How a Hobbyist Solved a 50-Year-Old Math Problem (Einstein Tile)

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Sources and further reading
arxiv.org/abs/2303.10798
arxiv.org/abs/2305.17743
cs.uwaterloo.ca/~csk/hat/
cs.uwaterloo.ca/~csk/spectre/
Chapters
0:00 - 0:49 Introducing a NEW SHAPE
0:49 - 2:20 Never repeating pattern
2:20 - 3:16 The 50 year old mystery
3:16 - 5:03 An amazing discovery
5:03 - 10:10 How do we know it never repeats?
10:10 - 11:40 Infinitely many ein stein tiles!
11:40 - 12:39 "Haters gonna hate"
12:39 - 15:01 An indisputable ein stein tile
15:01 - 16:04 Applications
16:04 - 17:59 Learn more about tilings
Creator - Jade Tan-Holmes
Animations by Tom Groenestyn
Music - epidemicsound.com

[@upandatom]

Sign up to Brilliant to receive a 30-day free trial + a 20% discount with this link! brilliant.org/upandatom/ Learn more about tilings here: brilliant.org/courses/beautiful-geometric-explorations/

[@curtisblake261]

I'm a member of Brilliant, but I signed up through a different KZbin channel. Am I supposed to sign up to Brilliant through multiple KZbin channels?

[@eternaldoorman5228]

There is a great book which was out of print for a while, but which seems to be available as a Dover book, It's Called "Tilings and Patterns" by Grunbaum and Shephard.

[@Amethyst_Friend]

The ideas in this are so beautiful. Especially the infinite fractal nature of the tilings. Thank you

[@Blackmark52]

You may want to rework your headline. I thought I was going to be watching a rerun, not a sequel.

[@fredashay]

6EQUJ5!!! This is fascinating!

[@C-Llama]

That is not the guy I was expecting to see singlehandedly solve a 50 year-old math problem. What a legend

[@YevhenSavchuk]

And then he found another tile

[@togfanatic3781]

when you write this comment he found another tile 😂😂

[@TheTruthKiwi]

Like, totally far out man✌️

[@StefanSteinerWA]

⁠​⁠@@YevhenSavchukand don’t forget the penultimate ein stein tile, the tile that is aperiodic without using its reflection.

[@yorthundir7343]

Given who solved the Poincaré conjecture, Dave seems to me like exactly the sort of person who would solve this kind of problem. Sometimes you need a somewhat eccentric person with a lot of time, passion and little to no connection to academia to come up with something novel.

[@binbadende]

As Dave got his fair share of praise in the comments and the real world, I like to thank and praise the "trained" scientists for supporting Daves' finding and setting his name even first in the publication. I like it!

[@hb1338]

They are mathematicians, not scientists.

[@binbadende]

​@@hb1338 whatever floats your boat mate.

[@derfzgrld]

@@hb1338 What do you think math is?

[@binbadende]

@@falxnecis are you just rephrasing my post or do you want to add something?

[@Jorge-xf9gs]

​@@binbadendeno

[@lucbloom]

The way he casually proceeds to discover more versions makes it apparent this discovery was not a fluke but a stroke of genius! So weird that a cut-up multi-hexagon didn’t prop up in an exhaustive search of some graduate’s geometric sim or something.

[@jayssoneer]

It might have been interesting to hear more about Dave's process of discovering this

[@edwardk3]

Ok. Your willingness to apply the word genius to somebody who is male is proof of where you are coming from

[@vaiapatta8313]

​@@edwardk3 unsure where *you* are coming from... OP isn't comparing men to women, he didn't claim women can't be geniuses. Emmy Noeter, for instance, is an undisputed legend in Mathematics.

[@tmac2744]

@@edwardk3 You jumping right into a feminist rant without regard for what was actually said shows where you are coming from. It isn't a good place either.

[@hashisgod]

@@edwardk3 he's right, a woman would never have figured this out as they very rarely ever do tiling

[@tineboes2726]

Can you imagine those mathmaticians while working on the first proof? “Alright guys, this is big. We’ve possibly got an Ein Stein tile, we need to spend all our time with this. This is a mathematic breakthrou-“ *phone rings* “Dave? What is it? We’re working on proving your tile!” “I’ve found another one.” … … … “What?” “I’ve found another one.” Dave is a goddamn legend!

[@607]

I haven't read the paper yet, but I'm honestly a bit confused why the spectre tile was not found by the people working on the paper.

[@tmac2744]

@@607 Something that academia is often loathe to talk about is falling into rigid thinking. They can often have the answers right in front of them, but because of built in biases due to education/training, they fail to see them until something or someone brings them to their attention. They get trapped in a "This is the way!" thinking process until jarred out of it. Another way to think about it that they tend to spend most of their time working on more practical shapes and applications. Ask them to design new wing shapes for a heavy hauler plane, a hyper fast supersonic plane, and a high efficiency wing for a solar powered plane for a world record attempt, and they can probably get you proto-type designs in a few hours. When you ask them for a non-standard shape that is not in any currently applicable endeavor, they have to first break out of the methods of thinking that make creating the wing designs easy before they can see those new shapes. Is this true of EVERY mathematician/academic? Absolutely not. However, the ones that can think in both manners easily are the minority. Additionally, sometimes they can be attacked for advancing an idea that is outside the norm because it can change how other areas have to be contextualized. This can make some academics mad, especially those that have built careers bolstering the current paradigm.

