A chiral aperiodic monotile

[ColinWright]

There was a rumour that if they found a chiral aperiodic monotile then they might call it a Vampire Tile, because it doesn't have a reflection.

Seems like they didn't go with that.

I also see that the old discussion has come up: "But what can it be used for?"

These sorts of things are pursued because they are fun, and there's a community of people who find it interesting. Is Rachmaninoff's second piano concerto useful? Is Bach's Toccata and Fugue in D minor (BWV 565)[0] useful? Is Rodin's "The Thinker" useful?

No. And for each of them there are people who Simply. Don't. Care.

So it is with Pure Maths.

The difference is that sometimes things people pursued simply out of interest or curiosity turn out to be useful. It might be decades down the line, but it happens, and you never know in advance which bit of maths they will be.

So maybe Chiral Aperiodic Tilings will turn out to be useful, maybe not. Maybe the work done to create them is what will turn out to be useful. Maybe not.

It's not the point.

[0] Interestingly, this might not be by Bach, and some claim it's not in D minor.

[dpflan]

You've reminded me of an older Quora post about: What do grad students in math do all day? Do they just sit at their desk and think? [1] Here are excerpts:

""" The main issue is that, by the time you get to the frontiers of math, the words to describe the concepts don't really exist yet. Communicating these ideas is a bit like trying to explain a vacuum cleaner to someone who has never seen one, except you're only allowed to use words that are four letters long or shorter.

...

This [research] goes on for several years, and finally you write a thesis about how if you turn a vacuum cleaner upside-down and submerge the top end in water, you can make bubbles!

Your thesis committee is unsure of how this could ever be useful, but it seems pretty cool and bubbles are pretty, so they think that maybe something useful could come out of it eventually. Maybe.

And, indeed, you are lucky! After a hundred years or so, your idea (along with a bunch of other ideas) leads to the development of aquarium air pumps, an essential tool in the rapidly growing field of research on artificial goldfish habitats. Yay! """

- [1.] https://www.quora.com/What-do-grad-students-in-math-do-all-d...

[pfd1986]

I agree with you that application is not necessarily the point (one of my mentors would always answer this questions with "what's a baby for?")

But in materials science / physics this has been a long standing puzzle: we know that hard polygons vibrating thermally should have a global entropy maximum (ground state) equal to their closest packing configuration. Can this ground state configuration be aperiodic?

So far, all quasicrystals discovered are either not in stable equilibrium or are in equilibrium at that pressure and temperature, but are not the ground state of the material (i.e. at infinite pressure, that QC would be unstable). The discovery of an aperiodic single tile shape means that, in equilibrium, this polygon should have a ground truth that is aperiodic. That basically settles this long-standing question.

[abetusk]

Just a note about applications, there is a pretty direct application of aperiodic tiles with quasicrystals [0]. Diffraction patterns of quasicrystals can have symmetry that can't occur with periodic tiling which explained some weird diffraction patterns people were observing.

One can imagine a scenario, occurring for metal or mineral creation or even in a biological setting, where only one shape is allowed because of some external constraint, including not allowing it's mirror.

[0] https://en.wikipedia.org/wiki/Quasicrystal

[hcs]

There is this footnote on pg 2:

> We might call this the "vampire einstein" problem, as we are seeking a shape that is not accompanied by its reflection

Also the glorious

> Lemma 2.1. There exists a Spectre.

[adhesive_wombat]

Example from the top of my head: some pretty "niche" (in the days of Galois, say) number theory is now the critical component in a large proportion of digital processing, cryptography and communications (e.g. forward error correction).

[codeflo]

This excites me much more than the original result, which I considered to use two tiles[1]. The fact that it’s a such tiny modification of the original result is crazy. Even if you don’t intend to read the paper, look at the illustrations of the hierarchical substitution algorithm at the top of pages 6 and 7, those are just beautiful.

