Today's episode of DONG is one you fried dough confection wanna miss.
We're gonna be squaring a donut.
Now I know what you might be thinking.  Doughnut?  Where's the nut?
I mean I see the dough but why doughnut?
Well as it turns out, when fried dough confections were first made
They didn't look like this.  They looked like this.
Little balls.  And one sense of the word nut, especially back in the day
was a small round thing, a nut.
So this was called a doughnut which means that this gets it's name from this
the doughnut hole as we call it today predates the doughnut, so in a way
this is all an allegory for the human condition.
We have in our centers, an emptiness and we've decided to give that a name.
And we've pretended like it's a leftover thing, it's unimportant.  When in reality
Our very name, our very existence
comes from that emptiness we no longer have.
An old fashion is my favorite.
But before we square a doughnut, we need to square a square.
Squaring the square is a problem, a puzzle that asks, can you tile
a square with squares?
Right here I have a square that is 16 units by 16 units.
One way to tile this with squares is to simply cut it in half
like this and cut it in half again.  Tada!
I have squared this square.  Each one of these squares has a side length of 8 and there ya go!
Problem solved, challenge completed.
A squared square.  And as always thanks for watchiiiiiing...again!
We've only just started!  This is a pretty uninteresting result.
It's trivial.  Take a look at this squared square.
This squared square has 90 degree rotational symmetry which is just fantastic.
The layout of squares is the same whether you look at it at any of these four 90 degree rotations.
However, even something as beautiful as this is not perfect.
This is a squared square but it's imperfect.  A perfect squared square
is a square tiled with squares such that all the squares inside have different side lengths.
Here we've got you know four squares with side length four.
We've got four with side length three.  Four with side length two.
And five with side length one.
Not perfect.  Now numberphile has a fantastic video about the history of this problem.
It wasn't even really tackled until the 1930's at Cambridge University.
Although it might feel like some Ancient Greek mathematical trouble
It's actually quite recent which is very exciting.
The very first perfect squared square discovered was discovered in 1939
by Roland Sprague and this is it.
Every single one of the squares inside this larger square
has a unique side length but it is compound.
This is a compound perfect squared square.  What does compound mean?
Well compound means that some of the squares inside the largest square
form a square themselves or a rectangle.
In fact this is all color-coded.  Right here all of these sorta yellow almost orange-ish
and some of the pink-ish squares make a rectangle.
We've got a rectangle here. We've got rectangles all throughout.
Which means this is a compound perfect squared square.
So the real question becomes, can you square a square
perfectly but not compoundly?
Meaning can you tile a square with squares whose side lengths are all different?
and also not make a square or rectangle inside the actual largest square.
Well, that is a lot more difficult.  Or is it?  Well, kind of except not really.
Because if a perfect squared square is not compound it is simple.
Simple means that no grouping of squares inside the square for squares or rectangles themselves.
It can be done.  Now this entire episode was inspired by
that I received from omega dot out of Australia.
What a beautiful puzzle.  This was sent straight to our P.o. box which you can use to also send me free stuff.
cha ching
This however is not a squared square.  It is a squared rectangle.
Which is still quite interesting.  This is called Stone's simple perfect squared rectangle.
Mathematician Arthur Howard Stone was the discoverer of this particular
simple perfect squared rectangle.
It's called an anisosquaric rectangle
where aniso means unequal.  The squares are all unequal.
And this is technically a rectangle.  It is a little bit taller than it is wide.
We have 99 plus 77 this way and 99 plus 78 this way.
However not a single grouping of any of the squares inside form a rectangle on their own or a square.
And they all have different side lengths.  So it is definitely simple and it's definitely perfect.
Now should I be showing this to you already solved?  Well I feel a little bit bad.  You could just like google it.
This is kind of an old puzzle.
Also it comes in the box solved and a picture of the puzzle solved is on the box.
So it's like really hard to not know how to solve this.
But the point is that today I wanna talk...how do you put this back?
Ah yeah let's start the video here guys so it looks like I'm a genius.
You should go check out Numberphile's video about squared squares
and squared rectangles because it is phenomenal but today I wanna talk not about
squaring a square, not even squaring a rectangle
like this puzzle but instead squaring a torus.
What's a torus?  Well the surface of a doughnut
is a two-dimensional torus.
Meaning I'm not talking about the stuff inside the doughnut, only the outside.
The surface.  If you imagine a planet that was shaped like a doughnut
with a hole in the middle, creatures living on its surface would be exploring and walking around and living
on a 2 dimensional torus.
That is that surfaces' shape.  Now interestingly there is a creature that lives on a torus.
Pacman.  If you've played pacman you know that on the game screen if you go
up off the top of the map you just appear down at the bottom.
And if you go too far to the right you just appear on the left side and vice versa.
That is what it's like to live on a torus.