[@truthseeker7815]

@@tmac2744, it's sad how scientists can be that without an actual scientific mindset 😔

[@kales901]

"Oh, hey, i found out the new tile he just made is related to it, and that there's actualy an infinite family of tiles!"

[@truthseeker308]

@@tmac2744 Your anti-academia bias is showing. The simplest answer is that the first tile was emailed to the academics on November 20, 2022. The second tile was emailed on December 6, 2022. The academics spent that time analyzing the first tile, to confirm mathematically that it was aperiodic. They were looking at the 'how', not the 'what'. That's quite reasonable, as that is their expertise, not Dave's. Dave's lack of formal training precluded him from working on the 'how', so he simply continued looking at the 'what'. And as a result, he found another shape quickly, before the academics had finished analyzing the first. He was running down the road while they were checking the road for its structural soundness. Both are laudable pursuits. Now as to the spectre, the third shape(the proto-spectre), Dave only noticed it based off of Joseph's model that morphed between the hat and the turtle. It was the academic expression of the relationship between the two discovered shapes that allowed Dave to discover the third shape. However, it still wasn't perfectly aperoidic either. But the academics now could apply their training to adjust the sides with the 'wiggles', which is a lot more complicated than was laid out here. While you're pointing out the 'oh the academics are too rigid, they don't have vision', you miss that the non-academic Dave used his unique ability to 'see' shapes to give clues to the academics to then discover more, and their ability to represent the new discoveries both mathematically and to morph those shapes with computer rendering then allowed Dave to think up a new shape, which they then modified, to create a final, perfect aperiodic shape. There were two halves to this equation: The intuitive driving the curiosity and discovery, and the logical driving the proof of what is there, and demonstration of the limitations of it as well.

[@daniellemurnett2534]

Genuine props to Dave for discovering a full hat trick of advancements in aperiodic tilings.

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@CLipka2373]

I see what you did there ;)

[@daniellemurnett2534]

@CLipka2373 I'm gonna be real with you the pun was unintended.

[@RC_Engineering]

He is incredible! No matter how much people complained, he just busted out an even better solution until it was perfect. Even with all eyes on it, he still found improvements before anyone else! Incredible

[@hb1338]

@@VeganSemihCyprus33 A large collection of uncorroborated assertions, made worse by the use of music where none should be present. Good luck finding subscriber #582.

[@jean-philippegrenier120]

it's amazing that Dave got credit, so many academics would have stolen/undermined his contribution. Bravo!

[@flashwashington2735]

Team work. David submitted. The others performed confirmation, Hat. Back to David non mirroring, 2nd iteration, Turtle. Third, weekly periodic. Wiggle sides, Specter. Don't forget the whiny moaners. I don't what's behind Dave's skills, but he spent years honing it in play. Much like mathematicians, physicists, and chemists of old. I trust that computers haven't completely robbed us of more to come. Thanks to team Up and Atom for bringing us.

[@Lu-db1uf]

Really? I can't think of any academic that would have stolen or undermined his contribution. What a strange thing for you to say.

[@plienair]

@@Lu-db1uf Maybe that Harvard prof who falsified data in their research on dishonesty. Not everyone in academia is humble and honest, although for the most part, they probably are.

[@tapwater424]

​@@Lu-db1uf professors getting majority of credits for the sole of a student happens all the time

[@giorgosgalanos9888]

@@Lu-db1uf How naive...

[@prydin]

I love stories of people with weird abilities. Without any formal training, this guy seems to have brain that’s somehow wired to find aperiodic tiles. It just goes to show how amazing the human race is!

[@codytownsend3259]

I mean. Ironically the more formal education you have the more unlikely you are to find NEW solutions or ideas

[@607]

I disagree. He just took the time to look into it, because he enjoyed it. If I understand correctly, he's been working with tilings for fun for many years, so it makes sense that he got a feeling for them. :)

[@mathmeetsmachines]

I could not agree more. There are many problems in math where most mathematicians (most probably rightly) think that only highly trained professional mathematicians will be able to make any significant contribution (e.g. Collatz). Here, this was/is different and this is makes the whole story very sympathetic. There was a lot of training and, as you call it, weird abilities but not from somewhere mathematicians would have expected.

[@jamesdakrn]

Dude has some great intuitions for shapes - so cool that people have some weird abilities that they're maxed out in lol

[@andoletube]

I love how Dave didn't just strike gold once, but three times and counting. It points to a way of thinking that isn't available to most mathematicians - and it seems to come relatively easily to Dave.

[@SeaShrimp]

No its called having too much time on his hands, mathematicians dont spend their free time (and definitely not the job time they get paid for) to look for solutions to useless meme shit like this....

[@andoletube]

@@SeaShrimp You're a bitter soul. Here's a guy helping mathematicians to further their understanding of complex geometry, and all you've got is resentment and insults. You're a sad case - but I suspect you already know that.

[@saltyda]

everything is useless until its not@@SeaShrimp

[@vaiapatta8313]

@@SeaShrimp the video itself points to practical applications in Materials Science.

[@LeesCreation]

@@SeaShrimp Literally just watch the video. It talks about mathematicians spending their free time looking for solutions to this problem This is not very sigma of you Shrimpy Simpy.

[@Bafflementation]

Dave's achievements make me think of Escher, another person who approached quite mathematical geometric concepts from an artistic standpoint.

[@leonais1]

Escher inspired Penrose who found the solution for two tiles.