[1] The authors discuss various historic definitions of tilings and whether reflections should be allowed or not (they argue that most definitions allow them). For me, the answer is simple: nature is chiral, you can’t reflect things willy-nilly. Puzzle pieces, bathroom tiles, even polygons in 3D rendering all have distinguishable sides.

[kzrdude]

Agreed, this is an indisputable improvement on Penrose tiles while the other one didn't feel like it.

[BlueTemplar]

And yet again, it's only available as PDF (rather than the standard HTML), which is super annoying when you have no desire to print it out, especially to view on a small screen. Nor that is seems like this document benefits in any way to be pre-separated into discrete pages (unlike for say, slides).

[abetusk]

The recently discovered 2D "hat" monotile tiles aperiodically but allows reflections (and must allow reflections to tile?).

The paper discusses a (2D) monotile whose shape will allow an aperiodic tiling without reflections.

[alreadydone]

I hope these tables help: https://twitter.com/Junyan_Xu/status/1663396061942038530

[jameshart]

Notably, it's the same authors: David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

[TrueDuality]

Hmmm... I wonder if these spectres are potentially a basis for a new form of cryptographic algorithms. Unique non-repeating sequences exclusively derived from a set of rules and an initial state in multiple dimensions sounds like a promising candidate.

[AlotOfReading]

I'm not understanding why that's different than seeded random sequence or the sequential digits of any irrational number? The latter guarantees a non-repeating sequence and can be trivially generated with square roots.

[scotty79]

Is that finally the single tile you can use to cover whole space aperiodically without flipping it upside down?

[Thomashuet]

Yes. Actually if you just want to cover the whole space aperiodically, you can already do it with a simple rectangle, it's just that rectangles also allow you to do it periodically and this new tile only allows aperiodically.

[rthomas6]

I wish I could buy ceramic Penrose P3 tiles to put on my floor or wall. They 2 different tiles instead of 1, but they're simple diamond shapes, and they tile aperiodically.

[sudb]

The discovery of the aperiodic monotile was what finally pushed me over the line to sign up for a ceramics beginners course funnily enough! Give me about a year...

[mabbo]

This feels like a niche market someone needs to go after. I would love to have a bathroom floor or a kitchen backsplash tiled in 'hat' in 4 colours (ensuring no two same-colour tiles touched of course).

[lordfrito]

Someone needs to start a Kickstarter for this. I'd contribute.

[CGamesPlay]

How could you address each tile, to create an aperiodic tile map? Would be a neat tech demo. Like HyperRogue, for example. https://www.roguetemple.com/z/hyper/

[robinhouston]

The most natural way to address the tiles is to use a sequence of indices based on the hierarchical construction.

Simon Tatham has written about this, in the context of the original hat tile: https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...

[zenorogue]

Addressing hierarchically constructed tiling is quite similar to addressing hyperbolic tilings. Both the "hat" tiling and "spectre" tiling already work in HyperRogue. (Spectre is not yet released but pushed to GitHub.)

There is a brief explanation here: https://twitter.com/ZenoRogue/status/1638997769141604360

[lying4fun]

You might be interested in this https://twitter.com/ZenoRogue/status/1663589972094353418

[onos]

Can anyone explain why people are interested in tilings?

[troymc]

Tilings are easy to understand and relate to... you can tile your bathroom floor with them for example. Despite that superficial banality, it turns out that it takes a lot of mathematical cleverness to construct and analyze them fully. This particular result is one that people have been chasing for decades and has involved some of the smartest mathamaticians in the world, including Conway and Penrose.

[xeonmc]

Earthquake resistance of buildings. See Incan usage of aperiodic masonry to spread out the frequency response of the construction.

Similar principle with Apple's laptop fan blades.

Same mechanism might be responsible for the Boson peak phenomena in amorphous materials and quasicrystals, where the macro-structure creates extra capacity for absorbing lower-than-lattice-frequency vibrations than what the crystal-structure alone predicts.

It's all about Fourier analysis.

