Discovery of the Aperiodic Monotile - Numberphile

An interview with Craig Kaplan, co-discoverer of the Aperiodic Monotile - the Holy Grail of Tiling. More links & stuff in full description below ↓↓↓
See our other video - the New Tile in Newtyle: • A New Tile in Newtyle ...
The first paper - An aperiodic monotile - David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss - arxiv.org/pdf/2303.10798.pdf
And the chiral follow-up - arxiv.org/pdf/2305.17743.pdf
Craig Kaplan at the University of Waterloo - cs.uwaterloo.ca/~csk/
David Smith blog post: hedraweb.wordpress.com/2023/0...
Numberphile is supported by the Simons Laufer Mathematical Sciences Institute (formerly MSRI): bit.ly/MSRINumberphile
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[@bradleysampson8230]

Brady is such a great interviewer. I miss Hello Internet.

[@volodyadykun6490]

Also more Numberphile Podcast pls

[@DefnitelyNotFred]

Yeah, hello internet was the GOAT…

[@oldcowbb]

many many moons now

[@Jivvi]

Definitely needs to make a comeback.

[@hellocanyouhearme]

I miss it too😢

[@QuantumHistorian]

Genuinely brilliant interview, by both both sides. Brady asks all the right questions, and Craig gives real answers to them. It's rare for it to bring the human element into the research without going too far one way or the the other.

[@Schattenhall]

I absolutely agree. Great format/style for a numberphile video - thrilling and captivating!

[@elmiraguth]

To me it seems like he's repeating a bit. I haven't paid 100% attention, but it feels like he asked him "How does this make you feel?" like four times, each time worded slightly differently. It was still a fun interview nonetheless.

[@QuantumHistorian]

@@elmiraguth Yeah, asked him how he felt _about different aspects_ of it. Which is... exactly how a half hour interview should be conducted lol

[@elmiraguth]

@@QuantumHistorian Should be? According to whom? I believe that such a long interview would benefit from more varied questions (or from being shorter).

[@QuantumHistorian]

@@elmiraguth And I believe it's best to pay 100% attention to something before making recommendations on it.

[@johnredberg]

For anyone trying to find the Japanese artist Prof. Kaplan is mentioning on several occasions, the proper spelling is "Yoshiaki Araki".

[@osmia]

+

[@MrCheeze]

Loved seeing the emails between the researchers, as they added more people onto the team. You can imagine what it must be like for a mathematician to get a mesage from your peers saying "We have a promising lead on the biggest open question in our field, and we think you're the ideal person to work on it." (In more cautious language of course, but they know exactly what it means.)

[@Schattenhall]

"I'm putting together a team" " you son of a bitch...I'm in"

[@jethin37]

I love how it was found by a random shape enthusiast. Just so cool that this guy could find it with awesome intuition

[@ferretyluv]

A recreational mathematician, just like Fermat.

[@adamsmith7885]

Not random. Dave. That man is a true mathematician

[@Max..Q]

If Craig is looking for a new quest... well, he can always go one dimension higher and look for an aperiodic monosolid.

[@fburton8]

Nice, but maybe it only works in even dimensions.

[@GerhardTreibheit]

stolz

[@bernardopicao267]

Or maybe a chiral aperiodic polygonal tile

[@thesenamesaretaken]

@@fburton8 well now I want to see a 3d projection of an aperiodic 4d hypertile

[@rickyardo2944]

@@thesenamesaretaken if the tile is made thicker and tiled into a plane and then layers of these planes are stacked, does that not count as a aperiodic polygonal tiling? just asking... thanks

[@rohitg1529]

Craig wasn’t my professor, but we had common office hours in first year and I went to visit him every week. He was a great teacher, and I never expected him to show up on numberphile before computerphile!

[@justin_5631]

Should interview this David guy too. Interesting to see a non-mathematician get real work done in math.

[@davidhuynh9996]

Exactly what I was thinking. He basically found the solution and then prof. Kaplan verified and made rigorous. I'd love to hear more about David's process.

