A New Tile in Newtyle - Numberphile

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We're in Newtyle, Scotland, to celebrate the discovery of an Aperiodic Monotile. More links & stuff in full description below ↓↓↓
Check opportunities with Jane Street at www.janestreet.com/join-jane-... (episode sponsor)
See an accompanying interview with Craig Kaplan - co-discoverer of both this tile and the subsequent chiral version - • Discovery of the Aperi...
This video features Ayliean MacDonald - linktr.ee/Ayliean
Extra footage from this video - • A New Tile (extra) - N...
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
Numberphile is supported by the Simons Laufer Mathematical Sciences Institute (formerly MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from The Akamai Foundation - dedicated to encouraging the next generation of technology innovators and equitable access to STEM education - www.akamai.com/company/corpor...
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1 июн 2024

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Комментарии : 343

@Nitnelav1994 11 месяцев назад

Love when math is so hot off the math presses that they immediately need a correction bit within the video

@fudgesauce 11 месяцев назад

Surely the "turtle" piece should have been called a "turtile"

@H34L5 11 месяцев назад

Brady's groan was the perfect response to 'rep-tile'

@monstronamaguederaz 11 месяцев назад

Ayliean a week ago: "This is genuinely a once-in-a-lifetime event". Ayliean today: "Oh wait a sec."

@layze23 11 месяцев назад

I wish I had as much passion about anything as Ayliean has about aperiodic monotiling. It's actually really wholesome and the energy is contagious.

@pilby457 11 месяцев назад

Ayliean is a great communicator and donning the space age outfit and background as Future Ayliean is a huge plus

@catcaptain6023 10 месяцев назад

Fifth Element

@alasanof 11 месяцев назад

It's funny having someone pass around lil tiles around town. I like that the people seem nice about it.

@scottcampbell96 11 месяцев назад

I especially liked the builder, who has probably thought about this problem many times. I wonder if he is actually relieved that someone has figured it out and he can stop wondering.

@CamerTheDragon 11 месяцев назад

It's cool how they found not only the previous tiling, the one mentioned here, but also the one more recently not requiring the flips so soon afterwards. Interesting to see how relatively simple these tiles are, it's just a matter of finding the right one and then proving it does tile aperiodically.

@davidmartensson273 11 месяцев назад

That feels like an example of the Bannister effect. Before anyone have done it, it can feel impossible and that creates a mental barrier. Once its been done, everyone knows its not impossible and more will succeed, just because they manages to try harder knows its not impossible.

@dougpowers 11 месяцев назад

Ayliean is like a punk math fairy. She's a great communicator.

@megnoliaedge6500 11 месяцев назад

_punk math fairy_ New life goal unlocked. ...or new D&D character, at the very least.

@DudeWhoSaysDeez 11 месяцев назад

On point

@MindstabThrull 11 месяцев назад

Combine with Dr Tom (can't remember his last name but I think he's at Tom Rocks Maths?) and I'm expecting all sorts of shenanigans in the mathematics world!

@Alexander_Sannikov 11 месяцев назад

this has always confused me, because you can trivially make aperiodic tiles out of right triangles. but the problem is not to find a shape that can be tiled aperiodiacally. it's to find a shape that can _only_ be tiled aperiodically.

@adamqazsedc 10 месяцев назад

more like, a shape that can never be tiled periodically

@adamqazsedc 10 месяцев назад

one more definition that I've checked: An Einstein (aperiodic monotile) is a shape that _forces_ aperiodic tiling

@zlatanibrahimovic8329 9 месяцев назад

a shape that doesn't tile at all is a shape that never tiles periodically.

@LineOfThy 7 месяцев назад

@@zlatanibrahimovic8329 big brain moment

@kellerkind6169 11 месяцев назад

Those L-shaped tiles have been part of a video with Cliff Stoll (Kline-Bottles) where he devided a cake in the same manner like 6 years ago. Brady should have remembered that 🙂

11 месяцев назад

The enthusiasm in Ayliean is so refreshing and fun to watch.

@PetraKann 11 месяцев назад

Love the ratio φ^4 to 1 That golden ratio seems to stick its beak into almost everything

@abigailcooling6604 11 месяцев назад

pi, e, phi, and i all seem to continually turn up places in maths that they have no right to be

@vanderkarl3927 11 месяцев назад

φ^4 is also equal to 3φ+2, since φ^2=φ+1 I love that beautiful number...

