhat interesting. I love musical mashups, so it’s satisfying to contemplate game mashups as well.

Thanks for playing!

]]>Late last year I met Brian Poindexter through a mutual friend from graduate school. Poindexter was part of a team that had been selected to create a large-scale art installation at Burning Man 2019. They were designing a pavilion that would be clad in wooden screens featuring Islamic geometric patterns. Poindexter saw some of my work in this area, and recruited me to create the geometric designs for the screens.

The pavilion as a whole went through several iterations over the next six months or so. By May it had settled down into a five-sided pyramidal tower with decorative fins. My “canvas” would be five large trapezoids (the pyramid was actually a frustum—it was to have a small opening at the top). One of the faces would have an arched passage at its base; the others would have oval windows.

The team sent me CAD files for the sides of the pyramid, which was all I needed to start drawing prototypes. I explored three main ideas for the design; here, I’ve got the luxury of showing you the two designs that we rejected, together with the one we eventually picked.

Zellij refers to a style of Islamic geometric patterns found in Morocco, based on assemblies of tightly fitting glazed terracotta tiles. The shapes of most of the tiles are related to the Seal of Solomon, an eight-pointed star formed from two overlapping squares. Designs often feature a large central star with as many as 96 points, with constellations of smaller stars radiating out from the centre.

I learned most of what I know about this style of design from Jean-Marc Castera’s books. Castera’s explanation of Zellij factors a pattern into a high-level “skeleton” that delineates large regions, and a few standard patterns of tiles that are then used to fill in those regions. This approach is convenient for my oddly shaped canvas—I had some freedom to choose a skeleton that put interesting features in eye-catching places. Above, a 16-pointed star hovers over the portal, surrounded by a hatched Seal of Solomon (reminding me just a bit of the Eye of Providence). Two more 16-pointed stars flank the portal near the ground.

In the end, this design was too busy. It might also have proven too difficult to manufacture, as this level of detail may not have been reliably cuttable from sheets of plywood.

In a second approach, I imagined the trapezoid extended to a complete isosceles triangle, and calculated that the angle at the peak of that triangle would be around 26.9 degrees. That’s somewhere between 360/13 and 360/14. Hypothetically, then, it might be possible to fill the trapezoid with a design based on 13- or 14-fold rotational symmetry. A “wedge” taken out of such a design could be stretched to fit the trapezoid exactly, with lines of reflection symmetry running along the edges of the pyramid—mathematically, a very elegant idea! The hopefully imperceptible stretching even tickles my interest in near-misses.

Now, 13- and 14-fold geometry are quite rare (though not unheard of) in historical Islamic art, so there wasn’t a lot of precedent from which to draw. Fortunately, there’s some lovely contemporary work on the subject to serve as inspiration. In 2012, Jay Bonner and Marc Pelletier published a two-part paper (Part 1, Part 2) on systems for 7-fold Islamic patterns; Jay later published a lot of this material in his comprehensive book on the subject (to which I contributed a chapter). As it happens, the second of their papers had exactly the design I needed as a prototype: a pattern with 14-fold symmetry, from which I extracted a slice that I fit to the panel shape above.

Mathematically, this design works quite well; as you can see in the drawing, the sides of the trapezoid are lines of reflection of the original design. However, the team decided that it didn’t work as well *aesthetically*, which in the end takes precedence.

In the end, we ended up using a variation of one of the first patterns I proposed, really just as an initial example for testing. The design is a excerpt from a repeating pattern of 8- and 16-pointed stars, adapted directly from one of the most famous Islamic geometric patterns in history, found for example in the Alhambra in Spain:

Interestingly, this pattern straddles the design traditions of the two rejected prototypes above. Its structure can mostly be accounted for through a kind of “polygons-in-contact” technique advocated by Jay Bonner, which he used for his 14-pointed design, but some of it definitely borrows from the Moroccan Zellij tradition. This is also, by the way, one of the patterns that M.C. Escher (and his wife Jetta) drew in sketchbooks while visiting the Alhambra, and which profoundly inspired his quest to create “regular divisions of the plane” from figurative animal forms.

After fine-tuning the placement of the design relative to the structure of the pavilion, and stealing the hatching from the Zellij prototype, the team in California decided to move forward with this plan.

At this point (around July), Poindexter et al. went more-or-less incommunicado, as they worked on the serious challenges of actually manufacturing parts that could be hauled out to the playa and assembled. Indeed, I heard so little over the next couple of months that I didn’t even know if they had succeeded! Happily, after the dust had settled (har har) on Burning Man 2019, we reconnected and I finally got to see the results of our work.

The results are quite lovely! Many great photos of the installation can be found online, for example at DesignBoom and Dezeen. There’s an official website for the project, which will hopefully also be updated soon with a post-Burning Man report.

I didn’t know a lot about the inspiration for the project while I was working on it. Later I learned that it had been envisioned by the prominent architect John Marx, while Poindexter and many others formed the “Playa Muses”, who put the work together and brought it to Burning Man. The project as a whole was intended as an homage to heroic women and female empowerment, which makes me even happier to have participated.

Without ever having been to Burning Man myself, I hope I have managed to make a small contribution to its mythology, as part of an artwork that explores decidedly mythological themes.

]]>The artist M.C. Escher drew many lovely tilings, which he called “regular divisions of the plane”. He worked hard to ensure that his tilings were of lifelike animal forms such as birds and fish. He filled notebooks with hand-drawn sketches of tilings, many of which later found their way into his woodcuts. If you’d like a detailed account of these notebooks and their mathematical connections, I recommend Doris Schattschneider’s book *Visions of Symmetry*; A Google image search is also a quick way to see a lot of the drawings.

One of Escher’s lifelong obsessions was the representation of infinity in a finite work of art. He experimented with several ideas for fitting infinitely many tiles into a drawing, culminating in his famous Circle Limit prints based on hyperbolic geometry. But that’s a story for another day. Before that, he drew a number of tilings where tiles form rings or spirals that diminish towards a point at the centre of the drawing. I call your attention in particular to Fish, Path of Life I, Path of Life III, and Development II. His remarkable final print Snakes executes a cunning transition from a hyperbolic tiling near the edge of a disc to a spiral tiling near the centre. Since then, others have created similar drawings. For example, you’ll see a few in a gallery of images by Jos Leys.

Naturally, mathematicians and computer scientists have worked on techniques for creating new tilings in this style. For example, you can see some abstract spirals emerging in a Bridges paper by Robert Fathauer. It seems to me that there has been an uptick in this sort of work recently—I have been sent a few different scholarly manuscripts on spiral tilings to review for journals. However, these manuscripts go through painful contortions to attempt to express the construction of these tilings in awkward, unnatural ways. It turns out that there’s a beautiful and exceedingly simple idea for turning ordinary planar Escher tilings into spiral tilings, an idea I’d like to share here for anyone interested in pursuing research on this topic. It’s an old idea too—I include it in Section 5.3 of my 2002 PhD dissertation, and I certainly don’t claim to have invented it (more on this later).

