I will never understand why hardware/home-improvement stores don't sell various kinds of aperiodic tiles. I'm sure I'm not the only one who'd absolutely tile a floor or a backsplash or a whole bathroom with them.
Perhaps these shapes only tile when mathematically perfect, and real-world imperfections prevent practical use? Then again, I’ve played with physical Penrose tiles and it seems to work.
This is maybe kind of pedantic (and something you surely understand, I’m just saying ~~just in case some viewers might not catch it~~ actually no, the real reason is just because I am compelled to be pedantic), but I want to note that the difficult thing isn’t a set of tiles which *can* be used to tile the plane in an aperiodic way, but rather to find such tiles which also *cannot* tile it in a periodic way. If your tiles are 2x1 rectangles, you can take the obvious tiling where you group them into squares, and then tile the squares in a periodic way, but the rotate just one of the squares. The resulting tiling of the plane with the rectangular tiles is not periodic, in that it doesn’t exhibit translational symmetry. So the difficulty isn’t “tiles which can tile the plane aperiodically”, but “tiles which can only tile the plane aperiodically”
@@massimomoro5895 linear symmetry is just symmetry under some group of translations, which is broken by the tiles that are turned the other way. It is still a tiling because it partitions the plane into copies of the tile which only overlap at the boundary.
This has been my favourite recent development in a long long time! I tried reading the paper but it kinda floated past me (I'm a lapsed professional mathematician). Would really love to see a breakdown of how it kinda works and why and what the continuum you mentioned is.
Very good content! The fact that this tiling uses mirror image for some tiles feels like a cheat on "just one tile" pattern 😅. Guess we can call it 1.5 tiling pattern
@@rosiefay7283 I understand that it was always allowed, and I suspect that doing infinite tiling without reflection or second piece is probably impossible
I just knew Penrose would get mentioned. In the early Eighties I read in Scientific American about tiling and fractals. I tried programming fractals on a Commodore 64. I was still not prepared for when in 1996 I visited Spain and saw the awesome tiling of places like El Alhambra. I even added a painted pattern to my bathroom walls when I got home. I got low marks for Maths at school, but have retained a life-long interest for another 50+ years. Even in about 2000, I was still creating patterns in MS Paint that could be tiled on my work PC desktop, in work idle time.
This is the first engaging video I see on this monster of a discovery! I'm SOOOOO surprised Matt Parker / Numberphile / Any other science channel haven't made a video on it yet. I was about to paint penrose tilings in my room, but guess I'm switching up now!!!
Your voice is so smooth to listen to, your enthusiasm so endearing, and the topic so interesting, that when the video ended I was hit with a mild shellshock. I was ready to just sit and listen for another twenty minutes.
This reminds me so much of a Vi Hart video! Those videos entertained me for a large portion of my childhood. I’m glad I found you and hope to enjoy more of your content!
It is a little bit of a cheat as there are two tiles in use, the tile and it's mirror image. For me that's pushing the definition of 'monotile' a little.
I remember seeing a numberphile video awhile ago that showed a tile that could do this, but it has multiple disconnected pieces. Great to know they found a single piece that can do it (albeit with some being reflected)
This makes me very happy! Thank you for sharing this news so clearly and enthusiastically! Congratulations to the team of discoverers and to the giants whose shoulders they stand on!
Truly groovy! But two quibbles: I don't think it's a hat; turn it upside down and you'll see a T-shirt. Also, as others on this thread note, if you have to turn the tile backwards, then that's two tiles, sort of. How do you do the tiling?
Absolutely - the actual search was for tilings that can _only_ tile the plane aperiodically. Penrose achieved that by basing his tiling on regular pentagons, which cannot (normally) tile, creating a tiling that doesn't abide by the normal rules of tiling. I don't (yet) fully understand the hat tiling, but the penrose tiling can trivially be made to be rotationally symetric, but if it repeated, then those rotational symetry points would repeat too, and you'd be able to find more points by rotating one point 72° around another point - but if you try that a few times, you'll find out that they'll never line up with each other.
Wow! Such a seemingly simple thing and yet it took years to come up with a single tile solution. But even if it is the same shape you still have to use it mirrored. Isn't that two tiles then? Still impressive to be able to cover a surface with just one shape AND the pattern NEVER repeats. Incredible! The connection with the aluminum alloy was interesting. There's math everywhere 😊
what's crazy is that this is such a simple idea. It really just combines the hexagonal and trigonal tiling and cuts out a kinda arbitrary but rather simple shape. (That said, this connection to those decidedly periodic tiles makes it, in a sense, less aperiodic than it could be. Patterns end up looking like hexagonal tilings with some variation. It feels less aesthetically pleasing, imo, than the penrose tiling) Of course the next question is going to be what if reflection isn't allowed? Rotation and translation only? Still possible, or is the reflection a necessary condition?
