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You're watching a Mathologer video and that
that probably means you're eating Klein
bottles and Mobius strips for breakfast
and you know that these tasty
mathematical surfaces have just one
side. Except, and only real mathematical
connoisseur seem to know this,
they are Klein bottles and Mobius
strips that have two sides.
Let me explain. Quick revision: this strip
of paper has two edges and two sides. To
make it into a Mobius strip what I have to do is
to bring the ends together such at the
edges combine into one long edge. Every
Mobius strip has just one edge but as you
can see something else happens here. As I
bring the ends together also the two
sides combine into just one side, so
this is a Mobius strip that has one side. Now
there are actually a couple of different ways
to bring the ends together by twisting
them to make this strip into a Mobius strip.
You can just do one twist and glue, that
gives you a mobius strip, three twists or five
twists or any odd number of clockwise or
counterclockwise twist that will yield a
Mobius strip. Now all these Mobius strips
have just one side. If you do an even
number of twists and then glue you get
one of these surfaces. They all have two
edges and two sides. Now these are not
Mobius strips
these are called topological cylinders
or just cylinders. Now I claim there are
ways to bring together the ends into
Möbius strips that are two-sided. Hard
to imagine how is that miracle possible?
Well, it turns out that the number of
sides of a Mobius strip or actually of
any 2d surface depends on which 3d space
which 3d universe it is contained in and
how exactly it is contained in this 3d
universe. Now most people think that 3d
just means xyz space what you deal
with in school or at university. But there are
actually infinitely many 3d universes,
mathematical 3d universes and we
actually don't know which one of these
mathematical universs describes the
universe that we live in. Now quite a few
of you will actually have heard that our
universe may be a pacman universe which
means that there may be a direction,
special direction. If i head off in this
direction and just keep going straight
I'll get back to where I started from. So
let's just assume we are inside a
mathematical universe that has this
property and let me introduce you to my
math cat maskot the QED cat.
Well, actually, there's a bit of dispute
at home whether it is a cat or chihuahua
but no matter
let's just launch it in this special
direction on its space surfboard and see
what happens.
So as the cat travels along the
surf board actually generates a strip.
Ok now keep going, keep going. Eventually it
gets back to where it started from, there.
And now it wants to turn the strip into
Mobius strip and you can see that to
create this one long edge what we have
to do is have to kind of flip upside
down and keep moving forward.
Ok, so we're creating a twist like this
and you can see the Mobius strip that
we've actually created here is just one
of those one-sided Mobius strips, so
nothing new here. Now, in a more fancy
universe something else can happen, so
let's just do this again. So QED cat
heads off again, we leave one of those
ghost images behind. It gets back to its
starting position but now something else
has happened, it's actually turned into
its mirror image. That's unusual and now
you see to create this one long edge, to
create the Mobius strip the cat has to
just keep on going, it doesn't have have to
do any of those upside-down acrobatics.
So just like this and we've created a
Mobius strip and obviously that Mobius
strip is two-sided. So if you put a
anti-cat on the other side and have
QED and the anti-cat run around on those
two sides they'll never meet. Once you've
got one of those strange mirror
reversing trips all sorts of other nice
things start happening so for example
the cat gets back to the starting
position, it's mirror reversed and it wants
to eat some of its cat food but actually
that's no longer possible
because the mirror reversing happens
at a molecular level and so the food and
whatever processes the food in the stomach
won't match anymore, won't happen. So to
unscramble itself the cat actually has to
either backtrack or just do a second
round and then it can eat. Also once you
have a two-sided Mobius strip like this
you can extend it into something solid
and this solid corridor is actually a
real 3d counterpart of a 2d Mobius strip.
A lot of you may have wondered
whether something like this exists, well
there you go. Finally what I want to show
you is how you can use a mirror reversing
path like this to create a one-sided
cylinder. Now, usually, cylinders are
two-sided, right? So now let's just look
at this situation again, we can now turn
this strip here into a cylinder by just
doing a twist, a twist like this creates
two edges, we're dealing with a cylinder
but as you can see this is a one-sided
surface. So at this point what you really
want me to show you is one of those
mirror versing path in our real universe
or at least in a mathematical universe
that i can hold in front of
you. But that's actually very hard to do
because no matter what you doing in
xyz-space round-trip wise you'll never
mirror reverse yourself. That also means
that we cannot have a copy of one of
those reversing universes inside
xyz- space, makes it hard to describe. But what
I can do is I can show you the analog of
one of those one-sided cylinders, a 2d
analog and for that I need a flat cat.
Now what's a counterpart of a cylinder
in a 2d world, it's just a circle. So what
I want to show you is one-sided circle.