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@barmacidic2257]

@@VeganSemihCyprus33bot

[@050138]

Escher is love.... Such a great artist!

[@SuperLifestream]

Hopefully Dave's achievements are lost through time. And he gets future credit for this discovery

[@maxday4991]

Dave seems like the epitome of the humble genius. A great video as always!

[@ohnonomorenames]

👍

[@fcsuper]

I don't know if humble is the right word for Dave. His responses would seem to come straight from an egoistical nature. He didn't come up with the final solution because he was humble about the first solution being wrong. He came up with the final solution out of spite for the first solution being called out as wrong.

[@ohnonomorenames]

So you're saying he is a lazy genius. He can solve problems that people have struggled with for years but only with the right motivation. Should we set up a prize for the person that can come up with the insult that can get him to publish a P vs NP proof?

[@maxday4991]

@@ohnonomorenames From Collins: Humble - "A humble person is not proud and does not believe that they are better than other people." Lazy - "If someone is lazy, they do not want to work or make any effort to do anything."

[@maxday4991]

​@@fcsuper You make an interesting point, I suppose I viewed it as his innate curiosity being presented with a new challenge, rather than as a direct response to criticism, though I'm sure any hate only boosted his determination.

[@RoyArrowood]

As a math enthusiasts in the middle of a home remodel, I think my tile guy is going to be extremely upset that I learned about this

[@leoisikdogan]

For those who wonder why the L tile does not qualify as an aperiodic shape: it is because you can tile it in a periodic way too. For example, you can put two rotated L shapes on top of each other to form a rectangle and tile it. The shape they propose cannot be tiled in any periodic way.

[@NiteSaiya]

Thank you! I spent the whole video wondering why we were shown an aperiodic pattern with a single tile but it didn't qualify as a monotile, and didn't have the mathematical literacy to look up the answer myself.

[@abcdefghilihgfedcba]

That’s literally explained in the video. Pay more attention next time maybe.

[@reinhardheinzwarfelr8215]

​@@abcdefghilihgfedcbatimestamp?

[@ryebread6790]

i know it's easy to put people down, but try not being cruel..."they explain that in the video, you missed it"@@abcdefghilihgfedcba

[@jhwblender]

Same here @NiteSaiya And thanks @leoisikdogan! Anyone have a timestamp for where this was explained in the video?

[@GinHindew110]

Dave is a case of someone reaching an abstract solution after a life of empirical work, actually pretty common for craftsmen, but he knew to contact the people who could distill it into formal knowledge A great example of reaching out to people with similar goals but different methods

[@BKNeifert]

Just to know. That's good enough for me. I don't need the flashing lights and neon signs. Fame is vanity, and so is fortune. To know... It doesn't matter if I get there first. Just simply to know. From there I can construct wisdom. And from wisdom I can shape those who come after me, so they can find it too. Life isn't a race. It's not a competition.

[@mcspud]

This is the best comment here.

[@darktimesatrockymountainhi4046]

I love your point here. Sometimes there are people who learn & understand something - in spite of not having “appropriate” training. Their thoughtfulness can overcome their lack of experience, making them useful collaborators. We would do well to nurture such people.

[@6ic6ic6ic]

​​@@BKNeifertKnowing is all well and good. But sharing opens windows of opportunity to people who will or can know more within different context. What might be mundane in your knowledge could very well be groundbreaking in others. Knowledge without sharing is always a waste of potential. It's not about fame. Other people put that on the knowledge and the person. Even if sought after or not. It's about outreach potential.

[@BKNeifert]

@@6ic6ic6ic When the LORD opens that door for me. One of my favorite stories, is Washington sitting in the Continental Congress. He's the only Founding Father with any experience as an officer, and he sits there, quietly, every day in his colors. He waits, waits, waits, and then is appointed his position by the congress unanimously. There's something shameful about pushing yourself and advancing your own goals. Rather, to humbly sit and wait for your peers to accept you, that's honorable.

[@puffpio]

One thing I love is that Dave is credited as lead author on the paper, even though the way the story played out he discovers the shape versus created the mathematical proof (eg inspiration was the lead over perspiration)

[@davidanderson2357]

I noticed that as well. Goes to show that not everyone who publishes a scientific paper is a raging egomaniac. I'm talking to YOU, Sheldon Cooper!

[@donaldkasper8346]

Dave made the discovery and the others confirmed it and ran with it. Others will always expand upon, generalize, and study the implications of a discovery. That takes using standard math theory, so anyone with that background can fill that role.

[@donaldkasper8346]

@@davidanderson2357 Dave contacted the others with his discovery which they confirmed with their mathematical proof, so they are not the lead authors by definition. Otherwise, the others had nothing.

[@joostdriesens3984]

@@donaldkasper8346I agree, even though the others might have invested more time building the proof, that assumption is actually irrelevant. And this might not even be true, perhaps Dave was only able to discover this, because he has been experimenting with shapes all his life. :-)

[@donaldkasper8346]

@@joostdriesens3984 Many people deal with shapes. He has a pattern recognition ability once he applied it to the geometry problem.

[@ZendorOne]

Damn the production work of this video was insane. I really enjoyed the visualisations a lot. Thank you for all that effort.

[@thegorn]

69 likes. Nice.

