A New Tile in Newtyle - Numberphile

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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[@Nitnelav1994]

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

[@fudgesauce]

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

[@H34L5]

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

[@layze23]

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.

[@monstronamaguederaz]

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

[@pilby457]

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

[@catcaptain6023]

Fifth Element

[@CamerTheDragon]

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]

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.

[@alasanof]

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

[@scottcampbell96]

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.

[@dougpowers]

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

[@megnoliaedge6500]

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

[@DudeWhoSaysDeez]

On point

[@MindstabThrull]

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!

[@kellerkind6169]

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 🙂

[@PetraKann]

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

[@abigailcooling6604]

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

[@vanderkarl3927]

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

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

[@cfgp]

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

[@alicec1533]

Idk, it just seems a bit _infantile_ to me 😏

[@matematicke_morce]

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

[@ravensiIva]

​@@alicec1533ok boomer

[@molybd3num823]

@@ravensiIva its a pun on tile, infanTILE

[@ravensiIva]

@@molybd3num823 oops missed that one wp

[@Alexander_Sannikov]

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]

more like, a shape that can never be tiled periodically

[@adamqazsedc]

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

[@zlatanibrahimovic8329]

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

[@LineOfThy]

@@zlatanibrahimovic8329 big brain moment

[@nerdiconium1365]

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]

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

[@user-sg4lw7cb6k]

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

[@sk4lman]

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]

go put the spectre next to it

[@tehlaser]

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.

[@johnflanagan871]

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!

[@woops9076]

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

[@Thimon88]

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

[@numberphile]

See the Craig Kaplan interview about discovering the tiles: kzbin.info/www/bejne/lYu2ZIKnnJZ0jpI

[@AylaKD]

first reply

[@SatishKumar-mo4hb]

Aapka mobile number dijiye

[@Cutflower]

Where’s the one with Dave smith???

[@osmia]

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

[@mriidulbhatia]

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

[@idontwantahandlethough]

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

[@CamerTheDragon]

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]

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.

[@tombaker8219]

The maths tattoos are so beautiful😻

[@FrankHarwald]

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

[@ianthehunter3532]

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

[@ianthehunter3532]

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

[@randy7894]

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

[@Malitz101]

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

[@Galenmacil]

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

[@AeonPrime92]

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

[@sandwich2473]

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

[@canalsoundtest]

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

[@michaelstoffregen2012]

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

[@DemoniteBL]

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]

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

[@darthrainbows]

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]

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]

@@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.

[@toadounetlovesyou]

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

[@thedofflin]

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

[@vanderkarl3927]

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

[@3ckitani]

Didn't expect the double upload

[@Arrman2]

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.

[@sophiusdynami3401]

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

[@engywuck85]

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]

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.

[@peetiegonzalez1845]

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

[@jacksonstarky8288]

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.

[@Drachenbauer]

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

[@jtharp]

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).

[@artswri]

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

[@me0101001000]

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]

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

[@me0101001000]

@@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.

[@Phobero]

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

[@sannekimenai639]

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

[@maitland1007]

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

[@arnauarnauarnau]

Loved the explanation, really cool

[@KipIngram]

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

[@kappascopezz5122]

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

[@WolfWalrus]

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]

I watch this exciting video and weep with joy

[@Sandvich18]

I was smiling the entire episode!!

[@orcu]

Great one!

[@renerpho]

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

[@Techmagus76]

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

[@jakethomas6123]

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

[@DDranks]

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.

[@jimmyzhao2673]

Wow ! What a time to be alive

[@DeFaulty101]

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]

love the future segment @matt parker would be proud!

[@thatonedynamitecuber]

Great puns exist everywhere, even in mathematics

[@GrantCulham]

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

[@brololler]

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

[@MrKYT-gb8gs]

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

[@CR0SBO]

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

[@leobrouk]

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.

[@Zaximillian]

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

[@DampeS8N]

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.)

[@henrikwannheden7114]

I loooove this video!

[@nicholas3354]

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

[@davidgro2000]

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]

@@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]

@@nicholas3354 Entirely possible. Pacific Northwest here.

[@dave20874]

Right! Totally a T-shirt.

[@kwanarchive]

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

[@josephpazar]

So AWSOME!!!

[@Rubrickety]

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

[@macncheez1248]

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]

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]

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"

[@ceegers]

11:01 that giggle

[@Bibibosh]

Aylein is the best!!

[@alexanderstohr4198]

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.

[@Tapecutter59]

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 :)

[@bivshiyministerr9424]

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

[@anhyzerkaiser]

Could we use this tiling as some kind of co ordination system?

[@scottcampbell96]

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…

[@TheMemesofDestruction]

7:41 - The Golden ratio is pretty cool. ^.^

[@pacobrezel]

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

[@hasko_not_the_pirate]

I totally would buy the Numberphile Monotile t-shirt.

[@Masterpdudley]

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

[@asheep7797]

It's finally been covered by Numberphile!

[@JustinDeFouw]

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

[@OrigamiMarie]

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.

[@ravenmcbride8622]

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.

[@PunmasterSTP]

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

[@BrendanGuildea]

25:12 … Best tumble weed Segway to a pun joke! (Though I like the Vampire!)

[@mdb1239]

David Smith the aperiodic monotile hero.

[@qtheplatypus]

Are there any frac-tiles (as in a tiling that is also a fractal).

[@JellyMonster1]

Thrilled that the turtle got a mention :)

[@TheDobstopper]

Bro wake up new texture just dropped

[@That-Guy-79]

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.

[@Veptis]

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?