A chiral aperiodic monotile
arxiv.orgThere 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.
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...
>> What do grad students in math do all day?
> ... by the time you get to the frontiers of math, the words to describe the concepts don't really exist yet.
A friend of mine[0] once said that the act of doing research in math is the act of inventing a language in which you can talk about the problem. Once you have that, the solution tends to come. But inventing the right language is really, really hard.
This might explain why so many research mathematicians end up married to (or in long term relationships with) linguists. His wife is a PhD in Spanish and linguistics, my wife's first degree is in French and linguistics, and I know perhaps three or four others in my immediate circle.
Anecdata, of course.
[0] Andrew Lipson: https://www.andrewlipson.com/
That makes sense as math is a fascinating exploration of the human mind and its ability to handle abstractions; and what is happening is one's mind even when not mathematical can be difficult to explain. A linguist's rigorous approach to language and understanding of language seems to pair well with a need to explain one's thoughts.
Thank you for sharing Andrew Lipson's site, digging his LEGO sculptures, for example: https://www.andrewlipson.com/mathlego.htm
Hm, challenge accepted: It uses a fan and a hose to suck up dust. You use it to get rid of dirt in your home. The dirt ends up in a bag. Now and then you must get a new bag, when the old one is full. If you have some tiny item or two that lies in the way of the dust, you will want to put that away, or else it can get lost in the bag.
But can I use it to rub the fleeb? [1]
Nice, I think you’re close to making it have a rhyme scheme… next challenge?
This reminds me of [Up-Goer Five](https://xkcd.com/1133/) - similar vibe but a different constraint (most common thousand words only)
A vacuum cleaner described in 'words that are four letters long or shorter':
a tool to suck dirt and dust into a bag
;)
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.
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.
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.
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).
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.
Agreed, this is an indisputable improvement on Penrose tiles while the other one didn't feel like it.
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).
HTLM may be the standard for many things, but it is not the standard for academic papers. PDF is the standard.
PDF is the optimal format for this use-case, mostly because of existing tooling which makes it very easy to make academic papers as PDFs. As far as I know no tools exist to make something comparable to an academic paper which would improve view-ability on a small screen.
What about GNU TeXmacs ?
http://www.texmacs.org/tmweb/home/welcome.en.html
(No direct relation to TeX or GNU Emacs.)
Though sadly this is not sufficient, we also have issues for HTML standardization for standalone documents, unlike, for say, PDF-A.
https://www.russellbeattie.com/notes/posts/the-decades-long-...
Ok then, PDF is a bad standard. HTML can do all the same stuff and more.
PDF is not a bad standard, the priorities of the people producing scientific papers as pdfs are just not the same as your priorities.
You can display it in HTML too: https://academ.us/article/2305.17743/
Thanks! arXiv Vanity fails on this one: https://arxiv-vanity.com/papers/2305.17743
Well, this is just a first version, enabling to quickly prove they are the first. The coming days, the internets will be filled with many more webpages than can be possibly read. In the old days, you had to wait many months before something went from found to officially published.
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.
I hope these tables help: https://twitter.com/Junyan_Xu/status/1663396061942038530
Notably, it's the same authors: David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss
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.
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.
The question is in reversibility. A typical PRNG is hard to exactly reverse, but easy to brute-force, given a few values from the sequence.
I think it may be way harder to brute-force a tiling, because the number of tiles grows quadratically relative to the distance from an initial configuration. I wonder how easy would it be to step back.
I think lattices are probably more secure though. It's a similar idea to tiling but you can run it in a huge number of dimensions, without having to change the underlying math.
Is that finally the single tile you can use to cover whole space aperiodically without flipping it upside down?
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.
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.
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...
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).
Could the chromatic number be less than 4 (for some or all hat/turtle/spectre tilings)? Note that for the lattice tilings by squares and equilateral triangles the chromatic number is 2, while for the tiling by regular hexagons it is 3. ZeroRogue has drawn dual graphs of hat tilings https://twitter.com/ZenoRogue/status/1639644061823819777 so you can just try to color them.
Which raises the question: how many colors of the inverted hat tile would you need? I don’t believe the inverted hats ever need to touch one another…
I suspect, given the 1:1 correspondence to a hex grid (where some of the hexagons map to an inverted/non inverted pair of hats) that they describe in the original paper that it would be possible to tile with just three colors of noninverted hat, and one color of inverted hat.
Someone needs to start a Kickstarter for this. I'd contribute.
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/
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...
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
You might be interested in this https://twitter.com/ZenoRogue/status/1663589972094353418
Can anyone explain why people are interested in tilings?
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.
