Can anyone explain why people are interested in tilings?
Game graphics. Aperiodic tiling allows us to render a natural-looking surface using a minimal texture graphic file.
A chiral aperiodic monotile
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Re: A chiral aperiodic monotile
#12Earlier quoted context omitted.
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?
Re: A chiral aperiodic monotile
#13I 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.
Re: A chiral aperiodic monotile
#14Earlier quoted context omitted.
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…
Out of curiosity, could you share the exciting unproven theorems in topology you referred to?
Re: A chiral aperiodic monotile
#15[deleted]
Re: A chiral aperiodic monotile
#16The 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
Re: A chiral aperiodic monotile
#17I 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.
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.
Re: A chiral aperiodic monotile
#18Can anyone explain why people are interested in tilings?
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...
Re: A chiral aperiodic monotile
#19Hmmm... 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.
Re: A chiral aperiodic monotile
#20Can anyone explain why people are interested in tilings?
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.