Most of the rest of this post will explain the math needed to create spiral tilings. You’ll need to be familiar with some geometry, trigonometry, and the complex numbers. Even if you don’t want to work through the math, be sure to scroll down to the bottom: at the end, I’ll introduce a fun web-based tool I created to let you play around with these tilings yourself!

We’ll start with the exponential function . Here is Euler’s number, approximately 2.71828. The function is standard in modelling situations with rapid growth, like populations of organisms or compound interest.

We can extend the domain of the exponential function to the entire complex plane using standard laws for exponents together with Euler’s formula (Euler has a lot of things named after him!). For a given complex number , we end up with

In other words, thinking for a moment in terms of regular Cartesian coordinates, a point will get mapped to a new point , which can also be thought of (via polar coordinates) as a point at distance from the origin, making a counterclockwise angle of with the axis. Note that the and functions repeat with a period of ; therefore, a vertical column of points spaced out by multiples of , like fenceposts seen from above, will all get mapped to the same point by the exponential function. In practice, we might as well factor out this repetition: we can draw pictures by starting with a single horizontal slice of the complex plane of height , and mapping its contents through . What will that look like? We can start to get a handle on how images distort by looking at what happens to straight lines:

The picture on the left shows a square portion of our horizontal slice of the plane of height , which becomes the picture on the right after mapping through the complex exponential. Let’s consider each of the lines shown on the left.

- Sticking with Cartesian coordinates, the green vertical line consists of points of the form for all values of . These get mapped to points of the form on the right. Or, letting , we end up with the set of points for all . In other words, a circle centred on the origin, as shown.
- On the other hand, the orange horizontal line has a varying and a fixed , giving . The range of the real exponential function is all positive real numbers, so the image of the horizontal line is a radial line extending out from the origin at angle . In our drawing the image of the orange line looks a bit like an infinite “wedge”; that happens because I’m mapping the full thickness of the line, and that thickness gets crammed into a tighter space closer to the origin.
- General diagonal lines don’t map to simple shapes like lines or circles. It turns out that they become logarithmic spirals, where the slope of the line determines the pitch of the spiral. Logarithmic spirals arise frequently in nature (the Wikipedia page linked here gives some examples).
**Public Service Announcement:**the so-called*golden spiral*is a logarithmic spiral whose pitch is related to the golden ratio. But not every logarithmic spiral is golden, and*nature doesn’t favour golden spirals*. You’ll often see the claim that nautilus shells are related to the golden ratio, but there’s simply no connection; they do grow (approximately) logarithmically, but there’s nothing forcing them to become golden spirals.

Notice also that although this mapping distorts space quite a bit, it does preserve *angles*: the angle at which two curves cross on the right is the same as the angle between the two original lines on the left. Such mappings are called *conformal*.

Already we can start to see how patterns of lines that repeat in the plane can be turned into interesting spiral patterns:

In general, as long as the top and bottom edges of the drawing on the left match up seamlessly, the exponential map will turn it into a seamless spiral drawing on the right. The effect is a bit like rolling the horizontal slice up into a tube and then looking down the length of the tube (but not really: things don’t shrink in perspective quite as they do with our mapping here). Looked at another way, if you have a pattern that repeats over the whole plane in some direction, you can rotate and scale the pattern so that the repetition is vertical and has length . Then we know that the edges of the slice will line up.

My introduction to the mapping of patterns through the complex exponential function came from the 1992 paper “Two Conformal Mappings” by Robert Dixon, which I also cited in my dissertation. Dixon gave it the more evocative name “the antiMercator mapping”.

A tiling of the plane is called *periodic* if there are vectors (i.e., arrows) in two distinct directions along which you can slide the tiling and have it match up with itself exactly when you’re done. For example, if your tiling is an infinite square grid, you can slide it horizontally or vertically by the edge length of a single square, and the tiling will look the same. We can even exploit this fact to make a animated GIF that loops forever:

These vectors are called *translational symmetries* of the tiling. The nice thing about symmetries is that if you perform several of them in sequence, the result must be a symmetry as well. In this case, if you have a set of vectors that are all translational symmetries, the sum of those vectors is also a symmetry.

To sprinkle in a bit of notation, let’s say you’ve got a periodic tiling, and let and be two translational symmetries of that tiling, as described above. (We’ll want to assume they’re not parallel, though that’s not strictly required.) Let and be two integers, not both zero. Then the new vector is a translational symmetry of the tiling.

OK, then, here’s the punchline: we can rotate and scale the whole tiling so that the vector is vertical and has length . That puts the tiling into a configuration where the top and bottom edges of our canonical horizontal stripe line up, meaning that we’ll get a seamless drawing under the complex exponential map!

We now have a recipe that we can apply starting from any periodic tiling together with integers and . Let’s look at some examples based on a humble tiling of squares. Unlike the previous diagrams, I’m zooming out a bit to show some of the tiling above and below the slice. The slice itself is bounded by the dashed horizontal lines. You can see that in each case I’m taking a different sum of horizontal and vertical square edges as my primary vertical repeat vector. The whole tiling is then oriented so that this sum of the two green vectors is vertical and has length . Something quite nice happens here. Consider the tiling, for example. As you walk along any vertical line in the slice, you’ll pass three grid lines parallel to and seven lines parallel to . This fact is reflected in the spiral tiling: if you look carefully, you can see three spirals of tiles in one direction and seven in another! The colourings below allow you to count them up. The pattern holds in general: a spiral square tiling based on will exhibit spiral arms in one direction and in the other.

Of course, there’s nothing here that restricts you to squares. Any periodic tiling of the plane can be combined with integers and and mapped into one of these spiral arrangements. Here are just a couple of examples; can you figure out the tiling and the values of and that were used to produce them? Note that I’m restricting myself here to simple geometric tilings, but this technique would work exactly the same on fancy Escher-like tilings with varying colours, internal details, and other artistic devices.

As I said, the idea of using the complex exponential function (AKA the antiMercator mapping) to produce drawings like these is far from new. The specific application to periodic tilings shown here isn’t new either, given that it appears in my dissertation from (gulp) more than 15 years ago. I don’t know of an earlier source for this small novelty, but I absolutely don’t think I came up with it myself. In fact, I believe it may have been the great tiling theorist Branko Grünbaum who pointed it out to me in person. I had the good fortune to interact with him while I was working on my PhD, and everything I’ve shown you here would have been immediately obvious to him.

I have one more trick up my sleeve. We can pass the spiral tiling through a different function of the complex plane called a *Möbius transformation*, a function of the form

These are beautiful and fascinating functions. I don’t want to get too deep into their mathematical properties here, so I’ll say intuitively that a Möbius transformation can *exchange a point at infinity with another point*. If I pass a tiling through one of these transformations, I can grab infinitely many tiles that are arbitrarily far away and pack them into a point that looks like the centres of the spirals you’ve already seen. Better yet, I can keep the spiral centre I’ve already got, giving me a picture with two infinities! Here are two examples, one based on squares and one based on hexagons.