[@roneyandrade6287]

I like how it looks like a Tshirt

[@letMeSayThatInIrish]

Was thinking the same thing. I'd call it a t-shirt tile.

[@asheep7797]

A torn-up t-shirt?

[@veggiet2009]

​@@asheep7797you might call it "high fashion"

[@triste4-21]

The other one looks like a pancho

[@rosiefay7283]

@@asheep7797 Or a shirt where one side's tucked in and the other side isn't.

[@geckoman1011]

What a wonderful interview. The guest was very generous to all involved, from his coauthors to the listeners.

[@ferretyluv]

I’m glad that David Smith got top billing on the article.

[@JellyMonster1]

Craig, really enjoyed the talk. Great to relive the moment. Fantastic journey. Many thanks.

[@osmia]

+

[@TheMeal]

Brady, you are amazing at interviewing. The window you open to the world's incredible nature is mind-blowing. Thank you for sharing with us.

[@Paul71H]

I've been waiting for a Numberphile video about this monotile! I've been interested in this subject since I read Martin Gardner's columns on Penrose tiles years ago. Thanks for sharing information about this interesting and important discovery.

[@PhilBagels]

Me too! I have a copy of that issue of SciAm where he talks about the Penrose tiles.

[@johndeaux8815]

I love that this hippie shape enjoyer created some shape and was like hey man I made this shape but it’s not working properly 😂

[@menso541]

The shape they called a hat was a t shirt to me lol. It's crazy to see these shapes and so clearly see how they could tile a plane. I wish I could see their reaction when they realized that they found it.

[@lightbeware9875]

Agreed! Looks like a v-neck.

[@Luper1billion]

To be fair, if "einstein" means "one stone" then even if you have to flip it, you only need one shape of that stone

[@3ckitani]

Didn't expect the double upload

[@felixu95]

Me neither

[@felixu95]

Me neither

[@jacobsparkstudios528]

Me neither

[@xongi9248]

I did

[@BrianDeBrain_]

Me neither

[@TheRuler89]

Wonderful closing words and beautiful interview. Thank you both very much!

[@Tesserex]

I want a t-shirt that just is the tile shape. WIth the point at the bottom and the asymmetrical wonky sleeves, and offset v-neck, but it would be amazing.

[@madarauchiha2584]

Respect for david

[@iwersonsch5131]

Now: What is the smallest number of edges that a polygonal aperiodic monotile can have?

[@bunnybreaker]

As a gamedev/artist/vfx geek, this is super interesting. Love this stuff 🖤

[@ferretyluv]

Someone on Reddit mentioned that an aperiodic monotile would make tiling in video games look more realistic. Like how water from above is just squares repeating and breaks immersion, a monotile could break it up more naturally.

[@Verlisify]

Great attitude to see the criticism of flipping the shape as another solution to solve

[@vernontemp]

Craig seems like a nice bloke. Happy for him.

[@szymonbaranowski8184]

why would it matter?

[@wesleydeng71]

Dave Smith is a genius.

[@JellyMonster1]

I've been called many things before but never 'a genius'. You are too kind.

[@radagastwiz]

UWaterloo content! Love it when I get to see someone local.

[@raytonlin1]

WATER WATER WATER! LOO LOO LOO!

[@OwlRTA]

thank mr goose

[@theBestInvertebrate]

​@@raytonlin1 nonsense, Waterloo STEM students don't do the chear.

[@TheAlison1456]

I don't know much at all about tilings but it's so much fun seeing how important and exciting this is. I love how you talk to guests, who are often academic (and frankly, typically stifled by strictness and disallowedness), but you as well as they are shown to just be normal people.

[@p1mason]

0:54 Did anyone else appreciate how Craig's background perfectly defined one of the kites that makes up the hat tile?

[@sicapanjesis3987]

Yeah it looked great tbh

[@stubbsmusic543]

Nice that you mention David Smith. You know, the guy that discovered this thing.

