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
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
So my main mission today is to actually
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
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