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Can you think of a shape that can completely cover a surface
but never in a predictable repeating pattern?
No one knew of such a shape existed until now.
Mathematicians have been searching
for this shape for over 50 years.
And in March of this year, it was finally found.
The excitement of this groundbreaking discovery
has rippled through the mathematics community.
A festival was even held in its honor.
I was lucky enough to be invited,
and now I'm sharing everything I learned with you.
I knew that there would be fascinating mathematics involved,
but what I didn't expect was that the story
of this discovery would be just
as fascinating as the discovery itself.
- It's a really fun, wild, and improbable story.
- Before we dive in, we need to understand
what a never repeating pattern actually is.
In mathematics, it refers
to what's known as a nonperiodic tiling.
Let's unpack that a bit.
A tiling is when you can take a shape
or set of shapes and arrange them
so that they cover a plane without any gaps.
Theoretically, you can extend this pattern
out to infinity in all directions.
Think of it like tiling an infinitely big bathroom floor.
A periodic tiling is when you can copy the entire pattern,
slide it across, and lay it perfectly onto itself.
This is called translational symmetry.
Periodic tilings are common.
And can be made of more than one shape.
Some shapes that make up periodic tilings
can also be arranged in ways
that do not have translational symmetry.
When we arrange this L-shaped tile like this,
it makes a periodic tiling.
But if we arrange it like this,
no matter how we slide it around,
we can't get it to overlap with itself.
Tilings like this are called nonperiodic.
And when we say never repeating pattern,
this is what we mean.
This got mathematicians thinking.
Are there shapes that can only be tiled non-periodically,
that no matter how you arrange them,
it's impossible to create a repeating pattern?
Such tiles are called aperiodic tiles,
and for a long time, no one knew if they existed.
Then in 1964, the first set
of aperiodic tiles was discovered.
- The original set of tiles that had this property,
there were 20,426 of them.
So different tiles.
And then rapidly, that set got cut down and down
until someone found a set of six tiles that could do it.
And then Roger Penrose's work brought it down to two tiles.
And then I think it was uncertain
as to whether it could go down to one.
- [Jade] Veritasium has an excellent video
about Penrose tiles.
I've linked it in the description.
Since the sixties, Penrose's Tiles have held the record
for the smallest number of aperiodic tiles.
- And then we hit the wall, right?
It was obvious that this,
the last step we wanted was to find a single shape
that can only tile aperiodically.
And for roughly 50 years, nobody was able to find one.
- And so the race for the single aperiodic monotile was on.
It was called the Ein Stein Tile,
meaning one stone in German.
And now, in March of this year,
it was discovered by a retired printing technician,
David Smith, who describes himself as a shape hobbyist.
- I'm just a bloke off the street, you know,
just playing with shapes.
- [Jade] David was playing
around with shapes on his computer
when he discovered a very simple shape
that behaved strangely.
It's called a polykite
because it's made up of eight kite shapes.
You can get it just by cutting up hexagons.
He printed out copies and began to tile them by hand.
The funny thing was he couldn't get them
to tile periodically.
I've never seen them.
That's weird, you know.
- [Jade] He felt he was onto something.
So on November 20th, 2022, he emailed Craig Kaplan,
a computer scientist from the University of Waterloo,
"Could this be an answer to the Einstein problem?
Now, wouldn't that be a thing?"
We now know that it was indeed a thing.
- He had reached out to me
because of this previous work I had done
that can use computer software
to compute Heesch numbers.
So it can try to surround and surround
and surround and see how far you can go.
And my software say you can go as far as you want.
As far as we could tell,
you never run out of the ability
to grow a patch of tiles.
That means there is some faint possibility that this is it,
that this is the aperiodic monotile people
have been looking for.
- While they were investigating its potential
as the Einstein Tile,
there was also the task of giving it a name.
On the 5th of December, Dave emailed Craig,
"Do you think it looks like a t-shirt, bird in flight,
some sort of boot, weird fish, or something else?"
Craig replied, "Family members think it looks like a shirt.
Maybe a hat if you turn it upside-down."
And so it was named the hat.
Now the money question.
How do we know that the pattern never repeats?
Sure, it doesn't look like it repeats
when you look at a segment.
But we're dealing with an infinite plane.
Maybe there is some ginormous translation
where the original tiling and its copy line up.
To be sure, we need a mathematical proof.
It was here that Dave and Craig called in reinforcements,
Chaim Goodman-Strauss,
an American mathematician working in geometry,
and Joseph Samuel Meyers,
a software engineer working in combinatorics.
They immediately jumped on board.