@tehlaser 11 месяцев назад

I kinda love that “humans” (well, Dave Smith, really) found this tile without knowing exactly what it was that “we” had for quite a while. Usually things go the other way: a proof by construction instead of just a raw counterexample. Those techniques often lead to unsatisfyingly complex, “messy” mathematical objects. The t-shirt tile just LOOKS like a fundamental truth of geometry, not some arbitrary, man-made technicality.

@cfgp 11 месяцев назад

i like how the tile community aren't shy of using puns to name everything

@alicec1533 11 месяцев назад

Idk, it just seems a bit _infantile_ to me 😏

@matematicke_morce 11 месяцев назад

@@alicec1533 I actually find them quite s-tile-ish

@ravensiIva 11 месяцев назад

@@alicec1533ok boomer

@molybd3num823 11 месяцев назад

@@ravensiIva its a pun on tile, infanTILE

@ravensiIva 11 месяцев назад

@@molybd3num823 oops missed that one wp

@user-sg4lw7cb6k 11 месяцев назад

Great discovery, and Ayliean brings it alive in a special way. The enthusiasm in Ayliean is so refreshing and fun to watch..

@sk4lman 11 месяцев назад

I was so excited when I heard about this a few months a go, i casted a 200lbs hat in concrete, painted it white, and have it on display in a local park, along with a few sheet of information on the maths :)

@aron8999 11 месяцев назад

go put the spectre next to it

@numberphile 11 месяцев назад

See the Craig Kaplan interview about discovering the tiles: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-_ZS3Oqg1AX0.html

@AylaKD 11 месяцев назад

first reply

@SatishKumar-mo4hb 11 месяцев назад

Aapka mobile number dijiye

@Cutflower 8 месяцев назад

Where’s the one with Dave smith???

@johnflanagan871 11 месяцев назад

With the little bit that hangs off the bottom of the t-shirt it looks like a baby onesie. “Onesie” would have been such a great name!

@nerdiconium1365 11 месяцев назад

So glad you guys finally got to cover this! I was mainly waiting for your definitions section since that’s what’s been somewhat lacking in other videos about this

@Milan_Openfeint 11 месяцев назад

I'd say the channel "Mostly Mental" explained it better and in more detail.

@woops9076 11 месяцев назад

I remember hearing about this a couple of months ago. Great to see numberphile covering this finally.

@Thimon88 11 месяцев назад

Almost 30 minutes of Ayliean. Get ready for some puns people! So much fun.

@tombaker8219 11 месяцев назад

The maths tattoos are so beautiful😻

@Malitz101 11 месяцев назад

I'd love to see those as floor tiles. I wish I'd see some somewhere.

@osmia 11 месяцев назад

This is the bestest explanation I've seen since I first heard of this new tile Much thanks

@michaelstoffregen2012 11 месяцев назад

Shared this video with my mom, I'm looking forward to the quilt being made based on the idea of hat like shapes.

@Arrman2 5 месяцев назад

I only just came across this video. Great to hear more about this amazing discovery! After I read the paper a few months ago I... ordered a pair of custom sneakers with this magnificent tyle pattern. I regret nothing.

@Galenmacil 7 месяцев назад

17:17 "Drawing skill engaged..." 🥰 So smart and elegantly beautiful you are Ayliean!

@canalsoundtest 11 месяцев назад

8:59 Turn on closed captions to see a special guest from Sonic 3 & Knuckles

@sandwich2473 11 месяцев назад

Oh my goodness there's two new newtyles 👀👀👀👀 Can't wait for the video

@toadounetlovesyou 11 месяцев назад

Very interesting mathematics, overshadowed only by Ayliean's puns.

@randy7894 11 месяцев назад

Animation guy and his family will have their holliday next year.

@thedofflin 11 месяцев назад

I'm so glad there's an unmirrored one! Now I can sleep well

@sophiusdynami3401 11 месяцев назад

The description of delight @23:53 is so beautiful! Its the best part of doing science...

@FrankHarwald 11 месяцев назад

8:40 love your Hilbert space filling curve on your wallpaper.

@ianthehunter3532 11 месяцев назад

Thanks for leaving your comment. I saw that pattern before on The Coding Train, but couldn't find his video on it.

@ianthehunter3532 11 месяцев назад

And now Steve Mould made a video about them, what a coincidence!

@3ckitani 11 месяцев назад

Didn't expect the double upload

@vanderkarl3927 11 месяцев назад

7:40 Phi (the golden ratio) to the 4th is the same as 3 * Phi + 2, owing to Phi's defining property that Phi squared is equal to Phi plus one. ɸ^4 = 3ɸ+2

@AeonPrime92 11 месяцев назад

The timing is amazing, I just watched the interview with Craig Kaplan a few hours ago.

