The result is actually even slightly better than what's in the video! The squiggly edges aren't actually necessary to prevent periodic tilings. What they do is prevent the shape from tiling with its reflection (which the basic straight-sided shape *is* capable of doing). Turns out all periodic tilings with the basic shape require the reflection as well, so if you're comfortable simply excluding the mirror image on the grounds that it's a different shape, then the non-squiggly version is already aperiodic! The squiggly version just gives the added bonus that you have no choice but to exclude the reflection (because the squiggles won't mesh), removing every last trace of ambiguity. Also I love that I found out about both the original tile and this new one from your videos! I'm a huge fan of your stuff :)
Except that it's always been acceptable to have some tiles be reflections of other tiles. This is a convention of tiling; it wasn't invented specifically for the subgenre of aperiodic monotiles. As Smith et al put it in their 30 May paper, the non-squiggly shape, the polygon, "admits only non-periodic tilings if we forbid reflections by fiat".
@@rosiefay7283 It's just a different mathematical question. Can we find a strictly aperiodic monotile? and Can we find a strictly aperiodic *and* chiral monotile? (no reflections) And it's awesome that we now have an answer to both!!
You are right. In other words if you belong the school of "Those hats and turtles (made of kites) do not count because their aperiodic tilings include reflections" the spectre responds to you by saying "Yes, but you can then exclude my tiling using reflections and I then give you a monotile that only tiles aperiodically!" Your objection melts away. The best thing you can say thus is that the spectrum of hats and turtles gives a complete answer to the question about connected monotiles that tile (only) aperiodically: the spectrum whose limits are chevron and comet.
ohhh my gosh i cant believe you actually interviewed craig kaplan for this the paper is awesome and this is just an absolutely wonderful video covering it ps: love how they keep naming the shapes after existing words. which gives rise to sentences like "Every tiling by Spectres is closely related to a tiling with a sparse distribution of hats lying within a dense field of turtles, and one with a sparse distribution of turtles lying within a dense field of hats." lmao
Nicely described and presented but I don't think some of the details are correct. It was my idea to use only rotations of the T(1,1) which I later called the Spectre, as I noticed it had more freedom after playing with Yoshi's aperiodic tile app. He was using both reflected and unreflected tiles and making a game out of it (he very slightly altered side lengths, so that T (1,1) would behave in the same way as hats and turtles). After that, Joseph Myers found out that combining hats with turtles and turtles with hats, one could simulate a Spectre tiling. Also, I just look for polygons that tile in interesting ways but I'm no mathematician and had very little to do with the final draft. All the best, Dave S.
@@815TypeSirius I thought about it (my children have said exactly the same thing) but it would have complicated everything. And remember that it was actually Craig, Chaim and Joseph that came up with the proofs, weeks, months later. I have used a lot of freeware programs on my computer over the years, so think it's now time to give something back.
@@JellyMonster1 Dave, I hope non-scientist, non-academic, people reading what you wrote here can appreciate how important what you have said is on at least two fronts: first, you describe the intense process of effective (as opposed to rudderless) curiosity search in a very clear way. Secondly, your non-humblebragging and deep understanding of idea ownership and teams gives me hope for people more generally. You have done something very special here, really. I speak for humans when I say thanks for that 😊 and I speak for myself when I say congratulations.
Came from Numberphile and this video finally helped me understand the hat/turtle spectrum! Plus the “dial” visualizations to explain the new proof are super cool
Growing up loving Math since I was 4, ma Dad got me an M. C. Escher book for my 9th birthday. I have been a fan of tessellations my entire life. The fact that they could be aperiodic is mind-boggling! And you are adorable, btw.
I'll be honest; going into this I did not know what to expect as I am not in the niche group of people who is following this. However: I AM SO FUCKING EXCITED FOR AN A-PERIODIC , NON-REFLECTING TILE!!! Thank you for introducing me to the fandom! Can't wait to see more content! Have a great day
Looking at the morphing between tile shapes and seeing how beautifully simple it appears to be, it's hard not to be a bit incredulous that this took so long to find. I suppose that's the wonder of finding answers: it's very hard work but looks easy in hindsight!
Just because they didn't call it a vampire tile doesn't mean you can't. Sure, that one tile is the "spectre," but the set of all tiles that can only aperiodically tile the plane could be called "vampire tiles."
This interesting story suggests that humans in colaboration groups might turn out to be as powerful as superintelligent AIs. And since so many people are worried about out-of-control AIs, that concept might be important for the future. There is also the possibility that a combination of AI thinking and human thinking will turn out to be more powerful than either one on its own. The latter possibility leads me back to the idea that the best defense against runaway AI would be to have a wide range of communities in which humans and AIs work very closely together. And since it's increasingly clear that there is not going to be an AI halt or moratoriium any time soon, my conclusion is that we immediately need to start building many diverse groups of humans and AIs cooperating to find solutions to many of the world's problems -- including the potential problem of AIs running out of control. As it happens, combining AIs and humans into thinking teams is very likely to be the best way to solve our most serious problems, such as global warming, loss of wild habitats, and depletion of fresh water resources. Creating diverse, multidisciplinary thinking teams has the great advantage that it does not sound like a hopeless task. No matter how scarily smart AIs eventually get, there will always be an advantage in combining cooperative (or "loyal" AIs) with humans.