- I mean, of course, it's a big deal to me.
It answers a longstanding question to everybody
that is involved in this has wondered about.
- Just eight days after joining forces,
the team derived a mathematical proof,
sealing the hat's identity
as the long sought after Einstein Tile.
Diving into a full proof
would involve some heavy mathematics and computer science,
so I won't go into all the details here.
But I really want you to get a feel
for the genius behind proving the hat's aperiodicity.
So what I'll do instead is give you an overview
of the main technique that was used
and show how the team applied it to the hat tiling.
By the way, this is far from the end of the story.
There are more surprises to come.
Infinitely more actually.
Okay. So at the core of the proof is a classic technique
in tiling called the unique hierarchy method.
In a nutshell, this method relies
on the hierarchical or fractal nature of a tiling.
Take this nonperiodic L tiling from before.
It starts with a single L tile,
then makes copies of itself to form a larger L,
then makes copies of that larger L
to form an even larger L and so on
until it covers the infinite plane.
This periodic square tiling is built in the same way.
Start with a single square tile
and copy it to make larger squares.
But there's a big difference
between the L tiling and the square tiling.
Any L, no matter its size,
belongs to only one L of the next size up.
This L belongs to only this one bigger L.
And that L belongs to only this one bigger L.
Try it for yourself.
Starting with this L tile,
can you figure out which bigger L tile it belongs to
and which one that belongs to?
Now try it with the squares.
It doesn't work.
This square could belong to any one of these bigger squares.
And that square could belong
to any one of these bigger squares.
This is the essence of unique hierarchy.
Each smaller tile belongs to one
and only one unique larger tile.
And that guarantees non-periodicity.
How? Well, one way to think of a periodic tiling
is that you can copy and translate it
on top of itself and it looks unchanged.
So now imagine copying a segment of our L tiling
and shifting it on top of itself.
Right off the bat, we can see it doesn't align perfectly.
But even without visual cues,
we can prove that it's different to the original.
As we know, in the original,
each L belongs to only one larger L.
But in our shifted version,
this tile belongs to two larger Ls.
You might think, okay,
well, what if we just make the translation bigger
so that it doesn't belong to both Ls anymore?
But that doesn't fix the problem
because this larger L belongs to these two larger Ls.
Because of the uniquely hierarchical nature of the tiling,
no matter how big of a translation you make,
there will always be a larger L
that belongs to two even larger Ls.
This means that it will never be the same
as the original tiling
where every L belongs to only one bigger L.
That is how we guarantee non-periodicity.
The researchers used this idea to prove
that the hat tiling is aperiodic
by showing that it must be built in a unique hierarchy.
They started by identifying
four reoccurring clusters of hats.
They realized that the entire plane could be tiled
with just these four clusters.
To make things easier to work with,
they replaced the clusters
with what they called metatiles.
Now the entire plane could be covered
with these four metatiles.
They began to focus on just the metatiles
and put the hats out of their mind for the moment.
They could always bring them back at any time.
They showed how these metatiles formed larger
and larger versions of themselves.
And just like the L tiles, the hierarchy was unique.
(upbeat music)
- Basically, you're essentially tiling
with enormous sized hats
or enormous metatiles, if you will.
You have these larger and larger structures
and any translation would be smaller
than one of these structures within the tile.
- While these guys were busy hashing out the proof,
Dave was busy too.
He'd found another aperiodic monotile.
He emailed Craig on December 6th,
"Probably not the right time,
but I came across another polykite of interest."
It's called the turtle, for obvious reasons.
After a 50 year dry spell,
suddenly two different Einstein Tiles.
As if that wasn't enough,
it was here that Joseph had an incredible insight.
- Joseph, I mean he's like...
Don't know I should say this.
He's like an artificial intelligence.
Yeah.
- The hat and the turtle were related.
By adjusting the lengths of the sides,
you can morph the hat into the turtle
and the turtle into the hat.
Not only that, but this gives an infinite continuum
of aperiodic monotiles.
Except for the shapes at the extremes,
every single shape
in this morphing continuum tiles the plane aperiodically.
For 50 years, we didn't have a single aperiodic monotile,
and now we have infinitely many.
The result was published in March, 2023,
just five months after Dave discovered the hat.
It took the math world by storm.
People could not stop celebrating.
- We finally have the, well,
the happy ending to this long story, this quest.
(tape rewinding rapidly)
- Except it wasn't the end of the story.
Some people weren't happy with these tiles.
And by some people, I mean Twitter and some of you guys.