[@adamsmith7885]

the man who discovered the shape twice. a true mathematician!

[@zmaj12321]

Amazing interview!

[@heretolevitateme]

As brilliant as this story is, as incredible as this interview is, the editing is pure joy. :D

[@louisreinitz5642]

I too am a shape hobbyist. I have not experienced this level of success

[@yahccs1]

Clever and amusing presentation design putting the videos shaped windows!

[@speadskater]

I'm really amazed about how easy the discovered monotile is no generate

[@tuna5618]

finally early to a numberphie video, and it's about tiling, honestly i see this as an absolute win

[@roskoced6598]

I love that the octo-kite is actually a symmetrical pentagonal bi-kite to which all three possible mirror image bi-kites are attached by each side type. And since the pentagon is symmetrical, two of these mirror image bi-kites have two sides to which they can be attached, while the third has only one. So there are four possible octo-kites that you could construct by this approach. I wonder if all four would be aperiodic monotiles, or just the one.

[@LeonardTavast]

It would be cool if these tilings could be used as texture assets in videogames. Then somewhat simple mathematic formulae could be used to make complex graphics.

[@Nerdule]

Aperiodic tiles have already been used as a texturing trick for quite a while - not using weird-shaped *mono*tiles, but several square Wang tiles with rules for what can connect on what side.

[@ZekeRaiden]

Not sure there would be much appetite for aperiodic tiling in computer graphics. It would be more complicated than triangle or square tiling, which is what everyone uses now, and as long as you keep your textures subtle you don't really have to worry much about the periodicity being obvious.

[@karlramberg]

@@ZekeRaiden In old games there would be visible artifacts is large fields of similar texture, like in for example grass. But it would be overkill to apply this tiling for that issue. It would be interesting to see if some one makes a board game like Carcassonne with this tiling.

[@szymonbaranowski8184]

why to use more complex part instead of smaller generic ones? nobody cared to even find this answer you got here just simply checking all possible diamond built tiles it means this discovery might be just art for art and all you people hyped about it just pumping empty balloon

[@welcometothemadhouse]

An interesting thing about the distribution of the reflective hats is that they seem to be 2 connected hats apart from each other?

[@nicksamek12]

When I read of the einstein I'd been waiting for the numberphile about it to come out! Exciting to hear that it was delayed because there's a new and better one.

[@pirobot668beta]

When I first saw the 'one stone' tile and heard what it could do, it felt 'broken' to me. Couldn't explain it, so a made a bunch and played with them. In very short time, I was making periodic structures in 30 degree increments. 'Specter' tiling fixed the problem; I can look at piles of tiles without getting a sick headache anymore.

[@waterloomath]

Great interview!

[@artswri]

Way beyond cool that these tiles are being discovered (and I'm around to see it happen!)

[@pepe6666]

its like finding a prime number in shapes or something. what a weird problem space. i aint never heard of this before

[@wyboo2019]

its just the fact that this simple shape that seemingly comes out of nowhere has a VERY unique property. these two (families of) monotiles have been out there in the space of possible shapes and its just never been found until now. why do they exist? what makes this combination of kites special?

[@as-qh1qq]

Superb interview

[@ehfik]

great story, what a time to be alive.

[@heaslyben]

Well, hats off to you all! That's great!

[@RaquelFoster]

Those arrangements of cardboard cutouts are really wonderful.

[@curtmcd]

Cute and creative video editing! Now time to work on an even simpler specter shape!

[@orterves]

18:11 - the careful distinctions between things like calculating vs computing, polyomimos and polyforms, is when you know you're listening to a passionate expert in a very specific field

[@MichaelDonlinAwesome]

Great vid Brady.

[@acidnik00]

Oh what a great time to be alive!

[@LeoStaley]

Finally you did this video

[@a88aiello]

Fantastic!

[@_1derscore]

seemed like an impossible problem, turns out to be the exact opposite, as a huge geometry fan, this discovery is HUGE for me i love this

[@xyzct]

Brady, please do a video on young Daniel Larsen and his amazing paper on Carmichael numbers.