AI is not going to solve the ongoing ecological disaster. billions of humans' habits would have to change drastically for any of those issues to be solved, which is not going to happen. AI is just another tool that capitalism's devotees will use to exploit the biofilm living on this rock. with my piece said, i'll fight with you against the ai overlords if we're both alive in the fantasy post-apocalypse :)
thank you for this. i saw you in the video with craig, and i appreciate you communicating the significance of this to a wider audience. I agree we're in a golden age for math art collaboration. i used to run naked geometry on facebook and elsewhere, and i'm more offline now, but it was wild just watching the interest in the subject from when i started to when i left was spirit-lifting. meet you at bridges some time!
thank you for teaching me. it was a lovely presentation and went in depth enough for me. I'll have to re-watch the 'spectrum between 2 shapes, at mid is the one true vampire that was Promised' section and give it my full attention
Already here. Instead of turning each straight edge into a curve, turn it into two edges by adding a triangular bump to, or making a triangular concavity in, the original polygon. What you get is still a polygon.
Thank you Ayliean for showing us the beautiful a/symmetry of these mathematical tile shapes. I never knew, and still don't. But myi nterest is peaked. Ta again and all the best.
description box here, looking forward to those links (especially playing with the shapes on the computer, I'm looking to incorporate the tile in a background shader to see what interesting organic animations can come out..) Great video and explanations! Thanks!
When will the David Smith paper get peer reviewed? I’m so glad that he, as just a hobbyist, got top billing since he discovered the shape and the other mathematicians just proved it for him.
Thanks for the explanation of what 'aperiodic' means. My understanding is 'no translational symmetry'. I'd now like to hear a 'simple as possible but no simpler' explanation of the proof.
At 2.57 in this video, the chaotic attractor fractal found the same monotile.. 4 years ago. I think some larger implications may be drawn from here. Can the fractal fill 4d space? A formal proof seems possible. Also, seems related to the Navier-Stokes problem.
has anyone ever looked into non periodic space filling stacked block quasi crystal structures? seems like that would be the next thing to be looking for, if 3d space even works that way. etcetera. ... ... :|
The curves are an arbitrary choice to force the edges to be "one-way" and can just as easily be replaced by any asymmetrical shape, such as a square prong/divot or a lightning bolt of any three unequal segments.
0:30 Can someone please tell me what the (relative) dimensions of that shape are? I'm preparing a workshop for kids (aged about 12-14) about this subject and would love to print that shape out for them to cut out and form a big shape with it.
- 2:26 Um, Lucy, you got some 'splainin' to do. How are two lines 180° apart not a single line? 🤨 - 4:42 I don't know about people in the past, but people right now are well aware that we've just entered a significant turning-point in history with the advent of A.I.
So ya know how some tilings are found in nature or Atom behaviors or whatever. What if scientists use this to FLIP atoms. Oppenheimer harnessed the power of atoms to build a bomb to end all bombs. What if flipping an atom is like perpetual motion machine for unlimited energy? Thanks for letting me cook in the comments.
Wait, so they considered the hat an aperiodic "mono"-tile despite it requiring reflections, but don't consider the spectre with straight edges an "aperiodic" tile, because you have to "arbitrarily" forbid reflections? What kind of logic is that? Somehow "forbidding reflections" feels way more natural to me.
The (1,1) Spectre polygon has the ability to tile periodically as well, so it doesn’t count as an aperiodic monotile. But with the lil curvy adjustments it can no longer tile periodically :)
@@Ayliean As far as I understood the paper, as long as you don't allow mirroring it can't tile periodically. I don't get, why the paper calls this restriction "by fiat", as if any other mathematical restriction would ever be natural. You can count a tile and its reflection as a single piece (as the authors of this paper seem to do and seem to perceive as "natural") or you can count them as two distinct pieces. For me, this is an arbitrary definition. It doesn't even make sense to me, to call one of these "more natural". But if we really want to pick a side, I would call "disallowing reflections" more "natural" and "allowing reflections" more "mathematical".
@@Ayliean The authors call the Tile (1,1) a "weakly chiral aperiodic monotile" - 'weakly' because it still admits a periodic tiling if flips are allowed. It is still a chiral aperiodic monotile - 'chiral' in this context just means not mixing left-handed and right-handed copies i.e. no flips. What the curved edges do is eliminates the possibility of a periodic tiling even when flips are allowed. The authors call this a "strongly chiral aperiodic monotile". So Tile (1,1) is definitely an aperiodic monotile - specifically a chiral aperiodic monotile. Hope this clears up any confusion! Love the video by the way.
