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

@volodyadykun6490

11 ай бұрын

Also more Numberphile Podcast pls

@DefnitelyNotFred

11 ай бұрын

Yeah, hello internet was the GOAT…

@oldcowbb

11 ай бұрын

many many moons now

@Jivvi

11 ай бұрын

Definitely needs to make a comeback.

@hellocanyouhearme

11 ай бұрын

I miss it too😢

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

11 ай бұрын

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

@elmiraguth

11 ай бұрын

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

11 ай бұрын

@@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

11 ай бұрын

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

@QuantumHistorian

11 ай бұрын

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

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

@osmia

11 ай бұрын

+

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

10 ай бұрын

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

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

@ferretyluv

10 ай бұрын

A recreational mathematician, just like Fermat.

@adamsmith7885

4 ай бұрын

Not random. Dave. That man is a true mathematician

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

@fburton8

11 ай бұрын

Nice, but maybe it only works in even dimensions.

@GerhardTreibheit

11 ай бұрын

stolz

@bernardopicao267

11 ай бұрын

Or maybe a chiral aperiodic polygonal tile

@thesenamesaretaken

11 ай бұрын

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

@rickyardo2944

11 ай бұрын

@@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

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!

I like how it looks like a Tshirt

@letMeSayThatInIrish

11 ай бұрын

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

@asheep7797

11 ай бұрын

A torn-up t-shirt?

@veggiet2009

11 ай бұрын

​@@asheep7797you might call it "high fashion"

@triste4-21

11 ай бұрын

The other one looks like a pancho

@rosiefay7283

11 ай бұрын

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

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

@davidhuynh9996

11 ай бұрын

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.

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

@ferretyluv

10 ай бұрын

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

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

@osmia

11 ай бұрын

+

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.

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

11 ай бұрын

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

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

Didn't expect the double upload

@felixu95

11 ай бұрын

Me neither

@felixu95

11 ай бұрын

Me neither

@jacobsparkstudios528

11 ай бұрын

Me neither

@xongi9248

11 ай бұрын

I did

@BrianDeBrain_

11 ай бұрын

Me neither

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 😂

Respect for david

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

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

@adamsmith7885

4 ай бұрын

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

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

11 ай бұрын

Agreed! Looks like a v-neck.

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

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

@ferretyluv

10 ай бұрын

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.

Craig seems like a nice bloke. Happy for him.

@szymonbaranowski8184

9 ай бұрын

why would it matter?

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.

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.

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

@raytonlin1

11 ай бұрын

WATER WATER WATER! LOO LOO LOO!

@OwlRTA

11 ай бұрын

thank mr goose

@theBestInvertebrate

11 ай бұрын

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

Dave Smith is a genius.

@JellyMonster1

11 ай бұрын

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

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

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

@sicapanjesis3987

11 ай бұрын

Yeah it looked great tbh

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

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.

Clever and amusing presentation design putting the videos shaped windows!

Amazing interview!

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

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

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

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

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

Great interview!

Finally you did this video

great story, what a time to be alive.

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

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

11 ай бұрын

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

11 ай бұрын

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

11 ай бұрын

@@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

9 ай бұрын

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

Fantastic!

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

11 ай бұрын

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.

Superb interview

Those arrangements of cardboard cutouts are really wonderful.

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

Oh what a great time to be alive!

Thank you

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

@wyboo2019

11 ай бұрын

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?

Great vid Brady.

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

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

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

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

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

well done you guys

That's interesting. Really cool in fact.

very interesting

Love it

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

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?

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

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

9 ай бұрын

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

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!

🔺 Bravo David Smith! 🔻 ☀☀☀☀☀☀☀☀☀

Canada on Numberphile! Hurray!

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

11 ай бұрын

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.

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

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.

Imagine tiling the entire maths building with one tile❤

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

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

11 ай бұрын

or, as David said, a bunch of kites.

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

11 ай бұрын

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

11 ай бұрын

@@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

11 ай бұрын

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

@SilverLining1

11 ай бұрын

@@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

11 ай бұрын

@@SilverLining1 hi~

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

Felicitations to him 😊

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

Genius editing!

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

awww new tiles!!

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

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

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.

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

watching this video made me miss the Numberphile podcast

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

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

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

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?

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.

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

11 ай бұрын

Yes, they're basically rule enforcements.

how many tiles are needed to build a minimum bigger tile of the same shape?

I'm in a "T-shirt" camp

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

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

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

I think that it'd be cool if the college/university that Prof. Kaplan works at would construct their future math building so that it's shaped like his "hat" tile!

There is a fundamental connection between the flipped-tile interval in the aperiodic tiling of "the hat" and the 3x+1 Collatz conjecture. I have discovered a truly marvelous demonstration of this proposition that this comment section is too small to contain.

@MerchantMarineGuy

11 ай бұрын

Cool story bro

@ChoChan776

11 ай бұрын

I must go milk the cow, but tomorrow I will prove this theory.

Sweet.