[@malcolmsavage7456]

well done you guys

[@lobsterrock4570]

its so amazing that it was discovered by a hobbyist!!!!!

[@stevenlangsford8624]

Thank you

[@seanbcusack]

are there tilings that go a long ways out and seem to be periodic or aperiodic but then change from seemingly periodic to aperiodic or the reverse? are there tilings that go a long ways out before they break and stop being able to tile at all? is there a maximum finite tiling that knowingly breaks? is it possible to construct a tiling that's unknowably periodic? i.e. it's impossible to prove if it's periodic or aperiodic?

[@frankharr9466]

That's interesting. Really cool in fact.

[@hareecionelson5875]

at Queen Mary's university in London, one of the walls has Penrose tile design

[@rtpoe]

I want to see photos of some of these things made with the tiles!

[@BryndanMeyerholtTheRealDeal]

How did you do the irregular curvy shape as a mask in the video?

[@codediporpal]

What's so weird about this is how obvious a potential solution it is. There are not that many combinations of kites from hexagons, and yet nobody tried them!

[@szymonbaranowski8184]

because they didn't try by brute force it's weird nobody else cared to use computing power to get this low hanging fruit that's why chatgpt will make us even more lazy cleaning up all low hanging fruits leaving only hard problems lol

[@CamerTheDragon]

Interesting to hear about the timeline of discovery and how fast it moved, especially from the hat to the spectre. In terms of does using flips count as a true monotile I feel like it depends. In a purely 2d space I'd say without flips is best, but physically in a 3d space I'd say with flips counts so long as the material you're using doesn't look different depending on whether the tile is flipped or not. So generally I'd say physically in a 3d space the hat is a monotile as is the spectre, but in terms of a purely 2d space I'd say probably just the spectre although it's up to interpretation.

[@ZekeRaiden]

Perhaps a simpler way to put it: up to chirality, there is at least one polygonal (straight-edged) aperiodic 2D monotile. If chirality is enforced, there is no known polygonal aperiodic monotile, but you can construct an infinite family of monotiles where the vertices are connected by congruent curves rather than straight edges. The "hat" is nice because it is polygonal, but it requires you to ignore chirality (or be in a space where 2D chirality is irrelevant, e.g. 3D space or higher.) The "spectres" are nice because they are genuinely monotiles (fully achiral), but you have to give up the straight edges. Now, the next question is: is there a polygonal aperiodic chiral monotile?

[@chalichaligha3234]

@@ZekeRaiden Yes, of course! The "spectre" is! At 27:43 Craig Kaplan says "You can modify the edges to do anything you want, whereas with the hat, the edges have to be straight lines".

[@starrmayhem]

@@chalichaligha3234 shh, don't tell them, i learn that it is disrespectful to backseat experts

[@SilverLining1]

@@starrmayhem if he didn't recognize the problem he posed was solved before he posed it by the very video he watched then he's not an expert

[@starrmayhem]

@@SilverLining1 hi~

[@johnchessant3012]

very interesting

[@PeterKelley]

What is the performance of these for a game board? Square tilings distort distance on the diagonal by root 2 to the centre of the square. Hexes are better but still distort at 2 away from the origin. What is the best periodic tiling where the number of shapes you have to traverse is closest to the distance between the centres of the shapes?

[@rmsgrey]

Hexagons are the best regular polygon tiling for that. I don't know if there's a better irregular shape - my intuition is not, but it is just an intuition.

[@mimidouceur5891]

🔺 Bravo David Smith! 🔻 ☀☀☀☀☀☀☀☀☀

[@advanzeelive]

It's a shirt that's not tucked into the pants on one side obviously.

[@josephpazar]

Love it

[@yohojones]

Canada on Numberphile! Hurray!

[@gabbiewolf1121]

It would be interesting if the tiling field could build on this work to create complete classifications of some sets of aperiodic tilings. Maybe future work in the field could discover important connections to other fields!