You are an incredible teacher and video editor. Watched 4 videos on this and still did not understand what was being discussed fully. After this video I get it and you made it so simple and fun. Cheers
Yes, it really does not count, but you could choose to use a more strict definition which doesn't allow for flipping the monotile and it remains an open problem whether such a tile exists that can tile the plane only aperiodically.
Aren't pinwheel tilings and sphinx tilings already aperiodic tilings done with a single tile? I would guess that what makes this one special is that it cannot be tiled periodically. Whereas pinwheel tiles and sphinx tiles can be arranged either periodically or aperiodically.
I mean yeah, it's a hat, but I think it looks the most like a tee shirt that's been half tucked 😂 I've been interested in topology and hyperdimensional geometry since middle school. I am so excited to see new mathematical discoveries being made! The larger tiling patterns look VERY fractal-like, is the correlation meaningful or pareidolic?
Fun video! I know Dave Smith (the guy who discovered the shape) and he was talking to me about his search for the shape and showed me it when he discovered it, but I didn't realise it was such a big deal!!
There are infinitely many ways of getting periodic tilings with regular pentagons and rhombuses, and they can be homogeneously but anisotropically deformed so that pentagons lose their regularity and rhombuses become rhomboids or squares.
I remember watiching the Veritasium video about the penrose tesselation and I was facinated by it, having a one shape, one color and no funny tricks, for an aperiodically tesselation just blow my mind
Wow what an amazing explainer. I was about to lose hope to understand what this breakthrough in tiling and math everybody is talking about. Thank you Ps: Don't rush and get a new "hat" tattoo, they updated the hat to a purer einstein "spectre" tile, now you can repeat it forever without the need for mirroring at all
What software are you using? I’m a chemist, and am studying water structure that follows 5-fold symmetrical quasi-crystal structure. That this shows us a spectrum of shapes that can tile aperiodically, makes me think there are other molecular structures that can be built, or already exist, and may explain certain phenomena like glass structure. So, which software is it?
I really like the 'we haven't finished maths yet'. I recall as a math Teacher's Assistant talking to a few of my students about the math classes I was taking in grad school. One of them was shocked when I talked about math research, because they apparently thought maths was 'done'.
My wife wants us to retile our kitchen. She's gonna be SO mad at me...
Doooooo it!
I have to remember this idea for my own home
$3.28 says you wont
Oooh there's a Matt Parker bit about that 😅
I was about to comment that I want this as a wall in my house 😂
I can’t get over how they called it a hat when it’s 100% a t-shirt
I was thinking the exact same thing! It IS definitely a t-shirt
Obviously it’s a pair of boots.
@@c2h680 a pair of boots makes sense too!
@Scott's Precious Little Account 2:21 "They call it the hat."
.... proportionally, more like a football jersey.
This is such a good, engaging, visual, quick explanation of this topic!
Thank you! That’s all I aim for 🥰
@@Ayliean Like how vihart would do it
I totally agree. @@Ayliean you have great talent for this!
I will never understand why hardware/home-improvement stores don't sell various kinds of aperiodic tiles. I'm sure I'm not the only one who'd absolutely tile a floor or a backsplash or a whole bathroom with them.
I wanted to sell regular pentagonal wall tiles under the trade name 'Futile'. 😀
Perhaps these shapes only tile when mathematically perfect, and real-world imperfections prevent practical use? Then again, I’ve played with physical Penrose tiles and it seems to work.
Grout lines would actually make it easier, not harder I suspect.
Good idea for someone who can 3d print in a material that can be kilned to tile durability :D or recycled plastics sealed against offgassing!
I imagine you could make a mold and use colored concrete for outdoor tilings. In fact, I plan to try!
This is maybe kind of pedantic (and something you surely understand, I’m just saying ~~just in case some viewers might not catch it~~ actually no, the real reason is just because I am compelled to be pedantic), but I want to note that the difficult thing isn’t a set of tiles which *can* be used to tile the plane in an aperiodic way, but rather to find such tiles which also *cannot* tile it in a periodic way.