@peetiegonzalez1845 11 месяцев назад

I would love to see Sir Roger's response to seeing this. I imagine he is absolutely over the moon.

@jtharp 11 месяцев назад

It is pretty interesting (to me anyway) that it is a prime number of sides (13) with the counts of concave and convex angles each being squares (4 & 9).

@DemoniteBL 11 месяцев назад

I wonder if this has applications in video game design, perhaps a way to stop textures from repeating, but idk how that would affect rendering.

@CookingWithCows 11 месяцев назад

Or maybe you can just use a quicker rendering model and make your texture better

@darthrainbows 11 месяцев назад

You don't need an aperiodic tile that can tile the whole plane for that, since render space is finite. You're also almost certainly dealing with non-flat spaces, which change the game significantly.

@kazedcat 11 месяцев назад

It is very bad for texture. Although it is aperiodic the structural pattern for "The Hat" is very strong. So even if mathematically it is aperiodic visually it has a very strong pattern. It is just like the L shape tiling it is aperiodic mathematically but visually the pattern is there.

@davidmartensson273 11 месяцев назад

@@kazedcat And giving it any type of non uniform pattern and getting that to match up to all combinations will be quite the challenge I think, especially if you want the edges to blend in and not be clearly visible.

@engywuck85 11 месяцев назад

Thanks for this great introduction! The excitement is so wholesome. Can’t wait for the update about the vampire tile. From a short search it seems to be based on the shape that was mentioned at 21:23 Just with asymmetrically shaped vertices in order to prevent periodic tiling.

@sk8rdman 11 месяцев назад

Not quite. From what I gathered the aperiodic monotile that doesn't use reflections can have straight lines, but the fact that it doesn't use reflections means that the edges can actually be any curve, as long as all edges are the same curve. The depictions shown do use a curve to demonstrate this, but it doesn't need any specific curve to work.

@jacksonstarky8288 11 месяцев назад

Always nice to have a topic that we know we're getting a follow-up video on before the first video comes out. I've been playing with Penrose tiles for years, and this is an impressive bit of progress.

@mriidulbhatia 11 месяцев назад

i cant believe you guys went to the middle of nowhere just for a pun lol

@idontwantahandlethough 11 месяцев назад

it does seem _very_ on brand for math nerds for some reason lol

@CamerTheDragon 11 месяцев назад

They should go again now for the even newer tile, I wonder if the people there would be confused why they've come back so soon lol

@Zveebo 11 месяцев назад

A pun is always worth putting in the effort for! But Newtyle is only about an hour’s drive from Edinburgh, so it’s not really in the middle of nowhere anyway.

@Drachenbauer 8 месяцев назад

For the penrose kites and darts there is an additional rule to prevent periodic tiling: Do not put the whide corner of the kite into the inner corner of the dart. Overgoing this rule create a diamonds shaped, that can tile periodic

@maitland1007 10 месяцев назад

Cool video. I'm in the camp that doesn't consider this a real mono tiling, but looking forward to the next video.

@me0101001000 11 месяцев назад

Mathematics always finds its way into other fields. As a materials scientist, I'm interested in seeing this applied in crystallography. I wonder what kinds of advanced materials we could develop using such a structural pattern. Surely it would be a poor conductor, and structurally very stable.

@ShankarSivarajan 11 месяцев назад

What is your intuition for why the aperiodicity of the structure implies poor conductivity?

@me0101001000 11 месяцев назад

@@ShankarSivarajan generally, conductors have a very periodic structure. The more axes of symmetry, the better the conductivity. And when you disrupt that symmetry, conductivity drops. Take iron for example. It by itself has quite a high conductivity of both heat and electricity. But both end up dropping as we add carbon into the structure to make steel. And on top of that, structural rigidity also increases. You can also play with this by looking at crystal twinning and boundary effects for mechanical, electrical, and thermal systems. There are anomalies to this rule, of course, particularly among nanomaterials, but I'm speaking from a very broad and general point of view.

@sannekimenai639 11 месяцев назад

Brady, I am stunned you didn't recognize that L-shaped cake! I remember Cliff Stoll making you a birthday cake like that!

@artswri 11 месяцев назад

Great discovery, and Ayliean brings it alive in a special way

@Phobero 11 месяцев назад

Aperiodic Monotile would be a great name for a math rock band

@KipIngram 12 дней назад

Wow - she is incredibly charming. Why have you not had her on as frequently as some of the folks you do? More, please.