Again, this isn’t a new idea. Escher experimented with it in his print Whirlpools, and others such as Jos Leys have created similar designs.

By the way, it’s very tempting to try to repeat this trick and drag a *third* infinity into the picture. Sadly, that simply doesn’t work. Can you see why?

If you’ve read this far (or even if you haven’t), now you have a chance to play with all of these ideas yourself.

Last year I released Tactile, a library for C++ and Javascript that makes it easy to write programs to manipulate and draw tilings. Part of my motivation in creating the library was that it would then be easier for me to write quick pieces of software to demonstrate or exploit ideas from tiling theory. And that’s what I’ve done here! I wrote a fun web-based editing tool that lets you draw tilings and experiment with transforming them into spirals. Tactile is based on *isohedral *tilings; happily, these are all periodic, and so they’re perfect for experimentation in spiral form.

**Click here to launch the tool in a new window**. It’s written to be usable from a touch device like a tablet (though I wouldn’t expect it to be usable on a small phone), but it will work fine with a mouse too. The many features deserve some elaboration. I explain the different controls below, using this annotated screen shot as a guide.

- There are four buttons. The “Fullscreen” button makes the spiral tiling take up the whole window. You can return to the full interface by tapping the button again. The “Animate” button moves the untransformed tiling automatically, making the spiral mapping rotate and scale. The “Help!” button brings you to this post (in case you found the tool without reading this post first). The “Save” button downloads the spiral tiling from your browser.
- This view lets you edit the shape of a single tile. Tap on a black edge to create a new vertex and drag it around. Long-press on a vertex to delete it. The grey vertices cannot be moved directly.
- Controls for the structure of the tiling. The top slider lets you choose a tiling type. There are 81 types encompassing triangles, squares, hexagons, and other base shapes. Each type uses a different number of additional sliders to control the arrangement of grey vertices on the left. Moving those sliders will change the overall configuration of the vertices.
- These two sliders are precisely the values of and discussed above, controlling the vector that will be mapped to cover the horizontal slice. You can actually set them both to zero; the program won’t crash, but it won’t draw anything.
- The regular Euclidean tiling. You can translate the tiling around by clicking and dragging, and see the effect on the spiral view. Horizontal and vertical translation map to uniform scaling and rotation, respectively.
- A view of the resulting spiral tiling. Tap on the view to switch between a regular antiMercator mapping with one infinity, and a Möbius transformed picture with two infinities.

The combination of the complex exponential function and suitable transformations of periodic tilings offers a vast playground for creating beautiful and fun spiral tilings.

Wow, this post ended up longer than I had planned. I hope the diagrams and pretty pictures helped it go down smoothly! For my part, this post and the accompanying software were at least in part a distraction from an upcoming grant deadline. Thanks for validating my brief escape from professional duties.

]]>Now, I do not have sufficient expertise in most of the science behind this work to unpack the whole article here; you might instead read the summary that appeared in *Nature*‘s “News and Views” section. My contribution relates to a few words that the journal chopped off of the end of the title due to space constraints. The original title read “…reversible assembly *and paradoxical geometry*“. The phrase “paradoxical geometry” refers to near-miss Johnson solids, a topic that I’ve studied for a long time; the “protein cage” referred to in the title is, from my point of view, a near miss realized at molecular scale (a fact that is mentioned only in passing in the News and Views summary). I helped to explain the geometry of the cage, and to some extent measure just how paradoxical it is. Here I will give a bit of background on this topic, leading up to a geometric view of the cage’s structure.

The five Platonic solids, the thirteen Archimedean solids, and the prisms and antiprisms are convex polyhedra with regular polygons as faces. But they’re not the only polyhedra with this property. For example, you can imagine a square-based pyramid that’s just tall enough that its triangular sides are equilateral. In 1966, Norman Johnson identified 92 additional convex polyhedra with regular faces, starting with the square-based pyramid; today these are known as the Johnson solids. They’re a motley crew of geometric oddities, which look as if they were assembled out of leftover parts from the Archimedeans.

In 2001, George Hart and I published a paper about polyhedra in which we included a brief mention of “near misses”: convex polyhedra with faces that are almost—but not quite—regular. We did not attempt to define near misses rigorously, merely saying that the angles were close enough that you could assemble a model from cut-out regular polygons “without noticing the discrepancy”. The paper offered these three examples:

The second and third polyhedra are easily exposed as Johnson solid impostors, by identifying vertices that could not exist in a Johnson solid. For example, the second polyhedron has a vertex surrounded by two hexagons and two triangles. A regular hexagon has an interior angle of 120°, and an equilateral triangle has an interior angle of 60°. So, if those faces were all regular, the angles around the vertex would be 60° + 60° + 120° + 120° = 360°. But that’s impossible! When you pack that much angle around a vertex, you force the faces there to lie flat, contradicting our assumption that this polyhedron is convex. The polyhedron on the left is not so easily dismissed. You need to grind through some trigonometry to prove that if the enneagons (9-sided polygons) and squares are regular, then the triangular faces are slightly isosceles and not really equilateral (I get an angle of about 63.1° at the triangle vertex nestled between two enneagons).

I’ve written about near misses elsewhere, including a 2016 post about a new one I had constructed. In 2017, Evelyn Lamb wrote a wonderful article about the more general phenomenon of near misses in mathematics, leading off with near-miss Johnson solids. She also had the good fortune to interview Norman Johnson, who noted that he stumbled on near misses while enumerating his solids, but cared about them only to the extent that he had to eliminate them from his list. Sadly, Johnson passed away shortly after the article was published.

Around the time that we wrote that 2001 paper, I found another interesting near miss, one that didn’t fit with the techniques in the paper. To construct it, start with the pentagonal icositetrahedron (PI), the dual of the more familiar (and Archimedean) snub cube. This solid is made from 24 identical shield-shape pentagonal faces:

Let’s focus on a single shield, as in the shape on the left below:

We can imagine slicing through each of the four upper corners of the shield, as suggested by the four dashed lines. If we do this correctly, the corners are truncated to create new edges that have the same length as the leftover bits of the original shape. The pointy bottom of the shield still sticks out, but by coincidence we can just about fit two more short edges into the point, as shown in the middle. The right drawing shows that these lines define a shape that bears an uncanny resemblance to a regular hendecagon (i.e., a regular 11-sided polygon).

Looked at another way, if we take a perfectly regular hendecagon and fit it by eye to the shield, we can achieve an amazingly good fit. In the diagram below, I’ve lined up the midpoints of the top edges of a shield and a regular hendecagon, and manually scaled the latter until it straddles the edges of the former. The close-up shows just how tight the fit is. You can just barely see the hendecagonal edge tilted relative to the shield edge.