[@rainbowseal69]

As a 3D artist, this is a great move because texturing 3D models would be more non repetitive, but it still depends on its practicality

[@LonelyRacoon]

Yeah Imagine if a game developer uses this shape to tile the oceans in the game. We'll no longer have the repeating waves in the game oceans lol

[@MartinMizner]

The games ussually code the space into square tiles for simplicity. Computing the perfect allignment of pixels and shapes on the edges of tiles would make the game more laggy.

[@rainbowseal69]

@@LonelyRacoon perfect. Although since the tile rotates it’ll need to applied in sheer so that the rotation remains within a horizontal influence.

[@Dr.W.Krueger]

Not practical.

[@AlvaroALorite]

Shit I didn't think about that you are right!

[@bramverhees755]

This is so fascinating. I would never have thought that new mathematical discoveries could be made by mere mortals at this point. I was certain that would be a “a computer calculated the aperiodic monotile” video. Go Dave!

[@rosiefay7283]

The way I understand it, a computer was involved, but I have no reason to doubt that Dave wrote the code, so Dave deserves the credit. Dave's program 1) tried to prove that the tile couldn't tile, by trying all ways to put tiles together with no gaps or overlaps, the point being to prove that every way fails. But with this hat tile, the program managed to keep going successfully, surrounding the first tile with 16 rings of tiles. No non-tilable shape previously discovered can be surrounded by any more than 6 rings of other tiles. 2) tried to find a way to tile it regularly, by putting some tiles together to make a bigger block that can tile by just shifting. But with this hat tile, the program tried all ways to put up to 64 tiles together, with no success. The previous record for a tilable shape is I think one that needs a 20-tile block. So all credit to Dave for finding that this shape is interesting.

[@jaapsch2]

Actually, Dave was using my software to explore his ideas. My Polyform Puzzle Solver is for designing and solving packing puzzles. It is a pretty good program, but not at all suitable for the things he wanted to do, which is why he got Craig involved whose software was. I got thanked in the paper, which was a big surprise as I was not aware of their research until the paper was made public.

[@racpa5]

Dave is not a mere mortal; he is a supremely gifted mortal. Yay Dave.

[@bramverhees755]

@@jaapsch2 That is very cool! A collaboration between talented programmers and insightful individuals, then. I can only ever hope to be the former, but I fear I’m neither 😅

[@jootpepet]

He could be a savant for all we know, he's not just a mere mortal thats for sure

[@OngoGablogian185]

Dave clearly always dressed like a legend, but now he literally is one. Very cool stuff.

[@milire2668]

or some sort of rockstar.. .. or hobo, jk :D

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@vaiapatta8313]

There are some areas of STEM where you can contribute even without a formal background; where curiosity, lots of experimentation, an encyclopedic knowledge of the subject and a keen eye can carry you pretty far. My warm congratulations to Dave for spotting not one, not two, but three solutions to a half-century-old problem. Not that the three other collaborators don't also deserve credit for the rigourous proof, an essential part of the full solution. But this is the type of problem where coming up with a good conjecture is the most difficult part, not proving it. It was mainly Dave's insight and observation (and hard work, presumably) that cracked this one. So yeah... congrats to all, but Dave's the man! :P

[@be.A.b]

It’s creativity through and through. Lots of people consider it the antithesis to formal science/math. He posited the solution before solving the problem.

[@edwardk3]

It would be better if it was a woman though. For equality

[@vaiapatta8313]

@@edwardk3 look up Marjorie Rice for a female example.

[@breni1518]

And I think it would be better if it was a child because younger people need recognition too

[@edwardk3]

@@breni1518 yeah but woman are oppressed by men who solve before they get a chance

[@reddmst]

This is easily the best exposition of the hierarchy argument. I've seen several videos on the aperiodic monotile but you're the first one to really help me grasp that part. Thank you!

[@rescuearch7802]

The amazing thing is that David Smith was not the first amateur mathematician to make a series of breakthroughs in tiling theory. In the 1970s, amateur mathematician Marjorie Rice discovered four different, previously unknown pentagonal tilings on the plane (among many other findings involving tilings). In 1999, one of those pentagonal tilings was made into actual floor tiles and used to tile the foyer of the headquarters of the Mathematics Association of America, in Washington DC.

[@jamescheddar4896]

i think math is mind numbingly simple in a way so you need to be less immersed to see something new

[@davidanderson2357]

After seeing this video, I do believe that foyer is due for a re-tiling.

[@itzakehrenberg3449]

I disagree altogether. @@jamescheddar4896

[@itzakehrenberg3449]

"Moneyball" doesn't tell the story of Bill James, the man you are thinking of. The movie tells the story of how the Oakland A's achieved great success with his ideas though. @@beaviselpastafarian5039

[@WobblesandBean]

​@@davidanderson2357 No, because that would diminish her work and her accomplishments.

[@GianmarioScotti]

The truth is, Dave is actually a genius, even though he has not the same mathematical training as his co-authors.

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@xxxkueckxxx]

He’s a computer science professor I think?

[@RC_Engineering]

For sure!

[@TymexComputing]

He can imagine things - its very important - the One-Stein thought experiment :) - btw he also looks as if he was skateboarding, driving hoover and smoking his lungs for the last 40 years :)

[@hamishadamson4628]

​@@xxxkueckxxx"a retired printing technician" - She told us that in the video.

[@HermannKerr]

Dave has an amazing visual way of thinking. It is a gift and it is a gift that is very useful in many fields not just in mathematics and bathroom floor tiling.

