Heesch Numbers and Tiling - Numberphile

[@corncolonel9171]

The sound effects make me really happy

[@f_f_f_8142]

If you like math and clicking sounds check out 3Blue1Browns latest three videos.

[@rcb3921]

I'm really struggling for some appropriate onomatopoeia. Splock? Maybe going to SplockaSplockaSplocka when it gets fast?

[@furiousxXxpyro]

Your profile picture does with me too

[@corncolonel9171]

@@f_f_f_8142 3Blue1Brown is amazing

[@JivanPal]

@@rcb3921, how about "clack"?

[@rosiefay7283]

I look forward to a follow-up showcasing Smith and Goodman-Strauss's aperiodic tile!

[@reversev9778]

I really hope they do it

[@wizardinthenorthable]

Came here to add my excitement. I do hope it is something they make a video on, I got sent the news by my brother and was surprised I hadn't heard from numberphile about it.

[@erizwidodo9926]

Yes, I can't wait for that too!

[@eatingsfun]

Yeeepppp

[@maiconfaria]

Hold my beer!

[@ZerofeverOfficial]

Can we just have a moment for how tight the animations on this channel are?

[@squeakybunny2776]

Number 4 has a bit of overlap which slightly bothered me if im really honest and the fifth heesch number animation is incorrect... Still satisfying tho

[@andioop6686]

Omg ZeroFever I didn't expect to see you here! *glad you also like numberphile heheheh*

[@ZerofeverOfficial]

@@andioop6686 OMG whats up Rocky!!! and yeah, Im addicted to math videos :D

[@xl000]

If you look closely, it wasn't that tight. But I doubt the animators will use some real animation software like SideFx Houdini to make the animations so that it never overlaps

[@TIMExBANDIT]

Heesch number shirts/hoodies would be hella cool

[@nzlawless]

that editing looks like a load of work, super well done and I would never have understood this without it. Massive props!

[@terrahyde217]

Casey Mann was my undergraduate advisor in mathematics 6 years ago. He's a really cool dude. And he does a LOT of work in tiling theory that turns out to be really interesting.

[@RC_Engineering]

They recently discovered a 13 sided shape that infinitely tiles the plane without repeating! I can't wait to see the video on it

[@RC_Engineering]

Still waiting!!

[@RC_Engineering]

We got it!!

[@Hyperdrive]

We got it!!

[@pickachu463]

Make a 10 hour version of tiling with those sweet sound effects

[@TruthNerds]

ASMR infinite aperiodic tiling…

[@alveolate]

go full production value and tile them with actual wooden/plastic blocks too.

[@nowonmetube]

Why does this make me laugh so bad 😂

[@markiyanhapyak349]

Ha • hA. 👌🏻 😅 👌🏻

[@Triantalex]

??

[@JJ-kl7eq]

Wow, this explains so much about The Dark Knight. Heesch Ledger was one crazy, unbalanced tile.

[@alephnull4044]

These are so bad lol

[@poznyakpoznyak]

He was one c(r)azy man

[@danfg7215]

too soon!

[@njdevilsforlifewoohoo5533]

I hate puns. None are funny.

[@pluto8404]

Jeesch. These puns dont fit

[@matthewb2365]

Last month, there was a new tile discovered that tiles aperiodically, and doesn't require a discontiguous tile. Can't wait for the upcoming video about it!

[@helpme6599]

Nice animations. Really helped illustrate the problem.

[@docdaneeka3424]

Nice illustrations. Really helped animate the problem.

[@jiminapemode5873]

Nice visual sequence, really helped draw the problem.

[@letMeSayThatInIrish]

Nice problems. Really helped sequence the illustrations.

[@jackofallmasterofnone8024]

Nice help. Really illustrated the sequence of problem.

[@alephnull4044]

Nice comments. Really helped demonstrate the gratitude of the viewers.

[@BeCurieUs]

The cheat is so pretty! I love this kinda of mathematics at play :D

[@tgwnn]

The opposite of the Parker Square, if you will.

[@andrerenault]

Parker-Taylor Tiling

[@Triantalex]

??

[@SendyTheEndless]

Those Heesch Shape Tiling sequences are extremely satisfying. Please, an entire channel dedicated to them!

[@TehNewV]

I love the look of the imaginary 4622 heesch tile. feels almost genuine in a way.

[@bgoggin88]

Cheat or not, Taylor's tiling is awesome!

[@anmax]

really thinking outside the box

[@Triantalex]

??

[@nolancoughlan4848]

A solution to the Einstein problem has been found a few days ago, pending peer review

[@ShankarSivarajan]

It has been found regardless of what "peers" think.

