Welcome to another Mathologer video. Today

I'll start by showing you something

absolutely amazing, a really nice

animation by mathematician Arnaud Cheritat

of an ingenious way to turn a torus

inside out.

Let me explain. A torus is just a

surface of a donut. It has been painted

green on the outside and red on the

insert. It can stretch, it can contract, it

can even pass through itself, just like a

ghost.

What's important is that all the way

throughout the deformation that you'll be

seeing the surface is always completely

smooth. There are no holes, there are no

creases, nothing terrible happens to the

surface. Now we've just arrived at a

critical stage of this inside-out

deformation. The torus is just folded in

on itself. Now we're going to take this

part which is overlapping and turn it.

Ok and you see some of the inside

starting to poke out and there's more

and more red here and at this stage

what you see is half red half green and

the rest of the deformation is really

running backwards what we've just seen just

on the other side of the circle. And

there the two kinks and annihilate.

Et voilĂ ,

at this point in time the torus takes

a bow and I hope you as impressed as I

was when I first saw this thing. So the

technical term for what you've just seen

is a torus eversion. This stage I want

to have a really close look at. So here

I've redrawn the cross-section just

after the critical move and does anything

here look familiar?

Well, you should see two Klein bottles.

So what we're really seeing here is a

double Klein bottle. If we deform this a

little bit more we can actually arrange this in a

really nice symmetric way. So a double

Klein bottle is a torus in some way.

Very strange. Even stranger when you think

about that Klein bottles don't really

have insides and outsides but somehow

when you stick them together in a double

Klein bottle they become a surface that

has two sides, the torus. Anyway because

of all this

I'll call our torus eversion

"The Double Klein Bottle Trick" and I will use

the Double Klein Bottle Trick to do all

sorts of ghost gymnastics in the

following. Now I've got lots lots of

Klein bottles, I even have a triple Klein bottle

but actually didn't have a double Klein

bottle, so I just ordered one from Cliff

Stoll who a lot of you will know from

Numberphile and actually Cliff

assures me I'm his absolute best

customer and in fact at the moment he

says you have more Klein bottles,

different types of Klein bottles, than me because I

just ran out of a couple. So that's

something new :)

Ok so a torus ghost can turn inside

out smoothly.

What about a sphere ghost? Can that turn

inside out?

I should really have done this video on

Halloween but somehow I ran out of

time.

Okay, so a straightforward idea is you just

take the North Pole, the South Pole and push

them through each other and just keep on

going like this but at this stage here

you hit a crease and that's not something

we want. We don't want creases, we don't

want any holes, smooth throughout, right?

And, if you try and try and try you actually

never really find anything that works

straightaway. So is it actually impossible?

Well let's just look at something that's

a little bit easier,

let's just see whether we can turn a circle

inside out. So here's a circle colored

green and red and try and try and try

and try and you actually find that this is

completely absolutely impossible and it is

actually fairly easy to see why it is

impossible.

So for that what I do is I put a race

car on the circle and actually use it as

a racetrack. The left side of the car is

always going to be on the green side of

the circle.

Ok now chase it around. What we're really

interested in is what the race car does

relative to its center so I'll just

highlight this. So it starts out like this

and then it does one clockwise turn

Now we're starting to

deform the circle and for any

intermediate stage there we check out

when we chase it around what sort of

turning does the car do. And it turns out it

always does one clockwise turn. So for

example here one clockwise turn.

That's even true for really

complicated tracks like this. So here you might

turn clockwise sometimes anti-clockwise

sometimes but overall you get one

clockwise twist and again it's very easy

to see why this would be the case. Maybe

if you're unsure discuss this in the

comments. Just one hint: the car always

starts and ends pointing in one direction

and everything kind of changes

continuously. OK, now let's have a look at

the inside-out turned circle. Remember the

racecar always has to have the left side

on the green side of the circle. So to

make this work here we have to turn it

around. Now if you chase the race car

around

what you see is well we still do one

turn but we do it in the

counterclockwise direction and so that

shows that it's impossible to turn a

circle inside out. You know, starting with

a circle you can deform, deform, deform

but at all stages we

basically get one clockwise turn, we

can't just jump all of a sudden to

a counterclockwise turn, that's impossible.

Alright sphere, obviously more

complicated and you certainly would think

it's not possible, too, but actually in

1958 a very high-powered mathematician

Stevens Smale showed that it's possible

to turn the sphere inside out and this

really counterintuitive result is

actually now refered to as Smale's paradox.

What may seem even weirder to a non-

mathematician audience is that the proof

didn't contain any pictures and didn't give

any clue whatsoever as to how you would

actually do this in practice. Now

subsequently quite a few ways have been

found and really people have been trying

to get really good ones happening for 50

years and here's four different ones

that are all very nice and all very

ingenious but you can see they're all

pretty damn complicated. I actually stop

them here at halfway stages and just to

wrap your minds around these halfway

stages is a killer. The original clips

are linked in from the description so

check them out especially that one up

there. It's called Outside in. It's actually a

half-hour documentary on

all this.

It's absolutely fantastic. So here are the

second halves of these eversions. Now this

one down here is actually buy Arnaud

Cheritat the person who animated the torus.

