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!
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 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
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!
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."
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.
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.
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.
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.
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. ... ... :|
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.
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.
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.
- 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.
great video. thanks Ayliean! my question is about the chevron; at about 2:28 and about 2:34 while cycling through the continuum of tiles, it really looks like the chevron tiles aperiodically, which would make it a spectre / vampire einstein tile from the announcement in March, which would be very noteworthy. the chevron is also unique in that continuum, in that it happens to be its own reflection already, unlike any other shape in the continuum, but it doesn't need any additional reflection. what am I missing here? thx
As mentioned in the video, it can make both periodic and non-periodic tiling patterns. The chevron tile is therefore not aperiodic. An aperiodic tile forces the tiling to be non-periodic. An aperiodic tile aka einstein will only allow non-periodic tiling patterns and cannot do any periodic ones.
@@gtziavelis It helps to appreciate how trivial it is to make a non-periodic tiling with periodic tiles. A domino rectangle, for example, can fill a grid completely randomly.
Well, now its just a matter of time before someone discovers a molecule that exhibits exclusively aperiodic quasicrystaline structure. Lets just hope that amount of time isn't negative, huh?
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".
what does the non-squiggly version look like?
@@proloycodes 3:00
@@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
I am a huge fan of the cardboard dial thing at 3:16
I need to see more of my life mapped to the cardboard dial thing!
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.
Aperoiodic monotiles are the foundation of material science and nano engineering... you should copyright these.
Is the 80's clothes pattern design Spectre?
@@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.
@@Simulera Thank you. As Popeye's grandpa once said, "I am what I am", or something like that.
OMG YAAAAS IT HAPPENED 💖💖💖💖
My exact reaction
can't believe you can find this comment in places that are not simply cancerous tumors on society
@@user-pr6ed3ri2k?
@oyoshimi3891 they mean that the typical girls who say "yasss" are extremely toxic
^ can confirm that this is what I meant
Is this a new tattoo idea? 😁
Let me tell you, I am so glad my tattoo artist has a 3 month waiting list!
@@Ayliean are you fighting for a spot against Tom Rocks Maths 😁
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.
Might be the least sketchy thing to come out of the back of a van.
you are a absolute delight Ayliean. I'm so glad a stumbled across hearing this news delivered with your enthusiasm. Keep up the great work
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
Thank you for explaining this in a way even I could understand, and for the new fixation on shapes 0.0
"We are having a moment" indeed! Love the shades.
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!
So, the invention of the time machine will be like this to the meta level. ;)
@@backwashjoe7864 "What do we want?"
"TIME TRAVEL!"
"When do we want it?"
"IT'S IRRELEVANT!"
@@macronencer 😂😂😂
Deserves so many more views. Great video, keep up the good work.
This mathematical discovery is truly gold. Great video.
Infinitely many??? I'm so happy!!! ❤🎉
Your energy is so, so good. The sheer love for the topic is just beyond. Completely great.
You are a genius teacher for such beautiful things. Please make many videos for us, explaining!
I really don't know how I ended up here but I'm glad I did
It looks so much more like a shirt than a hat.
love your passion for maths, keep up the amazing work!
Reflective is a fancy way of saying 'this is two different shapes but we cant get maths clout if we admit it'.
So is non-connectivity, looking at you Socolar-Taylor tile.
:O they found it so fast, wow
The video is stunning, btw, I love your style so much!
I’ve never heard of this before but I am psyched about this new aperiodic monotile 🔥🔥
I didn't know this channel even existed.... INSTANTLY SUBSCRIBED!!!!!!
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 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
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!
Please never change! Or maybe do, by getting more tattoos. They are awesome.
Very well made video!! Loved it.
I like how we suddenly all agree that before we thought it's not a "true",...
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."
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.
did you mean "piqued" ?
Is there a 3D analogy of this, like a mono-block that tesselates 3D space aperiodically?
Yes, sort of. en.wikipedia.org/wiki/Gyrobifastigium#Schmitt%E2%80%93Conway%E2%80%93Danzer_biprism
@@rosiefay7283Conway strikes again!
So impressive that it only took "us" 10 weeks to find this next step!
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!
This shows we still have so much to learn
Your channel deserves more subscriptions.
have no idea how did i get here, but SFOK YES I NEEDED IT 😢❤🎉
We have a blooming parasocial relationship. I'm thinking of introducing you to my friends.
I didn't know it had been found! Thanks for sharing!