As a final demonstration of the closeness of the fit, let’s compare angles in the shield and the hendecagon:

The angle *α* on the left is a property of the PI, and measures around 114.8°. The angle *β* on the right is exactly 180°⨉(7/11), or around… 114.5°. Again, an amazing coincidence, one that makes it possible to inscribe the hendecagon almost perfectly within the shield.

To complete the construction of a near miss, we can simply glue together regular hendecagons in the same pattern as the shields in the PI. The holes around the tops of shields can easily be filled with equilateral triangles, and the holes around the pointy corners of shields can be filled with clusters of four triangles around a square. Here’s the result:

The paper model on the right is actually easiest to construct, because you can let the laws of physics absorb and distribute the mathematical error inherent in the construction. To build a computer model, you must make explicit decisions about where that error should go. For example, the faces could be made slightly irregular, or slightly non-planar.

Again, I have no rigorous test for near-miss-ness that I could apply to this solid. But intuitively, the miss is very near indeed, and the paper model can easily be constructed without noticing the error.

As an aside, my goal at the time was to create interesting tilings of the sphere to serve as a scaffolding for drawing spherical Islamic geometric patterns. I eventually created a few 3D designs based on the techniques in the paper, and on the hendecagonal near miss. I then turned these models into 3D printed sculptures.

However, I didn’t have much else to say on the mathematics of near misses, so I put the shape on a short web page and left it at that.

Fast forward to late 2013. A team of biochemists had been working with a custom-engineered protein ring called TRAP, which they knew had 11-fold rotational symmetry. They observed that under suitable encouragement from gold nanoparticles, a set of TRAP rings would assemble into a round solid form, which they called a TRAP Cage. But the structure of this cage was something of a mystery, because they didn’t know of a simple geometric form made from pieces with 11-fold symmetry. I assume they searched the web for polyhedra with hendecagonal faces, because they eventually found me by chance. And sure enough, it looked like the TRAP rings in their cage were arranged like the hendecagons in my near miss!

I joined the collaboration, with the aim of building a computational model of the TRAP Cage, from which we could measure the likelihood that the cage could hold itself together based on the distribution of error in the model.

The TRAP Cage is a bit more flexible than the near miss. First, it is constructed purely from TRAP rings, so we don’t have to worry about fitting squares and triangles into the model. Second, the rings aren’t actually attached directly to each other as they would be in a polyhedron; instead, the ring has 11 tiny arms that stick out, and arms from neighbouring rings grab on to gold atoms to hold the structure together. This extra flexibility means that we can likely construct a model cage with even lower error than the original near miss.

I tested this hypothesis using numerical optimization, searching for a symmetric arrangement of TRAP rings for which the gold bonds would have the correct lengths predicted by chemical considerations. The goal of the optimization was to minimize the worst error in bond length, while also trying to keep the arrangement of TRAP rings as round as possible.

This optimization was easily able to find virtual cages where the bond lengths between neighbouring rings never deviated by more than *one part in a billion* from their ideal lengths. That’s an astonishing degree of nearness for a near miss. I’m not a physicist, so I can’t say exactly how near, but allow me to speculate. I’ve got to assume that this deviation is well within the chemical tolerances for atomic bonds, meaning that the TRAP Cage would hold together without ever running afoul of its own mathematical impossibility. Going further, I note that the ideal bond length was given to me with just two significant digits, suggesting that an error of one part in a billion is much smaller than our uncertainty in measuring the “true” bond length in the first place. In other words, we may as well round the error to zero in the messy real world.

The TRAP Cage is, then, a real-world near miss at molecular scale: you could build it out of regular microscopic pieces without, as I said earlier, “noticing the discrepancy”.

The article appeared in the May 16th issue of *Nature*. Even if you’re not a subscriber, they have provided a link that will allow you to read the full article. I tried to use my meagre 3D modelling skills to create a flashy image for the issue’s cover, but it didn’t pass muster. Of course, my blog is the perfect place to display the image.

I’m incredibly excited to see this curious corner of geometry find a real-world application. Perhaps this discovery will motivate me or others to establish a more rigorous theory of near misses. Indeed, Agnieszka Kowalczyk, a mathematics PhD student at Jagiellonian University, together with Bernard Piette from Durham University, are already researching other cage structures with “paradoxical geometry” and have found numerous examples (aided by the fact that these cages don’t need to be fully watertight like polyhedra). They force their polygons to join edge-to-edge but permit them to deform slightly away from regularity, and then work on measuring and minimizing that deformation. They’ll be presenting some of their initial findings at this year’s Bridges conference in Linz. I’m looking forward to seeing what else we can do with this work.

]]>As an example, Susan showed a picture of Mary Shephard’s piece “Wallpapers in Cross-Stitch”:

If you are familiar with repeating two-dimensional patterns, you will know that there are precisely 17 distinct pattern types, known as wallpaper groups. This sampler shows only 12. But as Susan pointed out in her talk, Shephard’s piece is nevertheless a complete symmetry sampler. In cross-stitch, the stitches are applied to a woven fabric (called “aida”), which has holes for stitches arranged in a square lattice. Rectangular grids are fundamental to woven cloth, and necessarily constrain the symmetries that are achievable in cross-stitch. In particular, the square grid cannot support threefold or sixfold rotations (consider, for example, that no three points in a square lattice can form an equilateral triangle), which prohibits the five wallpaper groups that include them (for the record, they are p3, p31m, p3m1, p6, and p6m in the traditional crystallographic notation).

But hang on a minute. The restriction to 12 groups follows directly from the structure of the aida, not from a mathematical law. And while it’s certainly true that almost every cloth on earth is woven using warp and weft at right angles to each other, producing a rectangular arrangement of holes, other weaves are possible. I started thinking about an article I had worked on in my past capacity as editor of Journal of Mathematics and the Arts: Paul Gailiunas’s study of the mathematical properties of mad weave:

Mad weave is *triaxial*: it’s made from threads in three evenly spaced directions. And look at the arrangement of holes left behind:

With a suitable stitching pattern, those holes would permit threefold and sixfold rotational symmetries!

Sitting in Susan’s talk, these two threads came together (so to speak). If I could find (or make) a suitable piece of triaxial cloth, I could use it as a kind of aida. Then I’d have to develop a new stitching pattern that’s as analogous as possible to traditional cross-stitch. At that point, I’d be able to create a hexagonal cross-stitch symmetry sampler. I resolved to attempt this project in time for the following year’s Bridges conference.

Obviously, the cloth would have to be very fine, so that stitches on it would be sufficiently small. I wasn’t eager to weave that myself by hand. So first, I looked into the manufacturing of woven triaxial fabrics. It turns out that they do exist. They have good engineering properties: they tend to be more isotropic than regular woven cloth (they behave similarly in every direction) and less stretchy (because they don’t shear). After a lot of exploration, I reached out to Sakase Adtech in Japan. They very generously sent me a few samples of their polyester triaxial fabrics. The weave they use is actually not the same as mad weave; it’s what Paul Gailiunas called “open hexagonal weave” in his article. But the large hexagonal holes in that weave are still in the correct arrangement:

I had never done any cross-stitching before, so I thought it would be prudent to start by creating a traditional square piece. I adapted a small piece of pixel art I had created a couple of years ago for a laptop sticker, leading to my first ever cross-stitch project:

I made a bunch of mistakes in transcribing the pattern, and you should see the horrific mess in back. But it provided a suitable introduction to the medium, enough to get working on the hexagonal piece.