[chriswarbo]

Tilings can encode arbitrary computation. For example, any Turing machine can be encoded as a set of Wang tiles. Some shapes can tile the plane; others can't (they inevitably get stuck, with no space to attach new copies without overlap). This is precisely the Halting Problem.

One famous application of this is to encode these shapes using complementary snippets of DNA, to perform massively-parallel computation at the nano-scale: https://www.nature.com/articles/35035038

[sapling-ginger]

Game graphics. Aperiodic tiling allows us to render a natural-looking surface using a minimal texture graphic file.

[thsksbd]

1. Its fairly pure math that you can see

2. Its important in technology and science (e.g quasi crystals).

3. They have an aesthetic some find pleasing.

[BlueTemplar]

One of the things this has consequences on are the physics of crystals and pseudocrystals.

[joshu]

Only slightly related: anyone know how to make porcelain or other normal tiles? If I wanted to redo the bathroom in these or whatever, can they be made?

[tiler2915072]

I’ve spent a decent amount of time figuring out to make porcelain tile. The most reliable way to do so and avoid cracking is to use a plaster mold and push slabs of clay into it. Cookie cutter and other slab methods produce too much cracking without heavy duty equipment or maybe a different clay formulation (I’m using standard porcelain which isn’t the easiest material)

[daniel_reetz]

They're either usually formed in a mold by pressing. The resulting "green" tile is sometimes stamped or patterned, and glaze is applied. Then they are fired in a kiln.

[twic]

Here's how encaustic cement tiles are made - with liquid cement, cement powder, a mould, and a press: https://youtu.be/XSp5Tj4yYNc

[gilleain]

I recently bought this book : "Arts & Crafts of the Islamic Lands: Principles • Materials • Practice" (Thames and Hudson Ltd, edited by Khaled Azzam).

It has a section on making similar (ish) geometric tiles, although the description is really for square tiles with the geometric design on the face.

From recent experience of drawing and reproducing tiles (including trying to draw the 'hat' monotile) I think the tolerances on your physical tiles would have to be quite small. Either that or make them as a more regular shape and cut out the correct tile from that larger one.

[isoprophlex]

You need pretty strict tolerances on the dimensions of this Spectre tile, I think. Normal Penrose tiles might be easier.

That said I have no idea in the first place how you'd get your hands on bespoke tiles...

[jdthedisciple]

If I was a billionaire I'd have such custom-made tiling installed in my home's hallways and terrace.

[wahahah]

I wouldn't see it being prohibitively expensive https://news.ycombinator.com/item?id=36121506

[firstlink]

I wonder if the authors anticipated this result when publishing their first paper, or if they were primarily motivated by "complaints" that their hat tile (and other tiles in the associated spectrum) required reflection. Certainly they mention this question, but my question is whether they completely anticipated it.

[ravi-delia]

I believe reflection is usually permitted in tilings- you certainly wouldn't wait around until you worked out a chiral tiling to announce. But it's a direct descendent of the hat tile, and I wouldn't be surprised if they already knew the direction to look before they published the original paper.

[yarg]

They may have been encouraged by the reactions to their previous paper, but I doubt it was anything close to a primary motivating factor - they're mathematicians, you can pretty much guarantee that they found the compromise at least as irksome as everyone else.

[gradschool]

Random thought/nerd snipe hazard: would a Wolfram-style classification of cellular automata running on this tiling kick up anything interesting?

[dvgrn]

There have been some experiments along these lines for "hat" tilings:

  https://conwaylife.com/forums/viewtopic.php?p=161571#p161571

As far as I know, HatLife hasn't been adjusted to make SpectreLife yet, but it's probably only a matter of time!

[jdthedisciple]

Seems useful for creating unique and modern designs.

[petters]

Can someone please let me buy actual tiles shaped like this? I would like to renovate my bathroom

[thriftwy]

I wonder if there is an aperiodic monotile that may tile a sphere.