@Sandvich18 11 месяцев назад

I was smiling the entire episode!!

@arnauarnauarnau 11 месяцев назад

Loved the explanation, really cool

@kappascopezz5122 11 месяцев назад

2:25 I don't speak animation guy but that one was easy to interpret as "yes I will, thank you"

@renerpho 11 месяцев назад

Your videos with Ayliean are always fun and engaging. I'm looking forward to the follow-up!

@WolfWalrus 11 месяцев назад

Quick thought about that "farmer and his four sons" puzzle The son who gets the middle tile must be really annoyed, because every time he wants to get to his new field, he has to go through someone else's, either one of his brothers' or the stranger who bought the top corner

@malcolmsavage7456 11 месяцев назад

I watch this exciting video and weep with joy

@orcu 11 месяцев назад

Great one!

@jakethomas6123 10 месяцев назад

I’ve seen alum crystals just like the H meta-tile. Also - multi-leaf clovers totally jump out at me, too!

@DDranks 11 месяцев назад

I got another answer to the farmer's field puzzle. I wasn't said that the pieces mustn't be continuous. Just split the L shape into three squares, split each square into four smaller squares and let everybody have a piece that consists of three separate squares located like the corners/endpoints of an L. The pieces are all exactly the same (separated only by translational symmetry), although not continuous.

@macncheez1248 11 месяцев назад

10:45 i noticed with this shape something that may be interesting. If you were to have an infinite plane of these shapes, as a fractal like environment, what would happen if you tried to continuously go towards the top right? Would you ever reach a “void” of no shapes? Or would it get wierd or something and force you to be in a bottom left L and go towards a middle L? Or is there a simple solution? Sorry if my explanation is a bit weird, I’m just curious.

@pyromig 11 месяцев назад

If it would be an infinite plane, that would mean that this plane is an infinite "L" shape. So if you were to go infinitely in a strait line you will continuously end up in larger and larger "L"... infinitely

@pyromig 11 месяцев назад

you also can imagine this as a Finite "L" that consists of infinitely small "L" shapes. And you are infinitely small inside and are moving infinitely slow so you will never escape or even get closer to the edge of a Finite "L"

@Techmagus76 11 месяцев назад

Luckily we are part of the right time period to see live such an aperiodic breakthrough.

@jimmyzhao2673 11 месяцев назад

Wow ! What a time to be alive

@Veptis 18 дней назад

So how many classes of monotiles are there? Retiles, periodic tiles, aperiodic tiles and that's all? Or might there be another class? Also do tiles extend to tile sets all the time. Maybe aperiodic tiling that is forced? as in there is no variations on tiling. It's always the same but some symmetries maybe? Or is that true due to the infinite and periodicity? As in any tiling you do... Is just part of the same Über tiling, just a very specific part of it. Meaning no matter what you do. You are forced to include every given subtitling eventually? Maybe a better formulations: does every aperiodic tiling include every single meta tiling? Or can you proof that an aperiodic tiling is possible by excluding a given meta tiling. Also how large can those be?

@thatonedynamitecuber 11 месяцев назад

Great puns exist everywhere, even in mathematics

@DeFaulty101 Месяц назад

My answer to that old riddle was just to devide each third into quarters, and to give each son the same corner of each third. This creates identical pieces of land, albeit disconnected.

@thegametrainer9004 11 месяцев назад

love the future segment @matt parker would be proud!

@GrantCulham 11 месяцев назад

That Hilbert Curve paper behind "future Ayliean" is great too!

@alexanderstohr4198 4 месяца назад

the join-together animation towards the end suggests these new tile set to be periodic into one specific direction - but not clear how the pattern will go on in the perpendicular direction. maybe such a way of visually describing it will shed more light into the subject than even the most well funded proof paper will ever do.

@dave20874 11 месяцев назад

Right! Totally a T-shirt.

@bivshiyministerr9424 11 месяцев назад

"Once in a lifetime event" that happened twice almost at the same time. Must be overexcited now :)

@Bibibosh 11 месяцев назад

Aylein is the best!!

@josephpazar 11 месяцев назад

So AWSOME!!!

@CR0SBO 11 месяцев назад

Hats are Turtles, and Mugs are Doughnuts, and we're all (essentially) spherical cows in a vacuum! I do love maths!