[@dapcuber7225]

could you link to the paper/proof?

[@Pomodorosan]

Oh yeah I saw that one first some time ago, then was surprised to see that odd tile shown in this video from 4 years ago. I can't find it anymore though.

[@victorcossio]

​@@dapcuber7225in recent videos in this Channel you find the video about it and the information about that paper

[@ubahfly5409]

​@@ShankarSivarajanWait why the contempt for peer review? Is the "establishment" not taking your grand theory seriously, Eric Whine-Stein? Lol

[@MrBrain4]

Love content about tilings! This is the area of mathematics I have done the most research in, and there currently seems to be very little overall information about tilings on KZbin.

[@ethanjensen661]

That's so beautiful. I like the sound of the tiles being placed together

[@yc1094]

Very cool to go back and watch this now out of date video!

[@joshua8483]

And now we finally have a single tile that periodically tile!

[@jeremysimmonds5213]

the animations are amazing please keep it up!

[@TheWhitePianoKeyProductions]

Yellow is touching the red at the top at 3:50 or is that okay?

[@ShayBowskill]

It's not okay. That yellow tile is supposed to be green. Time to riot!

[@TheWhitePianoKeyProductions]

@@ShayBowskill but even then it doesn't work though? or it's heesch number 5, so the red doesn't need to close it totally?

[@Party_Magician]

@@TheWhitePianoKeyProductions The Yellow is supposed to be a green, not the red. The red is in the right place, the whole shape is still enclosed in red

[@Mazsi1201]

@@TheWhitePianoKeyProductions it does have to be closed, i think if you changed the problematic tile to green then you could close the red boundary by adding one extra red tile where the newly green tile is touching the outside (by the looks of it it is possible to add an extra tile there, i think it would fit)

[@AdrenalineL1fe]

Parker tile

[@DelDelta_]

To any new viewers, you may be happy to know that an 'ein stein' has been found! There is a monotile that covers the plane, aperiodicly.

[@OlbaidFractalium]

The pattern made by mathematics is really beautiful.

[@alephnull4044]

THE pattern made by mathematics. I'd like to see that.

[@diegomo1413]

Eyy, it’s 2023 and they found a family of Einstein tiles!

[@poketopa1234]

Who would win: Centuries of the world’s smartest mathematicians VS a funny looking hat

[@walkingwriter4325]

Perhaps a dinosaur shape might work. You know, a rep-tile.

[@Monkey-l8s]

Walking Writer HAHAHAHAHHAHAHAHAHAHHAHAHAHAHAHAHHAHAH🤣😂😂🤣😂😂🤣🤣😂😂🤣🤣😂😂🤣😂😂🤣😂😂🤣😂😂😂😂🤣😂😂🤣😂😂🤣😂😂🤣😂😂🤣😂😂😂😂😂😂🤣🤣😂🤣

[@Pining_for_the_fjords]

I think a child-shaped tile would work better. You know, an infant-tile.

[@super_7710]

you guys lack (s)tyle

[@angelmendez-rivera351]

What a wholesome thread.

[@Monkey-l8s]

Ringo Garvin *pun* ny?

[@xenontesla122]

I love that boxy acrylic fractal sculpture in the background.

[@pruusnhanna4422]

MC Escher would love this.

[@mudmug1]

Escher and Penrose inspired each other

[@non-inertialobserver946]

How do they find those complicated tiles, is it some complex math or just trying different shapes?

[@ge2719]

there are methods of how to create shapes the tesselate perfectly. i imagine from the way they are talking about it there may not be say a specific way of finding a shape with specific hesch numbers.

[@StefanReich]

It looks like a deliberate construction really (the last one anyway)

[@logisthenewlinear]

Graeme Evans is exactly right. When it comes to shapes with specific Heesch numbers, we don’t know of any general method.

[@Pete-Prolly]

@@logisthenewlinear I dig your style! 😎 So "log," if I may call you that for short, are you "common" or "natural?"🍸🤔 (I sound like a perv 🤣 and I don't drink martinis, but I had to complete the cliché.)

[@unflexian]

computers mostly, the complex math comes when you want to take the jump from "this tile keeps tiling correctly in my program for as long as I've kept my computer running" to "this tile will never stop tiling ever ever never ever".

[@SaMadden99]

Killer animations man. It really brings the beauty of math to the forefront

[@blaholtzen]

i really love the animation in this one, and particualrly the litte tile sound, very pleasant

[@dudewaldo4]

Great animations!!! Really really enlightening. I wish you had showed why the penrose tiling was NOT periodic through the same illustration you did for the squares!