I needed to develop a stitching pattern that would produce little units as similar as possible to the X that forms the units of traditional cross-stitch. After experimenting with and rejecting a few alternatives, I settled on this pattern:

Here, three individual stitches in a prescribed order (“vertical, backslash, slash”) make up an asterisk-shaped unit. The stitches begin in the six hexagonal holes surrounding another hexagon and jump over that central hole, meaning that a number of holes have no stitches in them. That’s fine, of course, if the units are small enough, and this approach turns out to be the most convenient way to work with this fabric.

It was then time to design patterns. I decided to divide the sampler into a kind of hexagonal honeycomb (to emphasize the sixfold symmetry), and I created vector drawings in Illustrator for every cell. Each drawing shows the low-level hexagonal holes of the fabric, with larger, filled-in hexagons showing the intended units:

With all those pieces in place, the last step was very simple: hours upon hours of punishing, tedious manual labour. My close-up vision has taken a sharp nosedive in the past couple of years, so here I was aided by a *very stylish* magnifying visor with built-in LED headlamp. I’m quite happy with the finished result:

There are five large hexagonal cells, containing the five symmetry groups that are not possible using traditional cross-stitch: reading in rows from the top, p31m, p3m1, p3, p6, and p6m. The other three cells are each divided into three rhombi, showing the other nine wallpaper groups that are also possible with hexagonal holes: p1, p2, pm, pg, cm, pmg, pgg, pmm, and cmm. This fabric can’t show fourfold rotations, which leaves precisely three wallpaper groups (pm, p4g, and p4m) on the sewing room floor. That makes for a grand total of 14 groups, versus 12 for traditional cross-stitch. Take that, Mary Shephard!

A pedantic mathematician will no doubt observe that the sampler doesn’t actually have all the symmetries advertised here. In each individual unit, the stitches are layered, which strictly speaking eliminates nearly all possible kinds of symmetry. Pedant, I salute you! Indeed, if you take the layering of threads into account, and require every unit to follow the vertical-backslash-slash pattern, then I think you’re able to represent only groups p1 and p2. Of course, the same layering problem arises with traditional cross stitch, and we respond with the same simplification: for the purpose of evaluating symmetry we flatten stacks of thread into conceptual blobs of colour.

Very special thanks to Ryoji Sakai and Sakase Adtech for sending me samples of their triaxial fabrics, and to Teruhisa Sugimoto for providing assistance in reaching out to them. Thanks also to Veronika Irvine for providing feedback on both the mathematical and stitching sides of this project, and to Linda Carson for suggesting the oddly shaped matte, which was expertly cut by The Artstore in Waterloo. And of course, thanks to my wife Nathalie for teaching me the basics of cross-stitch, giving me access to her large stash of supplies, and not mocking my visor too much.

]]>I was very lucky as a graduate student to fall in with an advisor who was interested in applications of computer graphics to art and design, and who allowed me a lot of leeway to explore creative topics that interested me. I ended up completing a PhD about two main topics: Islamic geometric patterns and M.C. Escher’s tessellations.

I was also extraordinarily lucky to have a few opportunities to interact with Branko Grünbaum, who was a mathematics professor at the University of Washington (and who sadly passed away just a couple of months ago). He was a brilliant geometer who, together with Geoffrey Shephard, published *Tilings and Patterns*, which to this day remains the bible of tiling theory. It was Branko who first pushed me in the direction of studying the *isohedral tilings *as part of my research on Escher. The isohedral tilings are a particular family of tilings that are in the sweet spot for computer graphics: they’re simple to define and manipulate in software, and expressive enough to represent the sorts of interlocking animal forms that Escher invented. The isohedral tilings became such an integral part of my intellectual landscape that I began to use the word as an account name online (most obviously, because it was unlikely to have been taken by anybody else), and later chose it as the name of this website.

Over the 20 years since I first began studying computation on tilings, I have been contacted numerous times by other computer scientists who were interested in similar research topics and wanted to get a copy of my source code as a starting point for their own work. I always regretfully declined, for good reason. The library I created for my PhD was a mess—I created it by stumbling blindly through the mathematical ideas I needed, not knowing how complex the final product would be. To make matters worse, the old code relied on ancient 2000-era C++ libraries that are difficult or impossible to use today. Of course, I always hoped to have an opportunity to develop a new, streamlined version of the library based on the lessons learned through this first experience.

Today I’m happy to say that I’ve come full circle on the isohedral tilings. I’ve had a bit of time over the last few months to develop a new software library that streamlines and simplifies my earlier version. During my PhD I called my software Tactile, and I’ve decided to revive that name. The new Tactile library is slick, fast, lightweight, and easy to use. It’s been quite fun to play with so far, developing small demo programs that draw tilings, like the random tiling generator in the banner at the top of this page.

The library comes in two flavours: there’s a portable C++ version simply called Tactile, which should easily compile into any application. There’s also a Javascript version called TactileJS, which should make it easy to incorporate live tilings into web-based applications. The two links above will take you to Github repositories for the two libraries. They’re both offered under the permissive 3-clause BSD license, so feel free to use them however you like! Both repositories come with short tutorials and demo programs, which is hopefully enough to get started. If not, I suppose I’ll put together more comprehensive documentation in response to requests by users. I’m also planning to write a more technical paper about the development of this library, which I hope will include some information on how to use it in programs.

If you’re not a programmer, you can still use Tactile to play with tilings, just for fun. On a separate page I offer a minimalist interactive editor for isohedral tilings (see the instructions on that page for more information). That implementation will work well on a laptop or desktop, where you can drive it with a combination of mouse and keyboard. If you want a purely touch-based experience suitable for a tablet, you can also try fancier version at tactile.isohedral.ca. I hope these are fun!

I think I can say with some confidence that one of the direct inspirations for this cup was the death of Lt. Van Mayter in the Star Trek: The Next Generation episode “In Theory” (AKA “Data gets a girlfriend”):

The result is decent. The disconnected tip of the handle doesn’t quite reach the table’s surface, probably because of imperfections in the manufacturing process (the handle might have bent a bit before firing, or it might have shrunk by a different amount than the body of the cup). The handle also doesn’t end in a sharp wedge, as one might expect, because this 3D printing process can’t print very thin surfaces. Finally, the cup is a bit large for a typical espresso. That’s partly to make it big enough that I don’t need to compromise on its shape, and partly because some of the inner volume of the cup isn’t actually usable (because it’s meant to be tilted!).

