5 ms·
An Aperiodic Monotile (2023)
- bradrn 2y ago(2023)
- mkesper 2y agoThere are new materials on the linked page like follow-ups and interactive applications.
- zokier 2y agoAfter publication of Spectres, I don't know if there much interest anymore on Hats. Spectres are like Hats, but eliminate the need of reflections for tiling. https://cs.uwaterloo.ca/~csk/spectre/ https://cs.uwaterloo.ca/~csk/spectre/
- nhatcher 2y agoI think they are very interesting as a first step in the construction. I did a write up with some app you can play with a while ago: https://www.nhatcher.com/post/on-hats-and-sats/ https://www.nhatcher.com/post/on-hats-and-sats/
- Gare 2y ago> It also complicates the practical application of the hat in some decorative contexts, where extra work would be needed to manufacture both a shape and its reflection And people say that mathematical research has no practical applications
- foobarbecue 2y agoI, for one, would really like a spectre soccer ball. Seriously, though, I think the implications for mineralogy are interesting.
- robinhouston 2y agoI don't know whether anyone's designed a spectre soccer ball, but someone has designed a soccer ball based on the hat tile. https://youtube.com/shorts/_Rruxxrz9nY https://youtube.com/shorts/_Rruxxrz9nY
- foobarbecue 2y agoInteresting. I wonder why the five pentagons were needed. Is that because hat can't tile a sphere? Or some other requirement when assembling the ball?
- gilleain 2y agoWell you cannot tile a sphere with just hexagons, you need a minimum of 5 pentagons. Oh, from https://en.wikipedia.org/wiki/Fullerene https://en.wikipedia.org/wiki/Fullerene: "A closed fullerene with sphere-like shell must have at least some cycles that are pentagons or heptagons. More precisely, if all the faces have 5 or 6 sides, it follows from Euler's polyhedron formula, V−E+F=2 (where V, E, F are the numbers of vertices, edges, and faces), that V must be even, and that there must be exactly 12 pentagons and V/2−10 hexagons. " So I'm not sure.
- zokier 2y agoI would be really surprised if hat (or spectre) could tile sphere. Afaik most tilings of planes do not work on spheres.
- rini17 2y agoNext frontier: aperiodic tilings with irrational angles (meant, tiles having angles of x*2pi were x is irrational). Or are these proven to be impossible? Because both the hats and spectres are basically subset of triangular grid. Penrose tilings are subset of regular grid, too. Can we get rid of these underlaying regular grids.
- zem 2y agoit feels like it would be hard for those to tile at all, let alone aperiodially
- WCSTombs 2y agoNot really, since you can take a standard square tiling and apply a random shear transformation to it. With probability 1, you get a tiling of parallelograms with irrational angles.
- ganzuul 2y agoSpiral out and keep going?
- deleted 2y ago[deleted]
- joelthelion 2y agoSomething that is unclear to me: are hat reflections allowed? I think they are, but it would be good to have confirmation. In short, if you allow reflections, are the tilings still guaranteed to be aperiodic?
- shrx 2y agoThey discovered both variants, first the "Hat" and "Turtle" which require reflections, and then the "Spectre" which does not.
- gilleain 2y agoGood writeup of 'Combinatorial coordinates for the aperiodic Spectre tiling' from Simon Tatham here : https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-spectre/ https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...
- bluepoint 2y agoDoes anyone of if there are any consequences of the existence of monotiles in algebra or number theory?
- yayamo 2y agoInterestingly this was found by a “hobbyist tiler”, David Smith, who is the first author. He was interviewed on how he found it in this YouTube video: https://youtu.be/4HHUGnHcDQw?si=VsHLqVUdw6ihERg2 https://youtu.be/4HHUGnHcDQw?si=VsHLqVUdw6ihERg2