[@LeoStaley]

This is what I come to this channel for.

[@xenontesla122]

MATH NEWS!: An ein-stein tile has been found! “David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss have produced a single shape which tiles the plane, and can’t be arranged to have translational symmetry.” - The Aperiodical

[@TheMexicanPlatypus]

I love the animations in this video! Geometry videos are always great because they're simple to visualize in video form. One of my favorite Numberphile videos :)

[@nicholaspatrick4092]

At 3:46, a green piece at the top is miscolored yellow. The yellow piece is not touching the orange piece and is touching the red piece.

[@edenli6421]

Give them a break

[@alan2here]

A bit of a hack but cut slots in the islands and tabs that jut out of some connecting bridge like pieces that with a bit for force slot in firmly at 90 degrees, you'll find wood/perspex has a bit of give. Make sure the slots and tabs account for the width of the laser beam (or drill bit).

[@dhoyt902]

Brady Haran - the depth and breadth of your collection makes us all feel like Newton standing at the shores of Truth. You are the crown jewel of online inquisitiveness.

[@llegaremosalasestrellas3245]

I love your videos. They always show curious and interesting math problems. Today topic was really cool.

[@grivar]

8:05 "Here in the Soviet Union" You've invented time travel and haven't made a video about it yet???

[@Cashman9111]

he made it in an alternate universe

[@dreska255]

He did, in the future.

[@DackxJaniels]

At the time of the event it was the USSR

[@shruggzdastr8-facedclown]

Did we just have a Mandella Effect moment in this here video?!?!

[@blindleader42]

Nothing to see here. Move on. Actually, everything to SEE here. There's no mistake. It's a video. The image is integral to the statement.

[@111giodude]

Did Joan Taylor spend a summer in Santorini,Greece and get inspired? The example of a non-connected tile really reminded me of the island's shape!

[@nickkei2838]

Socolar-Taylor tiling is what Parker square tried to achieve. It went outside of the rule book, some people may call it cheating, but succeeded.

[@WilliamLeeSims]

Those animations are so on point today. Somebody spent a lot of time on those!

[@enderwiggins8248]

What a fun new puzzle to do during English class

[@jtknight4647]

I’m a contractor and found this very intriguing. Makes me think of all the Ogee patterns; are these shapes also considered Heech numbers? Also puzzles must also be part of this classification of shape/numbers?

[@cainmartin4131]

After watching this video, I managed to design my own tile with a Heesch Number of 1. It's based on a tetris piece with some semi circular tabs. It makes me think it must be easy to design new ones which work for the lower existing Heesch Numbers.

[@edmundharriss2288]

It is a lot of fun to play with. Casey Mann's research looked as polyominoes as well as these polyhexes.

[@glutinousmaximus]

Euler "Angels and Devils" and Penrose both did a lot of pleasing work in this area.

[@ln14517]

These videos are always so fascinating

[@criskity]

Heesch 6 was recently found.

[@rens1030]

Thank you and kind regards from Heesch, the Netherlands

[@andrewkiminhwan]

Tiling is my favorite subfield of mathematics, i used to be absolutely obsessed with patterns, islamic tiling, etc.

[@DeadCityJxro]

My only complaint is that these video's are too far between each other!? Did I say that right? My favorite channel!!

[@themobiusfunction]

NOTE : A shape with heesch number 6 was discovered in 2020.

[@tutoringwestern4970]

I like the numberphile videos that has interesting conjectures.

[@anselmschueler]

*have

[@borismize8739]

I am a simple man. I get a Numberphile new video notification, I click it.

[@pawebielinski4903]

6 has been found back in 2020!

[@Thomasikzelf]

Someone just discovered an aperiodic monotile, see: An aperiodic monotile David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, 2023. Note that this tile must still be flipped for it to work

[@deboogs]

Great episode. I have no idea how you could go about proving something like this.

[@ferretyluv]

According to David Smith, you just doodle shapes, cut them out with paper, and then just put them together.

[@user-xd1wl2cq5v]

that's a lot of tiles yeesch

[@diabl2master]

*heesch

[@alephnull4044]

@@diabl2master r/weesch

[@diabl2master]

@@alephnull4044 I got it

[@alephnull4044]

@@diabl2master but then why spell out the joke

[@DeadCityJxro]

This video is amazing after smoking HashHeesch? Sorry..

[@happy_labs]

Awesome animations! Must have been tricky to get some of these right.

[@Ometecuhtli]

And the sounds! Give me more of those sounds!

[@raymondstheawesome]

obligatory "parker tile" joke for the cheating solution to the ein stein problem

[@zimi5881]

I wonder how far we can get with this Parker square thing. Maybe in 200 years it will be legit idiom.