[@stevenbergom3415]

Can these shapes also tile non-planar surfaces eg. a cylinder, sphere or moebius strip?

[@smylesg]

I couldn't have come up with it, but the hat shape is really just two congruent inverted pentagons under two congruent overlapping rectangles with their opposite corners aligned.

[@ChoChan776]

or, as David said, a bunch of kites.

[@Birkguitars]

I really want to tile my new bathroom with the hat. This is a must. I need those tiles if only to bug my friends as they try to find a repeat, and fail.

[@CharlsonCKim]

is there a 3-D analog? what about in n-D?

[@MsAlleyZ]

I'm curious about the use of the hat as polygonal masonry. Earthquake proof walls??

[@pondermatic]

Is there a relationship between the aperiodic monotiles and the transcendental numbers?

[@michaeldunkerton3805]

What's going on with the arcs drawn on the Penrose Tiles and the Trilobite and Crab? Is it a guideline for how to place them to successfully tile?

[@andrewnotgonnatellya7019]

Yes, they're basically rule enforcements.

[@gb3551]

Someone please make a “hat” shaped cookie cutter so that professor Kaplan can safely eat (a bunch of) his hat!

[@FedeDragon_]

watching this video made me miss the Numberphile podcast

[@killymxi]

I'm in a "T-shirt" camp

[@fazergazer]

Yes: it would be quite challenging to prove a negative, but finding a single aperiodic monotile is demonstrating the positive.❤

[@danbutler7586]

Unfortunately tiles are usually only glazed on one side. An unglazed tile stamped both sides would work.

[@coulie27]

Felicitations to him 😊

30:13 What a wonderful thing to say and such a high note to end the video. Amazing interview, excellent questions and honest answers.

[@Yezpahr]

Von Neumann probes would build their circuitry and sensory through these shapes, rather than straight edges. The curved areas would allow for more ports/slots on the edges to connect these pieces for whatever data is needed. Theoretically speaking, of course.

[@geinling]

No visual tiling of the new shape? :(

[@StefanReich]

CLIFFHANGER

[@KaitouKaiju]

It does take time to make all those fancy animations

[@VideoNOLA]

Q: When can I buy these in ceramic for redoing my kitchen tiles?

[@arnoldmuller1703]

I wonder if there is a higher dimensional periodic tile that results in this aperiodic monotile via the cut-and-project method that is used to describe quasi crystals?

[@WrongParadox]

what about non-euclidean tiling? such as the surface of a sphere

[@user-bu5wr6jf5x]

It seems that there are 12 orientations used in the chiral aperiodic tiling. Is it possible to use only 1 orientation to make an aperiodic tiling? Or what is the minimum orientation needed to do that? :P

[@Emetris]

awww new tiles!!

[@pullt]

Is the hat they talk about related to the shirt tile they show?

[@fazergazer]

Imagine tiling the entire maths building with one tile❤

[@eviltreechop]

Genius editing!

[@alvarobyrne]

where to find the works Prof Kaplan talks about? i wonder and wander. Thanks in advance to anyone

[@erickmarin6147]

I wonder what happens with designer crystals in that shape

[@HoraceMash]

Invention or discovery? Either way: Faaaantastic!

[@fazergazer]

Should be possible to make a segment of 3D printer filament with preexisiting bracing that can interleave to create adamantine stability without resorting to custom Chiral space filling.

[@geoffreyoliver8925]

Does anyone know if with a given spectre tile there is only infinite aperiodic tiling; more than one; or an infinite number? I.e. if there was only one, no matter how you start, eventually you would generate the same tiling as from any other starting position. Is this an open question, or do we have the answer?

[@FenrizNNN]

Since there are multiple ways of connecting two tiles, my guess is that is is possible to make more than one different tiling, but I'm not 100% sure. You can, for example, connect them in a straight line, all having the same orientation, going off to infinity. (With the monotile on this video, specifically.)