[@Rexvideowow]

Massive credit to that Dave guy. I'm really glad this video touched on the fact that the scientific community will grab hold of this, because that was my very first thought. This is only the beginning.

[@francogonz]

Damn, i swear that Dave has a nice mathematician mind! Even if he only considers himself as an artist. Well done Dave, you just got immortal in maths ❤

[@nielskorpel8860]

Did Dave invent/construct all these shapes? The video seems to suggest so. Is that true

[@potatopobobot4231]

@@nielskorpel8860 no its not true

[@JellyMonster1]

@@nielskorpel8860 Half true. I didn't invent them but I discovered the hat and turtle after constructing them. I also recognised that the polygon in the middle of Joseph's evolutionary timeline could possibly tile without mirroring.

[@g.tucker8682]

​@@JellyMonster1You're the man!

[@stapler942]

​@@nielskorpel8860 This ties into (or tiles into) the age-old philosophical question about whether mathematics is discovered or invented. It's entirely plausible that such shapes have been used at some point for some purpose in our vast history, but the aperiodic tiling property was apparently overlooked until now. 😉

[@thegzak]

Let’s not overlook the phenomenal job Jade has done here distilling the essence of the proof for a broad audience - awesome video, as usual!!

[@upandatom]

thank you so much! :)

[@pingnick]

@@upandatom ​​⁠​⁠amazing interview with the einstein guy with no formal training however defined wow wow wow wild stuff who knows how people get inspired by other people considered living or dead even wow wow wow wow yeah the true einstein tile however defined now found indeed wow wow wow🧩♾♾♾♾♾☮️💟😍😘🥰🌈🗽🤯🤯🤯🤩🤩🤩🎬🎬🎬🤖🤖🤖🥳🥳🥳…

[@vipulpoddar8045]

​​@@upandatomis L tile periodic? I didn't get that part

[@markinipannini]

​​@@vipulpoddar8045 1:31

[@pingnick]

@@vipulpoddar8045L is both if I understood correctly as is one of the partial einstein tiles BEFORE it is squiggled to get highly technical!🤯

[@jessicahayman3729]

Great video! As a high school maths teacher, this is so inspiring to me. I will definitely be showing my students... Dave reminds me of all the brilliant, self sufficient hippies I know aren't willing to be cogs in the system. What an absolute legend

[@jabulaniharvey]

Major props to Dave for scoring a trove of examples of this geometric phenomenon, i wonder if this is related to the theory of six-fold symmetry in planar geometry

[@andrewkamoha4666]

15:01 "a lot of you wouldn't be happy unless I talked about possible *practical applications* " Amazing! She knows us very well !!!

[@antonnym214]

Thank you for reporting on this. Kudos to David Smith on never giving up! When people had complaints about one tile, he came up with more shapes until he shut them up for good. Way to go!

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@hb1338]

@@VeganSemihCyprus33 Trolling will not improve your chances of success.

[@Axodus]

He'll be remembered in the math world for generations for this, yet he's so humble, amazing.

[@hb1338]

Really ? He has produced an elegant solution to one of mathematics' more minor problems; only a very few text books will need to be updated. He will soon be forgotten, like so many other contributors to our knowledge and understanding.

[@algorithmgeneratedanimegir1286]

@@hb1338 Hell yeah, forget the bitch this ain't that interesting !

[@Axodus]

@@hb1338 His contributions will be remembered, and he'll live on through that.

[@ft.nufonia]

@@Axodus whos?

[@Axodus]

@@ft.nufonia David Smith's.

[@blutey]

It sometimes happens that someone with an unconventional background like Dave (a 'hobbyist') entering an established field of study for the first time, makes discoveries the established experts can't because that new person has a different way of looking at things, the experts with their conventional training and ways of looking at things aren't doing.

[@swagatpanda23]

Dave is not just a bloke off the street, he's an absolute legend. Great video Jade!

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@pwjaiter6277]

I’m starting my computer science bachelor tomorrow, it’s good to know that I can always watch this channel if I don’t understand a lecture😁

[@upandatom]

Good luck with CS!

[@Simulera]

There are all kinds of reasons that this tiling, even its existence, may be important to many areas of basic mathematics as well as applied mathematics. Its super that you found and decided to “cover” the aperiodic tile story. Please continue checking in on its progress! But I also want to just say what a wonderful collection of inspired and useful presentations all of your videos, both alone and taken together, create. You are very gifted at doing this important work of harnessing an often misused medium for good purposes. I am a scientist, old but still at work, and often watch your videos for amusement. But I also like to see how you explain certain things for importantly non recreational purposes - and try to understand how you might have come to the way you do it. Thanks for such good work and your channel.

[@mikew8100]

This story proves how modern 'academia', while amazing, can also act as a cage and box where confined and incomplete thinking exists. Dave was not in that cage. What a guy!

[@niezbo]

Dave is one of those guys who proves anyone can do something remarkable! And anyone can play with maths.

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@BillyViBritannia]

I'm pretty sure Dave is quite remarkable and his achievement proves nothing about "normal" people. The fact that he doesn't present himself in a certain expected manner doesn't mean he's less of a genius.

[@nobodyimportant7804]

He is one of those guys that has an exceptional natural pattern-matching ability. I wouldn't be surprised if he were on the high-functioning part of the spectrum. AKA not really a "normal" person. Recognizing what he had is the exceptional part. There are precious few people who can recognize a potentially great discovery that they stumble onto, trained or not. I bet more than one kindergartner has cut out that shape randomly.