If your tiles are 2x1 rectangles, you can take the obvious tiling where you group them into squares, and then tile the squares in a periodic way, but the rotate just one of the squares. The resulting tiling of the plane with the rectangular tiles is not periodic, in that it doesn’t exhibit translational symmetry.
So the difficulty isn’t “tiles which can tile the plane aperiodically”, but “tiles which can only tile the plane aperiodically”
honestly, oftentimes it's useful to have "pedantic" (rigorous) expressions, just for the maximum precision in communication. so, thank you drdca.
No need to call this pedantic. It is actually quite important detail, thanks for clearing this out.
I didn't realize. If i paused and thought about it then I'd see this coming but I didn't.
@@massimomoro5895 It covers the plain without overlaps or gaps. It meets the definition of tiling that we were given at the start.
@@massimomoro5895 linear symmetry is just symmetry under some group of translations, which is broken by the tiles that are turned the other way.
It is still a tiling because it partitions the plane into copies of the tile which only overlap at the boundary.
This has been my favourite recent development in a long long time! I tried reading the paper but it kinda floated past me (I'm a lapsed professional mathematician). Would really love to see a breakdown of how it kinda works and why and what the continuum you mentioned is.
Look for the National Museum of Mathematics CZcams video, "A Hat for Einstein".
I broke the Game.
Regular Triangle, in 24 tiles.
czcams.com/users/shortsdnGtToFlUFE
Very good content!
The fact that this tiling uses mirror image for some tiles feels like a cheat on "just one tile" pattern 😅. Guess we can call it 1.5 tiling pattern
This is not considered cheating. Reflecting has always been allowed.
@@rosiefay7283 I understand that it was always allowed, and I suspect that doing infinite tiling without reflection or second piece is probably impossible
An A press is an A press, you can’t call it a half.
Yeah but, if you have ∞ pieces of this shape 3d printed, you can cover an entire plane. As an tangible object you can actually do it as a unique tile
Yeah. Penrose tiles do not need to be flipped. So both these and Penrose's use two tiles each.
This is EXCELLENT work. What a great video. Thanks!
The fact that you can flip the shape over feels like a bit of a cheese to me, there ought to be an asterisk over 'aperiodic monotile' (*)
I just knew Penrose would get mentioned. In the early Eighties I read in Scientific American about tiling and fractals. I tried programming fractals on a Commodore 64. I was still not prepared for when in 1996 I visited Spain and saw the awesome tiling of places like El Alhambra. I even added a painted pattern to my bathroom walls when I got home.
I got low marks for Maths at school, but have retained a life-long interest for another 50+ years. Even in about 2000, I was still creating patterns in MS Paint that could be tiled on my work PC desktop, in work idle time.
I fantasize about MC Escher seeing Guastavino tiles/tesselations/vaults but he resisted getting into the Builders’ realm.
This is the first engaging video I see on this monster of a discovery! I'm SOOOOO surprised Matt Parker / Numberphile / Any other science channel haven't made a video on it yet. I was about to paint penrose tilings in my room, but guess I'm switching up now!!!
the part about an periodic element being structured as an aperiodic tiling was interesting!
Your voice is so smooth to listen to, your enthusiasm so endearing, and the topic so interesting, that when the video ended I was hit with a mild shellshock. I was ready to just sit and listen for another twenty minutes.
Fascinating! Some of those patterns are very reminiscent of Escher.
This reminds me so much of a Vi Hart video! Those videos entertained me for a large portion of my childhood. I’m glad I found you and hope to enjoy more of your content!
The proper response to “what’s the use of that” is a punch straight to the guts.
It is a little bit of a cheat as there are two tiles in use, the tile and it's mirror image. For me that's pushing the definition of 'monotile' a little.
I find it interesting that the tessellation pattern somewhat reminds me of a Mandelbrot fractal
I remember seeing a numberphile video awhile ago that showed a tile that could do this, but it has multiple disconnected pieces. Great to know they found a single piece that can do it (albeit with some being reflected)
I heard that it's impossible to have such a shape that is both connected and never needs to be reflected.
@@Tumbolisu The specter tile
Love the way you engagingly yet simply communicated mathematic principles which were able to be understood, especially since I did terribly at school
Next question: can it be done with one tile WITHOUT allowing reflections?
I absolutely love that "the hat" tiles out in a fractile pattern! 🥰
This makes me very happy! Thank you for sharing this news so clearly and enthusiastically! Congratulations to the team of discoverers and to the giants whose shoulders they stand on!
i like how instead of talking you telepathically transmit thoughts to me
Terrific, tantalizing telepathy: transferring thoughts, transcending traditional talk.