But the illusion is reasonably convincing. Or, perhaps more accurately, the concept is completely clear, so that even if you don’t *actually believe* that the cup is sinking into the table (and hey, you shouldn’t), you can at least suspend disbelief and enjoy the effect (not to mention some delicious coffee).

As always, the cup is available for purchase on Shapeways, with the caveat that it’s going to cost a lot more than a regular cup. Really, this one might be a fun candidate for mass production.

]]>The first of the two cups is inspired directly by the Moka Pot, the traditional Italian stovetop espresso maker. They almost all take the form of a kind of octagonal hourglass with a spout and a handle, an iconic shape that translates readily into a cup. In this case, I turned the top half into a small espresso cup:

There isn’t much more to say about the design—no hidden mathematical ideas or cute coding to solve design challenges. I’m reasonably happy with the design, though the flat unglazed bottom might not be to everyone’s liking. If you like the cup, it’s available on Shapeways, with the caveat that 3D printed porcelain can be a bit pricy.

Stay tuned for the second of the new cup designs.

]]>I have one final result to share regarding Heesch numbers, but I have to say up front that it’s a bit hairy. It’ll take me a moment just to explain what the problem is. I’ll bring in a lot more technical steps—I’ll “show my work”. I’m not sure my solution will have the immediate appeal of, say, polyominoes, to anyone who isn’t already heavily invested in this topic. Still, I’ll offer this last mathematical morsel in the hopes that it inspires further work on the subject. It’s also worthwhile in that I get to drop in a digression about some exciting new developments in the world of tiling theory.

The previous post discussed a paper by Agaoka, in which he presents a convex heptagon (a seven-sided polygon) with Heesch number 1. The end of the paper contains a selection of intriguing open problems for further study. My family of “ice cream cones”, also discussed previously, lays to rest one of those problems by demonstrating the existence of -sided convex polygons with Heesch number 1 for all . Just a bit further down the page, Agaoka asks:

Is there a convex pentagon with which admits an edge-to-edge corona?

Agaoka uses to refer to the Heesch number of a shape , and his inequalities demand that the Heesch number be positive but finite. So, using the language of the previous post, he’s asking whether there’s a finitely surroundable convex pentagon that admits an edge-to-edge corona.

And what’s “edge-to-edge”? Here’s where things get a little more technical. When we place two shapes next to one another, any time a corner of one shape sits somewhere along the edge of its neighbour we’ll call the result a “T junction”. We’ll then say that a configuration of shapes is edge-to-edge when it contains no T junctions. (To be absolutely precise, it’s nicer to call such configurations “corner-to-corner”; but when all shapes are convex, as they will be here, the two concepts coincide.) Now, thinking way back, the first example I gave of a finitely surroundable shape was Heesch’s original example from 1968:

This shape is certainly a convex pentagon with Heesch number 1, but the surround shown is *not* edge-to-edge: two T junctions are circled. That doesn’t automatically mean that this shape is disqualified from satisfying Agaoka’s problem. It could theoretically be possible to find a *different* surround that avoids T junctions, and all we need is any one edge-to-edge surround to satisfy Agaoka. But as it turns out, we can show that all surrounds of Heesch’s pentagon will necessarily introduce some T junctions. On the other hand, note that Agaoka’s heptagonal bamboo shoot* *does admit edge-to-edge surrounds (but for record, my ice cream cones do not).

We’re left, then, with the problem of finding a convex pentagon that we can constrain enough in shape to allow it to surround itself in an edge-to-edge way, without allowing it to tile the plane. At the outset, this is a lot to ask for. Finitely surroundable shapes walk a fine line between order and disorder: they must be sufficiently well behaved that they play nicely with a ring of neighbours, without being so friendly that the first ring permits a second, a third, and so on out to infinity. But the world of convex pentagons is a already small one, making this frontier zone between order and disorder especially narrow. And asking that the surround be edge-to-edge constrains things even further.

But we do have a small advantage in this context: this world is small enough that we can easily hypothesize possible solutions and check if they actually work. Basically, imagine some possible surround for a convex pentagon. That surround will imply a set of equations that partially or fully determine the pentagon’s angles and edge lengths. If those equations have no solution then the surround can’t exist. If there is a solution, we can construct the pentagon with those angles and edge lengths and check whether it tiles the plane. If we can surround it but not tile the plane with it, we have a solution.

At some point when pondering this problem, I started to develop a strategy based on a pentagon with three shortish edges and two longish edges, imagining a surround looking something like this:The initial shape is shown on the left, with its corners labeled counterclockwise through . My thinking was that I could attach copies of the pentagon to the three short edges, leaving gaps under the resulting “wings”. Those gaps would then be filled by slotting pointy corners up into the exposed vertices. The other angles would be carefully chosen so that the two copies covering the long edges and would also touch each other, completing the surround. I aimed for two copies under the left wing and three under the right, assuming that the asymmetry would help guarantee that the shape couldn’t be too simple. Note that the diagram above is really just a sketch to guess at the arrangement of neighbours in the surround—the tiles are all different shapes. We don’t yet know whether a shape even exists that is compatible with this arrangement.

We need to copy the vertex labels to the other tiles in the surround in order to determine their orientations (the positions of the vertices are clear, but each tile may be oriented clockwise or counterclockwise). Let’s try the following configuration:The labelling makes it clear that there are only two distinct edge lengths. Because the original pentagon’s is adjacent to a neighbour’s , we know that . Similarly, . Thus we can think of this pentagon as being an isosceles triangle with a quadrilateral glued to its base. Let’s abuse the notation so that the vertex labels will also refer to the angles at those vertices, and then split the pentagon as above, dividing angle into and , and into and :From these two diagrams, we can improve our understanding of the unknown angles considerably. First, because is isosceles, we know that . But at the bottom of the proposed surround we see that we have a vertex surrounded by an and two s, from which we know . Substituting, we find that , whence . Similarly, the pentagon’s remaining surrounded vertices imply three other equations: , , and . And because is a quadrilateral, we know . With enough persistence, these equations allow us to express all angles in terms of , yielding the following cleaned-up (but still imperfect) diagram:Now, we’ve still got a degree of freedom that we’d eliminate in ideal circumstances, to produce a single specific value for . And it turns out that we can, if we also use what we know about the edge lengths. Let’s focus on the quadrilateral:We know that the three upper edges all have the same length; in particular the triangle originally labelled is isosceles. If we add a diagonal from to (and use to denote its length), we can divide up the angles as shown on the right. Using one of the angles in the right-angled triangle, some trigonometry tells us that . We can also apply the law of sines to the isosceles triangle to learn that . We can arrange that equation (and remember that ) to obtain . Equating the two derivations above at last gives us an equation purely in the unknown :

This is far from pretty, but I happen to have a soft spot for trigonometry. By grinding through a few fun applications of trig identities, it’s possible to deduce that

We thus find that , and we can use that value to solve for and . Plug all those values in, construct the pentagon, and lo and behold it can be surrounded!So, is this the solution to Agaoka’s question that we’ve been seeking? It’s definitely convex, and definitely a pentagon. It has a surround, so its Heesch number is at least 1, and the surround shown is edge-to-edge. We need to clear one final hurdle: we need to prove that the shape doesn’t tile the plane.