[@whatisthis2809]

@@zimi5881 what does idiom mean?

[@fortidogi8620]

@@whatisthis2809 a phrase or saying that has such a widespread meaning that you can use it and your audience will understand what you mean intuitively. for example, "needle in a haystack" for a difficult task.

[@whatisthis2809]

@@fortidogi8620 Oh, okay thank you! don't remember ever posting this but hey, i know what it means now lol

[@Visocacas]

I was expecting this video to blow my mind with Heesch numbers for tessellating 3D solids, or n-dimensional solids. Still super cool!

[@eryqeryq]

Update: the Einstein problem has just been solved for a connected tile! :)

[@FelipeHoenen]

time to do a follow-up video about the hat, the turtle and the chiral aperiodic tiles!

[@rickyardo2944]

Be nice to see an episode on the the recent "Smith’s hat and turtle" tile

[@BenjaminAster]

Just wait a few more weeks, they will upload a video about that!

[@dimkadimon]

A tile with Heesch number 6 has been found recently.

[@IllPropaganda]

I always garner insight watching your uploads!

[@AnthonyDavid59]

Great video to follow. My 4 year-old grandson chose to watch this with me as he liked the colours. When the question came up about the Heesch number of a circle he called it. (He might have just called the shape zero :D )

[@SendyTheEndless]

3:50 I know this is 4 years too late but yellow is touching red at the top there! Was that a mistake? I imagine if you turned the touching red tile green, and then slotted a red one into that gap above it so it pertrudes outwards, that was the intended solution. EDIT: Also, you can probably guess why this video is trending again : )

[@eac-ox2ly]

Damn, these patterns are so fascinating

[@TruthNerds]

I should be commenting on the amazing talk and outstanding animations. Yet here I am, commenting on your little piece of advertisement: I am not easily scared, but when I first heard the story of the Dyatlov pass incident - I left the light on that night…

[@alan2here]

I've got this game on IOS, it's good. I think there mostly/all endlessly heesh though and tile like the rectangle. The goal is to use the fewest in the first two rings.

[@Axacqk]

Can this be used to design dendrimer molecules that won't grow indefinitely?

[@user-qf6yt3id3w]

I wonder if something similar could be used to design custom virus capsids. Only issue is that Heesch numbers describe 2D tilings and dendrimers and capsids are 3D.

[@karl-leopoldkontrus6544]

I personally believe, that you will find some Clusters in Nature with higher heesh numbers (some konfigurations of SiO2 seem to have heesh numbers over 1000 or so)

[@user-qf6yt3id3w]

@@karl-leopoldkontrus6544 Interesting. I'll look up SiO2 and Heesch numbers. I was actually thinking of proteins - I bet you'd find them with any Heesch number you wanted.

[@karl-leopoldkontrus6544]

Proteins would be very interesting, there are surely "numbers" which are similar to heesh numbers, but they have to be adapted somehow... a differnt repelling force for N - C and for C - C or hydrogen bonds , i think that some pc programs which try to predict forms of preoteins, use similar techniques for stacking like in the video (sry i am not a native speaker)

[@4ltrz555]

What about 3D tiles?

[@fittony]

cube should be infinite, but yes irregular 3D would be interesting.

[@DeadCityJxro]

Bruh! 4D!?!?!?!

[@axelasdf]

Scutoids

[@Soken50]

let's find a general rule to find 2D tiles for each Heesch (try and say that outloud a few times) number and worry about higher dimensions later yes ?

[@anononomous]

Can: Worms 🙂

[@roccoleader279]

I think there is heesh number 6, i read it somewhere and am in a discord server about finding heesh numbers

[@PINKALIMBA]

Aperiodic monotile / Einstein tile is found! And there are 2 shapes - the hat & the turtle.

[@johnkobilarcik4449]

Personally, I think that last result is genius. I love it.

[@dahemac]

In Adobe Illustrator the discontiguous (unconnected) shape would be called “compound.” Same as a doughnut with a hole except the hole does not overlap the other part.

[@charlottedarroch]

Does taking pieces of the Penrose tiling give larger Heesch numbers? In particular, if you take several tiles of the Penrose tiling which form a connected piece and call that piece your single tile, do you get very far with that? There are obviously infinitely many such pieces to be chosen, so it's my intuition that some of them would be relatively good tiles.

[@tghuffine6277]

Paused @ 4:28 to say that this shape reminds me of that British Man who says that all numbers are shapes and colors; and quoted Pi to the 10,000th decimal place.

