A chiral aperiodic monotile(arxiv.org) |
A chiral aperiodic monotile(arxiv.org) |
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.
""" 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...
> ... 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/
a tool to suck dirt and dust into a bag
;)
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.
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.
> 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.
[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.
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.
The paper discusses a (2D) monotile whose shape will allow an aperiodic tiling without reflections.
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 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.
Simon Tatham has written about this, in the context of the original hat tile: https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...
There is a brief explanation here: https://twitter.com/ZenoRogue/status/1638997769141604360
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.
That said I have no idea in the first place how you'd get your hands on bespoke tiles...
https://conwaylife.com/forums/viewtopic.php?p=161571#p161571
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.
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
[1] https://commons.wikimedia.org/wiki/File:Rhombus_Penrose_tili...
2. Its important in technology and science (e.g quasi crystals).
3. They have an aesthetic some find pleasing.
Thank you for sharing Andrew Lipson's site, digging his LEGO sculptures, for example: https://www.andrewlipson.com/mathlego.htm
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-...
Without a periodic crystal structure, it is likely to transmit light (see glass's transparency due to its amorphousness)
Being extremely hard and resistant they are used in applications like watch crystals, windows in grocery-store barcode scanners and armored car windows.
This is sounding like a more interesting project by the minute.
Also, they do have some inherent beauty. I mean an aperiodic tiling is crazy right? And with one tile?
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.