[@baganatube]

Dave has some exceptional abilities! He should be studied and funded!

[@liyana2810]

The video mentions that Dave is a shape hobbyist. So does it means "shape" is his special interest? I wonder if he has autistic traits. Indeed, many autistic people are actually genius, it's just we couldn't see their genius side before. (Sorry if i'm wrong). I believe if autistic people are given chance to enhance their special interest, there will be something.. I mean, God didn't create anything without purpose. Maybe one of the purposes is this, or maybe not. Just my humble opinion.. I apologize if I'm wrong about Dave or anything in this comment. It's not my intention to hurt anybody.

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@nathanmiller9785]

I agree there are many autistic people who have great intellectual contributions to offer, but from my understanding, the “savant” thing is a bit of a myth/misnomer- there are savants certainly, but I don’t think the data supports the idea that the autistic population contains “more geniuses” than a the average population as a whole.

[@leodf1]

On the contrary, he will soon be forgotten. That gang of u no hoo already made themselves 'co-discoverers'. And soon his name will be gone completely. Aside from appropriating other's ideas, re-writing history is another of their skills

[@craigslist6988]

@@leodf1 isn't there a new flat earth proof video you're supposed to be watching instead of this?

[@jackjhmc820]

Dave can visualise putting tiles together in his mind . surely that kind of skill is extremely rare? 1 in 100,000?

[@nervsouly]

I'm dyscalculian. I'm unable to comprehend any form of maths that goes beyond simple counting and at the same time there is a high chance I will make lots of mistakes at addition or substraction already. So it's pretty safe to say maths has been the bane of my life. BUT I think it's really awesome how so many people spent so much time and effort into discovering a unique shape and share it with everyone.

[@Xubono]

This is inspiring stuff. I have a long background in Computer Science and Applied Mathematics, which sees my recreational focus gravitate towards patterns, symmetry, etc in Art, Logic, Music and Science. I’ve just recently (at age 62) gotten into drawing. I love transcribing renditions of geometric objects, whether they be simple, have rotational, reflectional or translational symmetry, and then to try creating tilings. This includes 2d drawings of “impossible 3d shapes” and sacred / Islamic geometry (shapes which are constructed using only a compass and a straight edge) and many other sources of inspiration. Watching the animations in this video gave me chills running up and down my spine in waves, exacerbated by the current ambient room temperature of 9°C at 1:30 am. [I should have something something warmer to wear!] So THANK YOU Up and Atom for your continuing high quality and interesting content. My salutations to Dave and his collaborators for original, imaginative and spontaneous creativity at the pinnacle of excellence. And an incredible effort by all of you in combining talents to produce this masterpiece. I really am impressed!!! Also I am really cold and should get under my blankets in bed!

[@050138]

There is a lot to learn from you, sir!

[@hb1338]

Suffering for your art is no longer considered to be essential !

[@neochaft1587]

One of your best video ever and that's something! The in-person interviews, the storytelling, the explanations, the animated diagrams! Everything drips passion and effort. Thank you!

[@VeganSemihCyprus33]

This is the story of your enslavement 👉The Connections (2021) [short documentary]💖

[@peffken8834]

Most of the time such things are beyond my understanding, logic and intelligence. But here the subject is so well conveyed that I not only got an idea or feeling for the problem and the solution, but also thought that there must consequently exist many more tiles before it was mentioned here. Great job, thank you!

[@katiehettinger7857]

Dave is an example of why curiosity is a do thing. And active mind pondering a variety of ideas is a happy mind.

[@guidoferri8683]

He sure has an exceptional visual thinking ability

[@jurjenbos228]

There have been many publications about this, but you managed to actually explain the proof. Great work!

[@kc5402]

That's one of the best hat-tricks I've ever seen.

[@Amadioh]

Inspirational honestly, we should all be so great as this shape to not have to repeat ourselves

[@gouravsalla2204]

A great video Jade. Nicely reduced the complexity of the proof so that everyone understands easily. And a nice chronology of explanation from aperiodic tiles 1964 to Penrose tiles to current hat turtle Daves's eye and the spectre. Keep doing more ..Thanks!

[@BrunoLiegibastonliegi]

On 16:09 you say "he contributed to a ground breaking discovery". Actually, he didn't contribute, he made the discovery. Let's give the genius all the credit he deserves, please. The other guys just proved he was right.

[@GrifoStelle]

With you. They proved him right. Lots of work, for sure but he discovered it on his own with the exception of Dave's (?) contribution that pointed out every iteration of the shape was aperiodic

[@hypersapien]

I appreciate the paradox of a solution evading us for decades, which actually had infinite solutions.

[@VicJang]

Fascinating story! Dave is probably a genius. I hope he also gets into other fields. Maybe he'll solve some long standing problem!

[@Artaxo]

I really loved this video! Not only did Jade given us a very simple explanation to what appears to be a complex proof, the storytelling was also great and made us all love Dave.

[@Stella-se1lg]

Thanks for covering the story!!! I was once into Math as a kid but not as much nowadays. I still love to learn about not only how the Einstein problem is solved but also collaboration across the world really speed things up🎉 Also really appreciate your enthusiastic style. Instant sub😊

[@Nooticus]

This is such an excellent video, thank you for finally allowing me to understand this as someone who isnt a mathematician! I still can’t believe this actually could exist!

