I wonder if there is an aperiodic monotile that may tile a sphere.
A chiral aperiodic monotile
61–70 of 83 posts
Re: A chiral aperiodic monotile
#62Earlier quoted context omitted.
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…
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.
Re: A chiral aperiodic monotile
#63Earlier quoted context omitted.
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.
Re: A chiral aperiodic monotile
#64Can someone please let me buy actual tiles shaped like this? I would like to renovate my bathroom
Re: A chiral aperiodic monotile
#65There 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 usefu…
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.
Re: A chiral aperiodic monotile
#66There 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 usefu…
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).
Re: A chiral aperiodic monotile
#67There 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 usefu…
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.
Re: A chiral aperiodic monotile
#68This 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 reflect…
Agreed, this is an indisputable improvement on Penrose tiles while the other one didn't feel like it.
Re: A chiral aperiodic monotile
#69Earlier quoted context omitted.
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 al…
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.
This reminds me of [Up-Goer Five](https://xkcd.com/1133/) - similar vibe but a different constraint (most common thousand words only)
Re: A chiral aperiodic monotile
#70Earlier quoted context omitted.
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.
Ok then, PDF is a bad standard. HTML can do all the same stuff and more.