How do we know whether a given convex pentagon tiles the plane? I’ve dropped hints for a while that this question is interesting. Recall that for convex polygons in general, the answer is trivial for most numbers of sides: all triangles and quadrilaterals tile the plane, and no convex -gon does for . The answer for hexagons has been understood for about a hundred years: there are three families of convex hexagons that tile, and it’s easy to check whether any given shape belongs to one of these types.

Convex pentagons have a much more colourful history. Karl Reinhardt first enumerated the three convex hexagon tiling families in 1918. In the same work, he offered a list of five types of convex pentagons that tile, but did not claim the list was complete. Over the next hundred years, an additional ten types were discovered by a mix of mathematicians and enthusiastic amateurs. (Some of these types consist of a single tile shape). The 15th type was discovered only two years ago by a team including Casey Mann, whose name has come up a few times in this series; the discovery made the rounds online as an exciting mathematical achievement. Then, earlier this year, Michaël Rao announced that he had successfully conducted an exhaustive search over all possible ways a convex pentagon could tile the plane, a search that confirmed the list of 15 types already found and didn’t turn up any new ones. I don’t believe Rao’s article has been published in a peer-reviewed forum yet, but the consensus among tiling theory researchers is that his approach is valid. In that case, we know when a given convex pentagon tiles: precisely when it belongs to one of these 15 types. It was just my good luck, then, that the one day that I could attend the CRM workshop on tilings was the day when Rao gave a talk about this work!

So, we now have a simple test for whether the pentagon proposed above satisfies Agaoka’s problem. We check it against the 15 pentagon types, and discover……that it’s a perfect match for Type 6. Dang.

So does that mean it’s game over? Not surprisingly, I wouldn’t have built up to this point if I didn’t have one final ace up my sleeve. Looking back over the construction of the previous section, there are a few places where I left behind “knobs” I can now go back and twiddle to see if I get an answer I like more. In particular, when first building a hypothetical surround, I arbitrarily chose to put two copies of the shape under what I called the “left wing”, and three copies under the “right wing”. What if we tried other values?

It turns out that for any , we can put pentagons under the left wing and pentagons under the right wing, and repeat the rest of the construction above nearly verbatim. That revised derivation eventually leads to a new quadrilateral, fortuitously similar to one that we used to solve for the angle originally:Now, we could repeat the trig that allowed us to solve for , but there’s a shortcut: if we let , then we get the same angle relationships as before, now in terms of . Therefore we know that , or . From this new derivation we can construct—and surround!—a new infinite family of pentagons for all :I claim that these, at last, solve Agaoka’s problem. They’re convex pentagons with edge-to-edge surrounds. And for all , we can verify that they don’t tile the plane by checking against the 15 types of convex pentagons that do. As a bonus, we get an infinite family of pentagons, not just one!

It looks more specifically like all of these shapes have a Heesch number of 1. I could be wrong in two obvious ways. First, one or more of them might have a higher Heesch number. That would be amazing! I’d love to see a pentagon with a Heesch number higher than 1. And for the record, the shape would still be a win with respect to Agaoka, who wasn’t specific about the Heesch number, only that it be finite and positive. Second, if one of these shapes actually does tile the plane, that’s even more amazing! It would serve as a 16th type of tiling pentagon, and a harbinger of a flaw hidden in Rao’s exhaustive search. Both of these possibilities seem far more remote than the simple notion that I have found finitely surroundable convex pentagons with edge-to-edge surrounds, which nobody ever claimed were unlikely to exist.

So, that’s another of Agaoka’s problems settled, after quite a bit of scribbling. The problem that lingers most strongly in my mind is this: find a pentagon (convex or otherwise) with Heesch number 2, or prove that none can exist. More generally, if we believe Rao’s result and consider the classification of tiling convex pentagons settled, could we carry out a similar classification of *non-*tiling convex pentagons? That is, determine how many types there are of finitely surroundable convex pentagons, and label each type with its Heesch number. I’m not an expert on Rao’s method; perhaps it could be adapted to search for surroundability instead of tileability.

That’s also the end of this series about Heesch numbers, capturing a few new results I’ve come up with. (I may go back and update the table of Heesch numbers of polyforms if I compute more of them.) To some extent, this has been an experiment in research dissemination. I was unsure at the outset whether these results were sufficiently exciting to publish as an article in a math journal. On the other hand, I’ve always found that Heesch numbers have wide appeal to a general audience, and I enjoy the outreach component of my job a great deal. So it seemed natural to write these ideas up as a sequence of (more-or-less) accessible blog posts. I’m also interested longer term in popular writing on mathematics, and this was an opportunity to see if I could put together something in that form. Does the text work in this form? Should I create a more academic manuscript? Your feedback is welcome.

**Update [October 14th, 2017]: **A few weeks after publishing this note about my pentagon, I was delighted to receive a message from Teruhisa Sugimoto in Japan. It turns out that he had been working on a much more systematic and comprehensive enumeration of convex, finitely surroundable pentagons, resulting in an enumeration of 17 distinct types! My noble effort is now but one voice in a larger chorus (Category 3, to be precise). He wrote again recently to inform me that he posted a draft paper, “Convex Pentagons with Heesch Number” on his website. Sadly for me it’s in Japanese, but the diagrams tell a lovely story. Congratulations, Mr. Sugimoto!

Earlier this year, the CRM, a math research institute at the Université de Montréal, held a two-week session entitled “Combinatorics on Words and Tilings“. Because of ridiculous scheduling constraints, I was able to attend for *one day*. Thus it was that I found myself taking a full-day train ride on Tuesday, attending the workshop on Wednesday, and taking the train home on Thursday. Mind you, there was a side benefit: I was able to spend the evenings with my parents and brother in Montreal, and belatedly celebrate my parents’ 50th anniversary. And happily, it turns out that I chose a good day to be present at the workshop; more on that in the next post.

Also happily, I ended up spending most of the day sitting next to tiling theorist Casey Mann. Casey is the current Heavyweight Champion of Heesch numbers, having found the 5-hexapillar with Heesch number 5 that I showed previously.

As part of our idle conversation, Casey introduced me to a lovely short paper I hadn’t seen before: “An example of convex heptagon with Heesch number one” by Yoshio Agaoka. Responding to an earlier conjecture that every convex heptagon (7-sided polygon) would necessarily have Heesch number 0, Agaoka introduces the following shape, which he calls a “bamboo shoot”:

Agaoka claims that this shape has Heesch number 1. In fact, it’s not too hard to see why this must be so. First, unless I’m perpetrating a vicious deception, the configuration above shows that the Heesch number is at least 1, so we must simply convince ourselves that it could not be higher.