So I had a really really really close

look at pretty much everything that had

ever been written about this while

researching for this video and I

stumbled across something that I had never

seen before, a really original idea for

turning a sphere inside out based on the

double Klein bottle trick and this is by

mathematician Derek Hacon and he never

published it and maybe only a handful of

people know about this.

So my main mission today is to actually

let the world know about this amazing

idea,

you know, this is the way to go. So what

does he do? Well he takes the sphere and

he pushes in the bottom part. Now let's

look what this thing looks like inside.

Take off the dome here, another dome

sticking up inside. Take that off too

and then you see that the bottom part of

our shape is actually just half the

torus that we looked at before and

it's bounded by these two circles and

the domes are attached to the circles.

Now the idea is to just unleash the

double Klein bottle trick on this and

have the domes follow the movement of

the two circles at the bottom and while

they're doing this the contortions of the

circles sort of die out towards the top.

I just want to show you what that looks

like for the dome in the middle. So

there's the dome in the middle attached

to the circle. At some point in time the

circle is folded up like this, so the

dome would look roughly like that, so

it's got a flap and the flap dies out towards

the top of the dome. Later on we've got

this double kink. The dome would look

like this, roughly, dying out towards

the top and stays like this all the time.

Now we started with the circle on the inside.

At this stage, at the halfway stage the

circle is already half in half out and when

it now completes the eversion the circle

has moved all the way to the outside and

so now this dome here is on the outside

and of course if everything goes well

then the other dome has moved to the

inside.

And if you now take out the bulge we've

actually completed our eversion. Isn't

that neat? It's really really nice and

actually it sounds a little bit too good

to be true and there's a complication

but just to emphasize, the main

work that's done in this eversion is really

done by the double Klein bottle trick.

Now the complication comes about when

you look at what the outer dome does. The inner

dome deforms in a very tame way. So here,

for example, very tame, very tame.

Let's go back to this folded stage and

look at what happens to the outer dome

here. It's also completely tame but now

after the critical move you see these

two twirls appearing and it's a bit

harder to imagine what the dome now has to do.

If you're fixing your own bikes

whenever you're manipulating a chain it

happens that you get these two twirls

appearing and usually it's a bit of a

pain to get rid of them. Now they are

a pain here but actually in this sphere

eversion they are very, very useful.

OK now the twirls, what do they do? So they

travel around the torus and then they

annihilate on the other side just like a

particle antiparticle pair. Really neat, right?

So I'll provide a link to a

discussion of how exactly the outer dome

deforms in the description, so check that

out if you're interested. Anyway, this

version is my absolute favorite from now

on. There's a funny or sad story that

goes with this. Now, everybody who knows

anything about sphere eversions knows it's

very hard to visualize them. So Derek Hacon

was obviously very excited when he found

his way of turning the sphere inside out

and he submitted it to a popular maths

magazine and ... they rejected it because

they thought it was too trivial. How

silly can you get. Derek Hacon actually

died a couple of years ago so we

can't ask him which magazine he

submitted it to but hopefully the people who

rejected it will see this video and will

kick themselves now. Here's another

ghost, a double torus ghost and you can

also use the double Klein bottle trick

to turn that one inside out. That's also very

neat so I'll just show you. So let's just focus on

half of it,

the other half you have to

imagine it's still there, just to show

you the inside and the outside. Now we shrink

one half and unleash the double Klein

bottle trick on the big one. The little

one is just going to be carried around

like a fly sitting on the big one here.

So unleash it on the big one and at the

end of it the fly will sit on the inside.

Now to complete the eversion what you do is

you just reach in through the hole

and then pull out and the eversion is

finished and I've got a nice animation here

that shows what actually happens.

So you put your finger inside and pull

and as you pull that's what happens and

you kind of just rearrange by

some more deformation and you've turned

to double torus ghost inside out and you

can do this with all of these shapes, all

of these shapes can be turned inside out

smoothly, nice!

If you want to know more about

all is google Smale's paradox or sphere

eversion. Now you may think that all this

completely useless. Well I have to tell you

that this sort of ghost maths is often

really, really good for finding out when

certain things are not possible with

more solid counterparts of the ghosts

and the principle here is if the ghost

can't do it, the solid counterpart can

definitely not do it either and it's

often a lot easier to prove that a ghost

can't do something than to show directly

that something solid can't do it. So it

often comes in handy in this way.

Finally the Mathologer inside out

challenge. So to take part in the

challenge what you need to make yourself

is a ring consisting of four paper

squares. These paper squares have been

creased along the diagonals and

the ring is colored differently inside and

outside and you're supposed to evert

this by just folding along the diagonal

creases and the creases at which two

squares meet. It's a really nice

challenge. If you succeed, send me some

video evidence of you performing this

feat and i'll include your name in

a Mathologer inside-out challenge

hall-of-fame limited to 100 participants

in the description.

At the end of two weeks or something

like this I'll publish the best video I got

submitted on Mathologer 2. So no

submission please in the general

comments I'll just delete them. And

that's it for today. Well, it's really late

here and really the only thing left for

me to do is to go to bed. So before I do

this I'll take my special Mathologer

glasses case which actually nicely

turns inside out, put my glasses in, say

good night and I'll see you next time.