@leobrouk 11 месяцев назад

Where does the "spherical cow in a vacuum" come from? I know of a "spherical horse in a vacuum", the jocular definition of the horsepower unit.

@Tapecutter59 5 месяцев назад

For the farm puzzle I divided the remaining 3 quater squres into 4 squares each, then gave each of the 4 sons 3 of the small squares. I ended up with the same pattern :)

@DampeS8N 11 месяцев назад

Four leaf clovers are easy to find. Five leafers are harder. But the hardest is 6 because they look like two 3s stuck together and they are hella rare. I've only found two. I've never even seen a 7 but they _should_ be possible through a fusion of a 3 and 4 (which is how the 6s happen.)

@pacobrezel 10 месяцев назад

Am I the only only one hypnothised by the beauty of tiles

@henrikwannheden7114 11 месяцев назад

I loooove this video!

@scottcampbell96 11 месяцев назад

Seeing the map of Newtyle makes me want to buy some land in the shape of this tile. Or several neighboring plots. Or found a town where all of the plots are this shape…

@filipdahlberg4420 11 месяцев назад

Hmm would this be more stable in buildings? Ancient walls have irregular cut stone walls which leads to more resistant buildings(against earthquakes etc.) with this irregular tile could that improve on this?

@nicholas3354 11 месяцев назад

I thought four leafed clovers are just a myth. I looked for them alot as a kid, but never saw one.

@davidgro2000 11 месяцев назад

I'm guessing they are more rare in some places than others, where I live if you are actively looking it's often possible to just find some on the side of a road or in a field. If you do find one, the same plant will also often have more of them alongside the normal three leafed ones, or if you are extra lucky, a five leafed one.

@nicholas3354 11 месяцев назад

@@davidgro2000 Wow. Five leaves! Cool. Yeah, I don't think we have any four or five leafed ones in my town. I live in the valley in Northern California, but they may be elsewhere in NorCal.

@davidgro2000 11 месяцев назад

@@nicholas3354 Entirely possible. Pacific Northwest here.

@Rubrickety 11 месяцев назад

Pity about "hat". It could have been the Turtle and the Shirtle.

@MrKYT-gb8gs 11 месяцев назад

Hello, I have a software q for animation, did you guys make your own tool? If not, what do you use?

@kwanarchive 10 месяцев назад

23:15 "In plain sight". Nice unintentional pun. "In plane sight".

@JustinDeFouw 11 месяцев назад

My love of Tetris as a kid helped me solve The Farner Square problem right away. Sure the same for others!

@Masterpdudley 10 месяцев назад

The proof should be very easy! Just grab an infinite plane from your local hardware store and tile it in a finite amount of time… can’t believe they didn’t think about doing that

@Zaximillian 11 месяцев назад

Is THREE dimensional aperiodic shape possible? My instinct tells me that this shape could have real life applications.

@brololler 11 месяцев назад

are there blocks, to build an aperiodic 3d shape?

@ravenmcbride8622 11 месяцев назад

I wonder if there's a set of these with different powers of the golden ratio as their ratio of flipped to unflipped tiles. It would be cool to see if you can discover a proof for an infinite set of aperiodic monotiles.

@Faladrin 11 месяцев назад

So if they cannot tile periodically, does that mean there is only one tile? As in if you sat down and just plopped down a set of tiles that particular pattern of tiles in the end would only ever exist in one particular place in the infinite plane of these tiles? And if that is so couldn't you use that to do encryption? It would appear to provide a 1-to-1 mapping and you could have a pretty simple key. I'm thinking a rotation alone could be enough.

@mdb1239 8 месяцев назад

David Smith the aperiodic monotile hero.

@PunmasterSTP 9 месяцев назад

Oh man I've been waiting to see "The Einstein" covered!

@OrigamiMarie 11 месяцев назад

I have access to a laser cutter and I have a bunch of flat wood. I'm thinking of drawing details in the turtle tile and making a bunch of them to play with.

@Rabbit-the-One 11 месяцев назад

You think the triangle is a more basic tile than a square? Anyway, not why I'm here. So when I first heard this news in a RU-vid short, I commented that I expected to see a video from you and one from Matt Parker soon! I'm pleased to see that you have indeed made such a video.

@Rabbit-the-One 11 месяцев назад

That means you're up Parker. 😎

@That-Guy-79 11 месяцев назад

Dont know why but a Fibinoci spiral popped into my mind. I wonder if certain shapes follow a designated path where they'll be more likely to pop up in a predictable way. Im not a math kinda guy but nature has a way. And people love to look for a connection.