Note first that this shape definitely can’t tile the plane. Here, we can make use of one of my all-time favourite facts in tiling theory, one that I take great pleasure in repeating: no convex polygon with more than six sides can tile the plane, period! How can we possibly know so much about a shape’s capabilities based only on its number of sides? The theorem ultimately boils down to a variation on one of the most famous formulas in geometry: Euler’s formula. You know, *V* – *E* + *F* = 2, where *V*, *E*, and *F* are the numbers of vertices, edges, and faces, respectively, in a polyhedron. It turns out that the same formula holds in the plane, and further can be adapted to hold for an infinite tiling as the limit of ever-larger finite chunks of that tiling. In this case, if you tried to tile the plane with a convex polygon with more than six sides, you’d find that you were laying down too many new vertices to surround them all with tiles, a deficit that cannot be reversed as a patch of tiles grows to cover the whole plane. (On the other hand, a non-convex polygon can mate with a neighbour along many edges at the same time, potentially sidestepping this issue.) I find this theorem both astounding and infuriating—nobody’s going to tell *me *what I can and can’t do with a convex polygon!—and yet there it is, buried somehow deep in the firmware of the Euclidean plane. It also nearly settles the question of which convex polygons tile the plane. We know that all 3-gons and 4-gons do, and now we know that no *n*-gons tile for *n *≥ 7. That leaves just 5 and 6; more about them in the next post.

Knowing that the bamboo shoot doesn’t tile means it’s already guaranteed to be interesting, because its Heesch number is positive but finite. For convenience, let’s call such shapes *finitely surroundable*. In the search for high Heesch numbers, finitely surroundable shapes are guaranteed to be at least somewhat interesting stops along the way.

Agaoka completes his proof by noting that any legal surround of the bamboo shoot must contain at least these four neighbours:It’s possible to deduce the necessity of this configuration by remembering that the material that surrounds each vertex must add up to 360°. Consider the marked vertex above, for example. The bamboo shoot has an angle of 160° there; we must make up the other 200° using some combination of 60, 100, and 160 (the bamboo shoot’s angles) and 180 (which you can achieve by having a polygon edge partially overlap the vertex). It’s easy to see that the only possible solution is two 100° vertices. There are four ways to position adjacent shoots to cover this vertex, but the other three would lead to subsequent gaps of 140° or 40° at neighbouring vertices, neither of which can be closed off:It’s then possible to work from the legal partial surround above to figure out all possible ways to fully surround the central shape. Moreover, note that the two side-by-side orange tiles making up one of the “wings” leave behind another angle of 40°, meaning that none of these legal surrounds will itself be surroundable. Thus the Heesch number must be precisely 1.

Best of all, Agaoka ends his paper with a few additional unsolved problems. Among them, he asks a natural follow-up question: does there exist a convex *octagon* with Heesch number 1? In the rest of this post, I will answer a generalization of this question, and describe a family of shapes {*S _{n }*}, where each shape

The sides of the bamboo shoot suggest a way forward: let’s imagine that part of a shape will be an “arc” of short sides. Opposite them will be a single additional short side that allows a “fan” of copies of the shape to be placed along the arc. We end up with a shape that looks a bit like… an ice cream cone? In any case, we’re looking for a configuration that looks something like this:

The sketch shows a heptagon, but let’s assume we’re working more generally with an *n*-sided polygon, so that the arc will consist of *n* – 3 short edges. The rest of the polygon will consist of a single short edge sandwiched between two long edges. We’ll further assume that the angles at the ends of the arc, where the short edges meet the long edges, are also *A*. In that case, we already know enough to solve for the marked angles *A* and *B* in the drawing, based on two equations:

The first equation simply says that it should be possible to surround a vertex by an *A* and two *B*s. The second equation recognizes the the interior angles of an *n*-sided polygon must total 180(*n*-2) degrees. Substituting and solving, we find that

If we fix the short edges to have unit length, then a bit of annoying trigonometry is required to compute the lengths of the long edges, but it’s pretty easy to write a program that solves for the edge length numerically. I’ll omit the details.

The result for *n* = 7 looks like the shape on the left below.The middle diagram shows a partial surround in which we use up all the short edges, as planned. But the best part is that the gaps left behind along the long edges can be precisely filled with two more copies of the shape, yielding a complete surround, as shown on the right (note that all of the original shape’s vertices end up surrounded by two *B*s and an *A*). We know this shape can’t tile, and so it’s already guaranteed to be finitely surroundable (i.e., interesting!). Could it have Heesch number 2? By examining possible ways to sum angles to 360°, it’s possible to convince oneself that no surround of the central shape can be further surrounded—you’ll always leave behind unfillable acute angles as in the right-hand diagram above. The details are a bit boring, but if you want you can read my sketch proof. (Of course, I wouldn’t mind if one of these shapes could be further surrounded, as it means its Heesch number is 2 or higher!)

There’s nothing about the previous discussion that’s tied to the fact that the shape had 7 sides, and indeed, this construction generalizes to all higher numbers, yielding an infinite family of convex polygons, all with Heesch number 1. Here are the corresponding configurations for shapes with 8, 9, 10 and 11 sides:

The construction definitely *doesn’t* work for 5 and 6 sides: it produces shapes that tile the plane (most obviously, the construction for 6 sides yields a regular hexagon):But if we want to be sneaky, we can piggyback on this technique to invent a new pentagon and a new hexagon that have Heesch number 1. All we have to do is take one of the larger shapes above and divide it in half. Because the shape is symmetric, the two halves will be congruent, and so we already know that we’ll have at least as much surroundability as we had previously. The diagrams below show the construction of a pentagon from half of a heptagon, and a hexagon from half of an octagon.Here I’m being a bit cavalier mathematically. Perhaps other surrounds could be found for one of these shapes that permit additional layers of tiles, and hence higher Heesch numbers. Indeed, because the shapes on the right have fewer than 7 sides, we can’t even automatically assume they don’t tile. Can we prove they don’t tile by other means? As I said above, pentagons and hexagons are the only convex polygons for which we don’t have an immediate yes/no answer regarding tileability. Are there special rules that hold in these two cases? I’ll say a bit more about that next time. In the meantime, I’m confident that these sliced shapes both work as advertised, though I’ll retain a modicum of doubt.

The good news here, then, is that we appear to have a piece of genuinely new math: an infinite family of convex polygons, all with Heesch number 1. Having unlocked this particular achievement, I suppose I’ll finish by taking the opportunity to pose my own problem: find a convex polygon with Heesch number 2. Unless I’m mistaken, no such polygon is currently known.

**Update [November 12th, 2018]: **I recently received an email from Alexander Thomas, a mathematics PhD student at IRMA in Strasbourg. He wrote to tell me that when he was still in high school he also studied some of these same problems together with Christopher Standke, and found a different family of convex -gons with Heesch number 1 for all . His report is written in German, but the diagrams make it pretty clear that he has a working solution (see especially Pages 31–35). Nice work!