[@punditgi]

Jade is back! Woo hoo! ❤🎉😊

[@fcox7015]

Ready to fight in the next mortal kombat Tan tan Tan tan tuta Tan tan Tan tan tatu Tan tan Tan tan tura Tan tantara tura rarara

[@SteveGouldinSpain]

I guess such an unpredictable pattern could be more resistant to leaks than a periodic tile, say like the heat-shields on a Starship. I wonder if it would work over a 3D surface.

[@julianemery718]

That's... That's actually a fair point. No way for the shape to slide itself and others around. Though saying that, bestagons are also pretty good at keeping shape.

[@connorallen162]

What a wonderful video! So cool that you got to meet everyone involved

[@Dalroc]

Saw this when you uploaded it but skipped it because I had already seen several videos about it. Popped up in recommended now a week later and boy did I miss out. This was insanely interesting with the interviews of the discoverers!

[@genentropy]

Fantastic video. Great job Jade. Dave is clearly an amazing human. Massive kudos to him, and for the academic mathematicians who helped him with the formal proof.

[@laartwork]

Yeah the co-discoverers kinda latched on for the credit.

[@MeriaDuck]

Love this story and nice follow up on the story. I gave a workshop about this at a mathematics camp for young students (12-14 years). Very inspiring topic to present. I let them play with penrose tiles, then the L shape and finally with the newly discovered tile. The last one I made two colors, so they would discover the need to put some upsidedown. Was great fun!

[@MeriaDuck]

A heart ❤️ I'm honored! ☺️

[@ChrisPapewonderfulchris]

Really brilliant! Thank you, Dave! Thank you Jade for the making it understandable! 👏🏼

[@dogscratchedoor]

Astonishing story! I confess it's a little over my head at present. Going to follow up and try to learn more. The material science applications are massively exciting!

[@steveswoodworking2504]

Congratulations Dave! You not only hit the jack pot once, you kept doing it over and over!

[@kennethtaylor3467]

that is so wild. go from none to Dave saying "hey maybe this hat?" to "oh by the way, found this turtle?" the moment she said there was a connection between the two figured it was going to prove more but didn't expect a continuum afterward. i would've been impressed with that until the mention of the reflection was different (thinking of the z and s and the L and J in Tetris). this wiggly thing (OMG!) totally impressed!

[@kimes1137]

Looked forward to this so long! Its amazing!

[@tcaDNAp]

This is my favorite of the many videos about tiling I've watched recently! Every animation was unique and useful, so I finally feel an intuition for the proof too 🎉

[@tcaDNAp]

The script touches on every little part of this journey, including the many celebratory art pieces! Only Jade can make such a perfect introduction and deep dive 👏

[@apefu]

How have I never heard of this? This is dragon curve levels of fun! Also, how does one become a "shape hobbyist"? I don't see it being a hobby of mine, but I am very curious what leads someone into this hobby and what tools one would use.

[@JellyMonster1]

I used the Polyform Puzzle Solver By Jaap Scherphuis for the most part.

[@jaapsch2]

@@JellyMonster1And I’m amazed it got you that far! My program is pretty good for creating puzzles, but since it needs a board shape to be defined, and other limitations, it is not very suitable for exploring tilings of the infinite plane.

[@jootpepet]

Probably grew from jigzaw puzzles.. not hard to imagine at all

[@JellyMonster1]

@@jaapsch2 Hi Jaap. Even though you have to define a board shape, your program can still provide hints of any interesting/non-interesting tiles and has other uses. Many thanks.

[@stapler942]

Might be considered a cousin of "recreational mathematics" I suppose?

[@cheetah219]

I love how Dave proved everyone wrong and did it twice. And those mathematicians gave him first author! For those who don't know, first author is something that some researchers spend their whole careers trying to get as it meant they either funded research, led it or played an instrumental role to the point the project would be impossible without them. It's a huge honor for many researchers, so seeing Dave given that...pretty amazing. Some of my friends have been doing research for 20 years and still haven't gotten those honors

[@JellyMonster1]

I would been quite happy being last on the list of authors but the rest of the team insisted that I appeared first. Thanks for your comment.

[@maalikserebryakov]

@@JellyMonster1 you should put in a formal complaint and demand to be put in last 👍

[@staninjapan07]

Superb. Liked. Subscribed. I will need to watch this one again, and probably twice, to fully appreciate all that's involved.

[@garyw.9628]

Wow ! Great story, and very well told. Kudos Up and Atom

[@Madash023]

Amazing video! I remember hearing the buzz and excitement about this with the first discovery, and super nice to hear the mathematical proof behind it laid out and explained so clearly. Which is a testament to your teaching ability, since proofs seem to always be the hardest thing for people to wrap their heads around. And the animations in this video are wonderful too. Great work!

[@seb-astian-design]

Isn’t it beautiful how art can go hand in hand with science? Imagine how much more potential there is when combining those two ways of seeing the world. ❤

[@TheloniousCube]

Math is not science - it's art

[@seb-astian-design]

@@TheloniousCube it is!!

[@pietpaulusma5146]

Wow really good video! Love the way you set it up

[@bramfran4326]

This video is well-organized and easy to follow. Thank you!!

[@andrewfischer-garbutt2867]

The fact that someone solved a problem like this with no formal mathematical training. Is really something

[@arthurvandervelde]

There are no doubt many problems out there unsolved by experts, which remain unsolved only because they have not yet been examined by a genius with the raw talent, the interest, and the determination to solve the problem.