I see thanks. I guess, hearing about other famous hard problems, I am used to seeing comments like: “if problem X could be solved it would unlock lots of other important areas of math.” Wondering if this tiling area draws attention for its own sake alone or if the problems here are similarly vectors to attack larger issues.
Math has problems where everyone suspects the proof will open doors and give valuable insight- I'm mostly plugged into topology where there are a lot of those. There are also problems that aren't interesting except in their difficulty, which drives the creation of new techniques and tools- Fermat's Last Theorem isn't particularly useful, but the effort to prove it created a vast body of spinoff work. But you also see problems that are more passive, waiting for someone to approach them with new firepower. Tilings are more like that- a testing ground for new techniques, and a way for mathematicians to keep their wits sharp.
Also, they do have some inherent beauty. I mean an aperiodic tiling is crazy right? And with one tile?
Out of curiosity, could you share the exciting unproven theorems in topology you referred to?
One big name is the Borel Conjecture- a very productive aspect of topology is developing finer and finer tools to detect differences between spaces. The Borel Conjecture essentially states that for a certain class of spaces, a well used and loved tool is equivalent to an extremely strong tool.
I was thinking more historically though. The development of those tools was driven by specific problems- classifying the behavior of higher dimensional spheres, determining if genus uniquely classifies spaces (not at all, but I believe people were once hopeful), even knot theory is an outgrowth of this kind of research.
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.
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
Game graphics. Aperiodic tiling allows us to render a natural-looking surface using a minimal texture graphic file.
Not really. Aperiodic tilings, while lacking symmetry in a strict sense, still tend to look quite regular and repetitive, and thus non-natural.[1] Aperiodicity alone doesn't guarantee the "natural" look, which is unsurprising because natural textures are not stitched together from tiles either.
[1] https://commons.wikimedia.org/wiki/File:Rhombus_Penrose_tili...
I think it is computationally easier and more effective to blend two relatively prime periodic rectangular tilings?
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.
One of the things this has consequences on are the physics of crystals and pseudocrystals.
Hmmm... now that we know what shapes these tiles take, maybe we can look for or design molecules with the "same shape" (and yes I know molecules aren't 2D planar objects, but some can be approximated as such, e.g. the benzene ring) and with the right molecule-to-molecule attractions, such that they naturally arrange themselves into these aperiodic tilings.
TRANSPARENT ALUMINUM!?
Without a periodic crystal structure, it is likely to transmit light (see glass's transparency due to its amorphousness)
Aluminum Oxide (aka Sapphire, Ruby) and ALON are already transparent despite being crystals.
Being extremely hard and resistant they are used in applications like watch crystals, windows in grocery-store barcode scanners and armored car windows.
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?
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)
any results? What if I made delrin molds?
Plaster is better for ceramics because it helps to evenly dry the clay. Can use delrin to make a plaster mold though!
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.
Here's how encaustic cement tiles are made - with liquid cement, cement powder, a mould, and a press: https://youtu.be/XSp5Tj4yYNc
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.
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...
The grout when tiling allows for a fair amount of fudging things, though from experience exactly dimensioned tiles are MUCH easier to work with.
With periodic tilings you can uniformly scale them to make room for grout of any thickness. Intuitively, it seems like aperiodic tiles would have to be manufactured slightly too small to leave room for grout, and the tiling would only work if the grout was exactly the right thickness. I wouldn't want to lay them.
Oh that's an interesting point actually. I hadn't considered the tiles needing to be designed around the grout width. Still, I don't think you'd need to be particularly exact with the tiles or grout. In fact it would probably be best to eyeball it and make sure you work outward from a single spot. Trying to tile by e.g. starting around the outside edges and working inwards is bound to get you in trouble with compounding error.
This is sounding like a more interesting project by the minute.
On reflection I've realised my point was nonsense. Sorry.
It's not nonsense. The fact that aperiodic tiles like these need to be designed for a specific grout width is non-obvious and quite interesting.
If I was a billionaire I'd have such custom-made tiling installed in my home's hallways and terrace.
I wouldn't see it being prohibitively expensive https://news.ycombinator.com/item?id=36121506
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.
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.
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.
Random thought/nerd snipe hazard: would a Wolfram-style classification of cellular automata running on this tiling kick up anything interesting?
There have been some experiments along these lines for "hat" tilings:
As far as I know, HatLife hasn't been adjusted to make SpectreLife yet, but it's probably only a matter of time!https://conwaylife.com/forums/viewtopic.php?p=161571#p161571
Seems useful for creating unique and modern designs.
Can someone please let me buy actual tiles shaped like this? I would like to renovate my bathroom
I wonder if there is an aperiodic monotile that may tile a sphere.