Truly groovy! But two quibbles: I don't think it's a hat; turn it upside down and you'll see a T-shirt. Also, as others on this thread note, if you have to turn the tile backwards, then that's two tiles, sort of.
How do you do the tiling?
I'm really stoked for the math fandom right now, you're all over here doin stuff--gold star!
Such a fantastic subject explained in such a fantastic video... Simply beautiful
wouldnt quadrilleteralls tile the plane aperiodiclly if u lined them up like normal then shifted each column along by a random amount
Absolutely - the actual search was for tilings that can _only_ tile the plane aperiodically. Penrose achieved that by basing his tiling on regular pentagons, which cannot (normally) tile, creating a tiling that doesn't abide by the normal rules of tiling. I don't (yet) fully understand the hat tiling, but the penrose tiling can trivially be made to be rotationally symetric, but if it repeated, then those rotational symetry points would repeat too, and you'd be able to find more points by rotating one point 72° around another point - but if you try that a few times, you'll find out that they'll never line up with each other.
Super video! Easy to understand and fun! Great job
Kudos! A very informative, concise and entertaining explanation.
Wow! Such a seemingly simple thing and yet it took years to come up with a single tile solution.
But even if it is the same shape you still have to use it mirrored. Isn't that two tiles then?
Still impressive to be able to cover a surface with just one shape AND the pattern NEVER repeats. Incredible!
The connection with the aluminum alloy was interesting. There's math everywhere 😊
Aluminium*
@@thirddiversiondeep Sorry, English is my second language.
Is it only called aluminum in American English? It's aluminium in Swedish.
@@electronicgarden3259 Aluminum is used in american english and aluminium in british english, but both spellings are correct.
@@ttmfndng201 Thanks. I like the European way 😀
@@electronicgarden3259 Correct!👍😀
Finally someone who explains tessalation is a easy to understand way. Great Video!!
Gorgeous in so many ways...
Another fantastic video. Brings a whole new meaning to ' a night on the tiles'
Wow. This is a lovely and succinct way of explaining a topic that can be so difficult to visualize!!
The happiness this gives me is unsurpassed. An aperiodic monotile. This is peak elation.
Those freehand tile illustrations are awesome. I might give that a go! Thanks for the cool content!
This concept is sooo badass. I love the little cardboard TV, too.
Very nice video! Love it.
If it also uses a mirrored version, aren't those two different tiles?
Very well presented!
This was really interesting! It was a lot of information provided very briefly, but it never felt overwhelming. Brilliant and engaging, thank you!
Very cool fact about the Penrose tilling right at the end.
I thought of this channel when I saw the discovery. I was sure this would make some great math art
this video is so high quality, informative and entertaining you managed to get the big three I LOVE IT great job
what's crazy is that this is such a simple idea. It really just combines the hexagonal and trigonal tiling and cuts out a kinda arbitrary but rather simple shape. (That said, this connection to those decidedly periodic tiles makes it, in a sense, less aperiodic than it could be. Patterns end up looking like hexagonal tilings with some variation. It feels less aesthetically pleasing, imo, than the penrose tiling)
Of course the next question is going to be what if reflection isn't allowed? Rotation and translation only? Still possible, or is the reflection a necessary condition?
Come on! That's the Julia set! Amazing.
Never would've guessed how fascinated I would be by this-- thank you!
"We haven't finished maths yet." I love it
You are an incredible teacher and video editor. Watched 4 videos on this and still did not understand what was being discussed fully. After this video I get it and you made it so simple and fun. Cheers
I love this video. The topic, the spirit of Ayliean, This is is such good presentation visually. 1000 out of 10
Does using backwards tiles really not count as using a second set of tiles?
Yes, it really does not count, but you could choose to use a more strict definition which doesn't allow for flipping the monotile and it remains an open problem whether such a tile exists that can tile the plane only aperiodically.
@@nonyobisniss7928 no it's known, it's called the spectre
@@MichaelDarrow-tr1mn True. I'm guessing I wrote my comment before that tile was revealed.
This is peak CZcams for me. Thanks for making interesting content.
Whoaa, this is super cool!! Thanks for making such a great video!
amazingly good explanation of such a complex subject,
thank you so sooooo much for this great effort you put into it!
Well done- thanks for that.