[@georgeallen7487]

Not just that he found the shape but knew about it's relevant properties.

[@TheNefastor]

Perfect illustration of why we say math is the universal language. You don't even need to learn it to (re)discover it.

[@aaaab384]

@@georgeallen7487 its*

[@KravMagoo]

@@aaaab384 You could have also drawn attention to the random period in the op, right before a linking verb.

[@jenniferelkins]

First video of yours I'm seeing. Amazing! You have a new subscriber. ❤

[@notheories9012]

This was actually awesome! Can't believe Dave found one and then continued to discover more. This man is a shape genius!

[@danlhendl]

That puts a whole new slant on the primordial struggle for all Puzzle enthusiasts

[@paintballercali]

I hate when I lose my t-shirt for 50 years.

[@joshuabarbeau9159]

You are brilliant. It takes a smart person to understand complex ideas, it takes a genius to be able to explain those complex ideas in easy-to-understand terms. Thank you for your videos.

[@zqzj]

Dave is a hero! Proving that we don't always need math to discover the improbable.

[@BrianPeiris]

There was such a frenzy of news and videos during the first discoveries. It's helpful to watch an excellent summation of it all, now that the dust has settled. Thanks!

[@DeclanMBrennan]

I remember when Penrose tiles showed up on the cover of Scientific American in my school library and I had fun making kites and darts out of cornflakes boxes. And now I've lived long enough to see *two become one* .

[@klaasbil8459]

"a ginormous translation" - what a word! Kudos to you for just casually slipping it in

[@landspide]

I love how one of the most common names and surnames found something so uncommon. Not to mention a hobbiest!

[@joec-hd6dc]

Fascinating story of discovery! A problem that some of the greatest minds in math and physics could not solve but a bloke just playing around with shapes figured it out. Absolutely inspiring!!!!! Way to go Dave👍. Thanks for posting Jade your a Rock Star ⭐.

[@vivekdabholkar5965]

Dave is a true genius indeed! Long live Dave, we need minds like yours

[@n8-cre8s95]

That is so awesome! Great work

[@100vg]

Interesting how David named the Undisputable Einstein Tile "Spectre" and amazing that a non-mathematician came up with all of them just by working with geometrical shapes. How picky and petty that mirror images were balked at, but Dave did it anyway, giving them what they demanded. I don't think I'd want to tile any floors or walls with it, except for maybe a framed display commemoration for its creation and creator, but Dave solved a 50-year mathematical question which provides possibilities for science, engineering and technology. One possibility could be Run Flat Tires for vehicles. They exist, but this could make them better. Maybe a Graphene organization for cell phone touch screens. I enjoyed your video and the story of development. Thanks

[@msclrhd]

To expand on the idea at 8:55 of two Ls being part of only one larger L... If you take two adjacent small L tiles, for them to be a translation of each other, as you move up in size, that larger L must also be adjacent *in the same way*. In other words, as you increase both e.g. up and to the right, the result would be a larger L with the same adjacency property as the smaller L. For the L tiling, this works up to point, but then you hit an L in a different orientation (e.g. down and to the right), so you cannot form a larger L at that size. -- The L you are working with has shifted from being the inner part of a larger L construction to the outer part. As such, it cannot be adjacent to a different L, it is adjacent to the inner part of a larger L. Therefore, the tiling is aperiodic.

[@helldad4689]

I love how you can tell that tile guy's brain is wired in a *completely* different way from everyone else's. Like he is not "doing math" as anyone would understand the term but he's still running laps around people who made careers of "doing math" about this problem.

[@hb1338]

No. He took a geometrical but non-mathematical approach to the problem. When he found a solution he asked mathematicians for help with the maths. Dave did the visualisation, the mathematicians did the maths. All are worthy of credit for their contributions.

[@gunnermac3570]

Thank you for breaking this down into understandable segments.

[@acctsys]

A mini gallery of the shapes tiling an area, from penrose tiles, the hat, the extremes, and the spectre sounds good to me.

[@mctuble]

This guy just sits there for hours on end morphing shapes in his brain and understands shapes to such a degree. In a world of 3d shapes he said you guys just visit here... i live here.

[@jeffneuhaus8475]

This video made me feel smart because I think I actually understood most of it. Bravo!

[@technoneko8108]

I cant believe someone would name their kid "one stone" only for that kid to grow up to be one of the most renowned genius.

[@u.v.s.5583]

It is such a great success story. Pros sometimes lump all weird hobbyists into the crackpot category because there are so many of them (crackpots). Dave here is the real deal though. Kudos to him, and also the profs who took him seriously. And thanks for the great video!

[@remo27]

It helped that he seemed humble and yet educated (himself, probably self taught )enough to know when he : 1) Might have found something others would be interested in and 2) Knew what kind of expert to contact to verify his discovery. Since they collaborated at least two more times, I'd say Dave must be easy to get along with and those three Academics must not be the snooty kind or the greedy (taking all the credit for something he first discovered )type. To me, this shows the 'best' of science and mathematics where everyone does their duty and they work together to advance knowledge.

[@philshorten3221]

If someone has to be the last human to solve a maths problem before AI takes over, I hope its Dave!

[@BKNeifert]

AI will never solve math problems. It's limited to what exists. It can't think beyond. A thing like Chess, it can do. Solve PvNP, it can't.

[@hb1338]

@@BKNeifert Not only that, there are vast areas of mathematics where algorithmic proofs are not possible - that is a consequence of Gödel's work.