Now, usually, circles are two-sided right.
Two-sided circle with respect to this 2d
world that the cat is living in. Now instead of
off using this off-the-shelf 2d universe
I use a Mobius strip universe, that's a
2d universe. The cat's living inside it, the
circle is part of this universe and I'm
going to chase to cat around it but
what's really important here is actually
too
emphasize that a real mathematical
surface has zero thickness just like the
xy-plane inside xyz-space has zero
thickness. So that Mobius strip has zero
thickness, the cat sliding around in it
has zero thickness.
Let's just see what happens when it runs
around the circle.
Ok so it's completed its roundtrip
and as you can see with respect to
this 2d universe its living in it's
actually mirror-reversed itself and it
seems to be locally on the other side of
the circle but, as you can see, when we do
a second trip around it actually gets
back to the beginning and what this
means is that this circle here has just
one side. On the other hand, if I take
away the Mobius strip and surround this
circle by this ring here, then the circle
is actually a two-sided circle. So what
that also tells you is that without the
2d context it actually doesn't make any
sense to ask how many sides this circle
has. Just kind of floating with in 3d
space
it doesn't make any sense to ask how
many sizes this thing has and similarly
if you've got a surface you need a 3d
context to be able to ask and to answer
how many sides one of these surfaces
has. Otherwise it just doesn't work. For
example, we could put something like this
in four-dimensional space and just have
it floating there, it doesn't make any sense to
ask how many sides one of those things
has. Now I also promised you some
2-sided Klein bottles. How do we get those?
Have a look at this.
So QED this flat so it can't really see
a Mobius strip but it wants to play with it
anyway, so it can do this a la
pac-man. It's not ideal but it's good
enough to visualize what's going on.
Ok, so what you do is you just kind of
draw a flat rectangle and QED can run
around in there and then you just
indicate how the ends are going to be
glued together with arrows like this so.
The arrows here basically tell you that
these two points get glued together and these
two points get glued together and so on
and now a Klein bottle is actually just a
Mobius strip whose edge has been
route to itself in a certain way and
that certain way I can actually show you
very easily also with arrows, goes like
that. So we have to do is, we have to glue
these two points together, we have to
glue these two points together, and so on
and that will give you a Klein bottle
and well since we have 3d beings I can
actually show you this construction in
space. So here I've got a Mobius strip. Now I'm
just going to bring corresponding points
of the edge together, like this, and there
you've got your Klein bottle. Now
obviously once you've found one of those
mirror reversing paths and a two-sided
Mobius strip it's pretty easy to imagine
that we might be able to extend this
strip into a two-sided Klein bottle and this
is exactly what happens. All right, now
we've got two pictures of a Klein
bottle here and just like QED can use
the square to describe a Klein bottle we
can use a solid cube to describe a solid
counterpart of a Klein bottle, so
basically a solid Klein bottle and this is also
done by these fancy arrows. What the
fancy arrows show you is how opposite
faces of the solid cube are supposed to
be glued together. For example, these two
points get glued together, these two
points, those two points, and so on, should
be pretty obvious and actually this
solid Klein bottle there is one of
those mirror universes and if you have a
really close look you can see that this
here is a two-sided Klein bottle within
this 3d mirror universe, very very fancy,
very, very pretty.
It's an absolutely beautiful Klein bottle
much nicer than the one that I showed
you before. The one I showed you before
has this strange sort of
self-intersection which is really annoying.
This one doesn't have any of this so so
much much nicer in this respect.
Ok, now I learned about all the stuff for
the first time from this book here, The Shape
of Space by Jeffrey Weeks.
This is an amazing accessible introduction
to two- and three-dimensional universes
manifolds.
I really recommend it to everybody here.
Jeff's also created some amazing pieces
of software, totally free that you can
download from the website I'll link in
from the description and they allow you
to you play chess on Klein bottles on
tori, play pool, all kinds of other
things but he also has pieces of
software that allow you to fly
around in strange 3d universes.
So, for example, here's a view of a very
small version of this solid Klein bottle
universe which just basically has space
for one Earth and as you kind of look
around because of the way it kind of
connects up to itself you can actually
see yourself, see Earth over and over, not
only Earth but also the mirror image of
Earth and you know I leave it to you to
kinda figure out how exactly this works
how how exactly the pattern of Earths and
mirror Earths comes about.
Now there's a lot more to be said about
all this, e.g., four dimensional stuff. I may
say it's a little bit about this in the
description. Also I'll definitely come
back to these strange 3d universes, make
another video about that but for the
moment I just like to say thank you very
much for all your support throughout
2016 and Happy New Year to all of you
and I'll see you again soon.