Aren't pinwheel tilings and sphinx tilings already aperiodic tilings done with a single tile? I would guess that what makes this one special is that it cannot be tiled periodically. Whereas pinwheel tiles and sphinx tiles can be arranged either periodically or aperiodically.
I mean yeah, it's a hat, but I think it looks the most like a tee shirt that's been half tucked 😂 I've been interested in topology and hyperdimensional geometry since middle school. I am so excited to see new mathematical discoveries being made! The larger tiling patterns look VERY fractal-like, is the correlation meaningful or pareidolic?
Fun video! I know Dave Smith (the guy who discovered the shape) and he was talking to me about his search for the shape and showed me it when he discovered it, but I didn't realise it was such a big deal!!
Amazing. Great video.
I have been wondering about this for years and had no idea what to even look up. ❤
this is so interesting to watch,, i love it!!
Thanks for the video Ayliean, helps a lot!
There are infinitely many ways of getting periodic tilings with regular pentagons and rhombuses, and they can be homogeneously but anisotropically deformed so that pentagons lose their regularity and rhombuses become rhomboids or squares.
Whatever branch of math involves fractals tesselations and things of this nature is exactly where I need to be
Great explanation of such an interesting topic :D
Oh wow. There was the Taylor-Socolor aperiodic monotile, but it was disconnected, which was less than satisfactory. This is a super nice monotile.
The problem with it was that it isn't one tile. Being disconnected means that it isn't one tile.
@@rosiefay7283 That is a fair assessment.
Would have appreciated links in the description -- yay new shape!
Mathematicians: Damn this problem is really hard.
Aluminium-alloy: Hold my beer...
I remember watiching the Veritasium video about the penrose tesselation and I was facinated by it, having a one shape, one color and no funny tricks, for an aperiodically tesselation just blow my mind
I think this might be my new favorite channel! 😃
Wow what an amazing explainer.
I was about to lose hope to understand what this breakthrough in tiling and math everybody is talking about. Thank you
Ps: Don't rush and get a new "hat" tattoo, they updated the hat to a purer einstein "spectre" tile, now you can repeat it forever without the need for mirroring at all
Great video!! You earned a subscriber! Would be awesome to go down into the more math behind it as well or how it works
Thank you for great explanation
love this video! great explanation
This immediately reminded me of the dragon curve fractal from Jurassic Park
it looks like such ha simple shape at first glance! it makes you wonder how many times this shape has been created by just pure chance
What software are you using? I’m a chemist, and am studying water structure that follows 5-fold symmetrical quasi-crystal structure. That this shows us a spectrum of shapes that can tile aperiodically, makes me think there are other molecular structures that can be built, or already exist, and may explain certain phenomena like glass structure. So, which software is it?
Getting New-Vihart vibes from this vid. Keep up the good work :D
I'm not good at math (yet) but boy am I obsessed with how amazing and satisfying it is. I loved this video!
it's just a matter of definition. the chiral shape is a different shape, so it isn't one, it's two shapes
this is the first video i saw from this channel and i have to say - and i mean that as compliment - your style reminds me of vihart.
A superb video ! Well done !
Great video. I would love to know how much shapes are created/discovered. There's more to it than trial and error, but what? 🤔 Would love to know!!
YOOOOOOOOOOOOOOOOOOOOO NEW SHAPE DROPPED 🗣🗣🗣🗣🗣🗣🗣🗣🗣 🔥🔥🔥🔥🔥🔥🔥🔥
Didn't know Scott Manley and Vihart had a baby ❤
My mind is so totally blown now.
Just wanted to say this was a lovely video! Great work, I love this kind of stuff!
That’s so cool! Is it related to fractal geometry?
Mad ventriloquist skills!
I thought that this was a new proposal for the periodic table, man I need glasses...
Need more info on that "continuous spectrum of tiles", one the ends of that interval looked like a symmetric einstein?!
great explanation! very cool
2:35 am i the only one who thinks this looks like a dragon curve (or something from that family?)
The complexity bell curve for mathematics has shapes at one end and shapes at the other end, with numbers, letters, then symbols in between
This channel deserves more subscribers.
IT LOOKS LIKE THE JULIA SET
I really like the 'we haven't finished maths yet'.
I recall as a math Teacher's Assistant talking to a few of my students about the math classes I was taking in grad school. One of them was shocked when I talked about math research, because they apparently thought maths was 'done'.
Mindblowing. I have never paid attention to floor tiling patterns.
Kids wake up, new shape just dropped!