On the surface of a donut.  If you think about it
a creature...oh this one looks much nicer.  That's got a nice hole right there.
Hey guys this is a thumbnail right here huh?
Mind blown!  Doughnut uhh square the doughnut challenge!
Point is Pacman lives on the surface of a donut.
Topologically speaking because of pacman is here and pacman goes up
What's gonna happen is that pacman is gonna go into the donut
back out from the bottom and come back to where he began.
If Pacman goes to the right he'll circle all the way around and come back to the left.
So if you could flatten the surface of a donut
you would have Pacman's universe and it would in fact be a quadrilateral.
Think of it this way.  Take a donut and cut it just like that.  Okay?
See that cut?  Now straighten it out which I can't do with this real donut
but straighten it out into a cylinder. Here look at this animation.
Okay now that it's a cylinder slit it lengthwise and unroll it.
Ahh how perfect.  But of course the topological properties are still going to remain even though we have flattened it.
Now what I have here
is a square I've drawn on a piece of paper but it's not really a square
It's a flat two dimensional torus.  What does that mean?
Well it means that I should like finally swallow. One second.
okay!  Mouth is empty.
This square is actually a surface that is a flat two-dimensional torus.
So if I have anything on this surface and move up, it will appear down here.
And if I move anything to the right it will come out over here.
The challenge is fitting this square and this square
on our flat two-dimensional torus.
Here's how we're gonna do that.  And by the way this tiling of a torus was discovered
100 years before the squaring of the square was tackled by Cambridge University.
In about the 1830s Henry Perigal discovered this tiling.  Now this is what you do.
First of all you take this square and you can probably already see that it's cut into pieces.
And I'm going to move it across out surface.  Now as you can tell it can't really be there because this side is glued around to this side.
So this half of the square will peak up from here.
We have not, I repeat we have not actually cut this square.
Instead these two edges are glued together and our original square is still whole.
The next thing we're going to do is move the square this way.  Now these two pieces
will have gone around the torus and they'll come out on the other side just like Pacman.
Oh my gosh look what we have now!  We have a slot for our final square.
We have a perfect simple squaring of a donut.
Right there. Now this was discovered in 1830 but there's an even earlier way to create a perfect simple squared torus
that's more amazing.  And it was discovered in the year about 900 by Al Nayrizi of Arabia.
I'll show you why it's my favorite after we put it together.
This is the square.  It's not a square. It looks like a square but this is a flat
two-dimensional torus that I will be tiling with these two squares.
You can probably already see a bit of what I'm going to do
because of where these cuts are.  The first thing we're going to do
is start with this...whoopsie dropped the corner
We're gonna start with this larger square
and we're going to position it on our torus like this so that this part of the square
leaves what looks like an edge but actually is glued to this one.
Goes around the torus and comes out over here.
Then we're gonna move the thing down enough.  You should be able to see that, that corner is also not there but it's around
Up here.  Beautiful!  Okay now we wanna take this square
and we can fit in pretty well right there huh?
Oh look how beautiful this is!  This line on our torus goes around
and comes back like that.  Again these squares have not been cut.  They are whole
and the tile the surface of a donut.  It's just beautiful.
But it gets better.  Of all the squared tilings of a torus Al Nayrizi's is my favorite because it contains within it a visual proof
of the Pythagorean theorem.
Yeah.  Let's take a look at that.  So here we have our tiled flat two dimensional torus.
Let us say that this side right here
Is length C.  Well then that means the area of this shape right here, this white shape
equals...the area is c squared.
But look closely and what do ya see?  A right triangle.
The Pythagorean theorem tells us that c squared equals a squared plus b squared.
where a and b are the other two sides of the right triangle.
We have a right triangle here and we can decide to label them a and b.  Let's call this length right here
Alright let's call this one...beautiful...we'll call this one A.
And then the other side that is not the hypotenuse or a is b and b is this length right here.
There we go.  B for blue, a for pink, and c for the side of the large square.  Couldn't be easier to remember.
So we know that c is the side of this square and the square's area is c squared.
What are a and b?  Easy, let's put these squares back together.
We will move this piece down here and that's one of our original squares.
It has side length A.  Let's put this one back together.
And by goodness, B is its side length so it's area is b squared.
The other square's area is a squared.  Combined the areas are a squared plus b squared
and combined their areas equal the area of the larger square who's area is c squared.
a squared plus b squared equals c squared, the Pythagorean theorem.
Now that's what I call
the square the donut challenge.
Thank you, Al Narizi of Arabia.
Beautiful, beautiful work.  And such delicious work. Now I have links down below
where you can play games on a torus.  Thats' right.
Why stick to a euclidian plane when you could play a word search that wraps around in both directions.
Or you can play tic tac toe on at torus in three dimensions.  It is tasty!
They are extremely awesome.  Check those out and as always, thanks for watching.