I posted a YouTube short
about the Einstein Tile earlier this year
and a reoccurring complaint was that it doesn't count
as a single tile because it uses its reflection.
I do understand this complaint.
If you wanted to tile your bathroom floor with it,
you'd need two different tiles made.
So people were arguing
that this wasn't really the Einstein Tile.
- How did I personally feel about it?
Well, I guess on some level,
it's like, well, come up with your own tile then, buddy.
- That, I mean, hater's gonna hate.
I don't know, right?
- I was a bit cheese off about it
'cause there was a lot of people
that were just basically moaning.
- But whether or not you think a shape
and its reflection is one or two shapes,
doesn't matter anymore.
Because just five days after the paper was published,
Dave found another aperiodic monotile.
And this one does not need its mirror reflection.
- For those who thought the hat was two tiles,
this is one tile.
They've got nothing to moan about.
- It was hiding in plain sight,
yet nobody recognized it except Dave.
- It was in front of everyone's eyes.
- [Jade] You know how I said
that all these shapes tile aperiodically,
except the ones at the extremes?
They tile periodically, which is not that exciting.
Yet, somehow this particular shape caught Dave's eye.
- I kind of put it together in my head, you know,
assembled it in my head,
and I think, hang on,
this has more freedom than the hat,
which is understandable 'cause it's periodic.
And so I thought, well,
maybe I should just cut them out
and play with just upfront or back.
You know, don't flip them.
And I kind of did that.
- And an aperiodic pattern emerged.
So here, we have a single tile
that doesn't use its mirror reflection
that tiles the plane aperiodically.
However, when you include its mirror reflection,
it does tile periodically.
So the researchers said
that this shape was weakly aperiodic.
Ideally, they wanted a shape
that does not tile periodically with its mirror image.
But they solved this problem by wiggling its sides.
Now it no longer fits with its mirror reflection
and still tiles a periodically.
If you wanted to get your bathroom floor tiled,
you'd only need to get one shape made.
So here, we have an undisputable Einstein Tile.
It's called the spectre.
- [Interviewer] Why did you call it the spectre?
- Ah, yeah.
Well, I think because it was
in between my brother's, my oldest brother's death.
A spectre, it's like a spirit or a presence,
and I felt, I could see it,
that this thing was looking over me.
It was just this figure.
It's looking over me.
- This is a groundbreaking mathematical discovery
and I would be happy to leave the video here,
but I knew a lot of you wouldn't be happy
unless I talked about possible practical applications.
It turns out that apart from making a cool bathroom floor,
aperiodic patterns are being used to make new materials.
- Physicists will be quite excited
about a few properties of it.
- Materials with periodic lattice have been used
and studied extensively in material science and engineering.
And now, scientists are experimenting
with aperiodic lattices.
They found that they behave
in different and surprising ways.
We're only a few months in,
but already they're finding that they may be more elastic
and less prone to failure than their periodic ancestors.
We are still discovering applications
for the Penrose Tiles
and they were found over 50 years ago.
Given time, who knows
what other cool inventions the Einstein Tile could lead to.
In the meantime,
if you did want to get your bathroom floor tiled,
here's what Dave has to say about that.
- People would come in, yeah,
I'm gonna tile my bathroom with these.
I think why?
Why are they wasting their money?
- Dave is one of the most inspiring parts of this story.
Without any formal training in mathematics,
he contributed to a groundbreaking discovery.
It just goes to show that anyone can dive into
and even influence the world of math.
Because they're so visual,
tilings offer an accessible gateway into math.
I've become a bit obsessed
and I'm loving this beautiful geometry course on Brilliant.
If you've been captivated too, I highly recommend it.
In this course,
you'll encounter a variety of intriguing tilings
and see how they solve real world challenges.
You'll gain the knowledge to craft your own tilings
and uncover intriguing properties
like why only three regular polygons
can form irregular tiling.
As someone who learns best through visuals,
I particularly value Brilliant's emphasis
on visual learning.
Their interactive questions stimulate curiosity,
challenge you just the right amount,
and reward you with a sense of achievement
as you tackle increasingly complex problems.
Whether you are keen on understanding neural networks,
grasping the essence of calculus,
or diving into data analysis,
Brilliant has a course for you.
It's crafted for lifelong learners,
students, professionals, and the insatiably curious.
Dive into Brilliant for a full 30 days entirely free
by visiting brilliant.org/UpAndAtom
or clicking the link in the description.
The first 200 of you to sign up via this link
will receive a 20% discount
off Brilliant's annual premium subscription.
Who knows?
Your journey into a groundbreaking mathematical discovery
could start right here, right now.
(upbeat music)