[@Axman6]

Yeah if you could make those tilings as t-shirts, that'd be great - the tiling with Heesch number of 5 looks fantastic.

[@pokestep]

This guy is adorable, more of him please

[@kmktruthserum9328]

its funny that a lot of these videos you do were just things i used to do in grade school for fun when a teach was lecturing

[@KirbyTheKirb]

I´m really curious about how they found these tiles. Especially the tile with the Heesch number 5.

[@altejoh]

Parker square tiling. Kind of a cheat, but close enough to an answer.

[@FredRandall01]

They gave it a go.

[@petercarioscia9189]

MC Escher and and David Fathauer have been making these tessilations for decades. It makes some fantastic artwork, and it's easy to make a basic tessilating tile shape with a simple visual formula...it might not be a masterpiece like Eschers work, but it will work. Take a shape, like a rectangle, a take a semicircle notch out of one side, the repeat that shape with a mirror transformation on the opposite side. So the semicircle notch on the top rightside, would be a semicircle bulge bottom leftside. Then make a triangle notch on the top line, and repeat as a triangle bulge on the bottom line. Simple tessilating tile. Edit: use more complex shapes and transformations and you can make a tessilating swan or turtle shape.

[@KenWangpiano]

Dude I just saw you on some of lowko's videos...

[@KatarzynaMatylla]

Do a part 2 about the einstein hat! :)

[@StonedWidowOnDoom]

6:24 No wonder a person from Tasmania comes up with this solution. Sticking to a tile you are separated from but can not go without it anywhere. I like :D

[@dropkickedmurphy6463]

Since this video was made, there is a new WR holder of 6. The tiling was made by Bojan Bašić.

[@swfreak258]

Update: There now is a tile with Heesch number 6

[@InvntdXNEWROMAN]

Killer animation for this video.

[@ThAlEdison]

Re: laser cutting. It depends on how many tiles overlap the same point. If you can find a set of tiles that are analogous to the single tile then you could theoretically make 3D models where the discontiguous pieces are held by contiguous tiles used as a base. If the base is clear, you should be able to see the aperiodic tiling.

[@edmundharriss2288]

Yes there are ways to make them. Though making the bases of neighbouring tiles not intersect might be tricky. Also this pulls the problem into 3d (and it is a 2d tile). These are as much aesthetic concerns as anything else.

[@ThAlEdison]

@@edmundharriss2288 I can think of ways of dealing with intersection, but it becomes trickier and less nice looking the more tiles cover the same spot. And yes it's 100% exploiting 3D to cheat the tiles into existence. And it would likely require more than one type of cut, so while it's a single tile in a specific 2D slice that we're trying to visualize, it'll likely require multiple 3D models. So more cheating.

[@Zarunias]

Do you have to cover the diagonals (is it sufficient that the edges are completely covered or do you need to completely cover the vertexes too)? In your square example it seems that you don't have to, but in the example for Heesch number 1 you covered it.

[@oegunal]

Intuitively, the square "covers the vertices" if you consider that there are eight squares around the center square. The diagonally placed four squares cover the vertices. I'm just not sure what vertex would be considered formally "covered" or "not covered" in this case.

[@BobStein]

Very observant. At 3:05 the tile on the left covers a vertex that it might not otherwise have to for the 1st layer.

[@BobStein]

@@oegunal You might say (formally) that a vertex is not covered if there are any outside points infinitesimally close to it.

[@diabl2master]

@@BobStein In the squares case *and* the teardrops case, there are outside points infinitesimally close to "vertices"

[@oegunal]

@@BobStein yes, this seems a reasonable definition. The way I was thinking of it was: if you can draw a line segment of any non-zero length anchored at a vertex (or any point on the circumference) that does not intersect with any outer tile, then the point is not covered. Which is basically the same thing.

[@m1m1snake]

Damn, that Heesch number 5 is amazing, makes for some nice art.

[@Fatabuna]

Heesch number for a square: infinity Heesch number for a square according to my local tiler: about two

[@qcard76]

On the tail end of an all night studying session for Neuroanatomy... much needed break for the brain!

[@benjaminnewlon7865]

Which is better, the clicks of the tiles in this video, or 3b1b's clacks in his sliding blocks video?

[@cogoly]

I think there is an error in the animation for the Heesch number 5. In the top part a red tile touches a yellow tile. Shouldn't the red one be green?

[@pietervannes4476]

the yellow one should be green

[@rednecktash]

7:32 it's just a hexagon with pieces removed and nested inside the nearby hexagons in a perfectly symmetric hexagon tiling

[@jay-tbl]

How did the existing tiles get discovered?