I like the placeholder comment in the description box.
I wish I was unique enough to be an aperiodic monotile.
You could only couple with yourself if that were the case though 😀
@@lokanoda so no change then?
Get pwned me!
You are, we all are, except...twins,trips,etc
Very cool and an excellent presentation!
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.
Phantastic job!
Heck yeah. Maths! 🎉
Awesome!!!
So, that will be your next tattoo?
The spectre tile looks like the pokemon Misdreavus
David Smith is the hero.
i love your videos ☺☺☺
Thanks for the video. Where can I get the program to make the tiles ast 1:57 ?
^ This
Spectre?
Looks more like a BEAGLE
The reflection and the 180 degree "angle" make me more upset than I should be.
you are living in the perfect time for you
OMG!!! This is huge!!!
3:45 it so beautiful
Cooooool 🤩 So what's next? 3-dimensional aperiodic tiling shapes? What would they be called?
Did you 3D print those tiles?
Amazing news! Aperiodic non-reflecting polygon next?
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.
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.
ruclips.net/video/fDSIRXmnVvk/видео.html
So cool!
The roof tiling at 00:12 has overlaps. It’s not on a single two dimensional plane.
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.
Looks like a pig with wings. So the condition “when pigs can fly” has finally been met!
Nice story 😄
I say chirality doesn’t take away from the fact that it’s the same shape.
Finally! How have we managed with out one for so long?
She is just simply lovely. Was she not on numberphile with her mum?
Yeah, that was me on Mumberphile talkin trefoil knot hugs 🥰
2:14 _..de SimpLes₺ mono₺ile, _*_WiNs!_*
_(CİA:..HEADBANGİNG İN₺ENSİFİES!!)_
So if I understood well there is not a true (doesnt need reflection) aperiodic polygon mono tile, right ?
I still say it’s not a hat, but a t-shirt
It looks like a winged pig!
Yes, that's what I saw too ! Like they'll discover an aperiodic monotile when pigs fly, and there it is !
3:22 last syllable of both names is "ki" pronounced "key", not rhyming with "sky"
Ah, but now is there one with fewer edges? :)
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. ... ...
:|
With this a-periodic, non-reflecting tile... how many colors are required so that no two of the same touch?
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.
Or, is that a class of shapes and do the exact dimesions not matter?
Looks like 3, 2, 1 would work.
Math people are artists
Next question: is there "true" aperiodic monotile without curved edges?
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.
So is the tile with squiggly lines the Spectre, or is the the straight lines? I think one should be called the Vampire to differentiate
_Puzzle!!_ ✓
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.
where can i find a true aperiodic 3d cristal or sponge model for 3dprint, is there a aperiodic tetris game? :)
Where can I find a CAD file for the true monotile dxf or step or stl ?
- 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.
does this exist in higher dimensions as well?
Next: True aperiodic tiles that isn't ugly as fuck.
As soon as you use the English language in a way that isn't ugly as fuck.
great video. thanks Ayliean! my question is about the chevron; at about 2:28 and about 2:34 while cycling through the continuum of tiles, it really looks like the chevron tiles aperiodically, which would make it a spectre / vampire einstein tile from the announcement in March, which would be very noteworthy. the chevron is also unique in that continuum, in that it happens to be its own reflection already, unlike any other shape in the continuum, but it doesn't need any additional reflection. what am I missing here? thx
As mentioned in the video, it can make both periodic and non-periodic tiling patterns. The chevron tile is therefore not aperiodic. An aperiodic tile forces the tiling to be non-periodic. An aperiodic tile aka einstein will only allow non-periodic tiling patterns and cannot do any periodic ones.
@@jaapsch2 makes sense. it's gotta be exclusively aperiodic, and the chevron is aperiodic but not exclusively. thank you.
@@gtziavelis It helps to appreciate how trivial it is to make a non-periodic tiling with periodic tiles. A domino rectangle, for example, can fill a grid completely randomly.
Why did they call this shape a Spectre?
A bit off topic, but do you live in your car?
Well, now its just a matter of time before someone discovers a molecule that exhibits exclusively aperiodic quasicrystaline structure. Lets just hope that amount of time isn't negative, huh?
Seems borderline useless :) has any tile maker produced them so they can be put in bathrooms?
Why was this reccomended to me?
👍💪✌
Looks more like a t-shirt..
So... the monotile is just Spain?
it shouldn't be called a hat. it's clearly a t-shirt.