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.