Okay, so can you connect Klein bottles together, and if so, what do you get? Well,
I take a Möbius loop
Right, I give a -
half twist, get a Möbius loop. What if I give it -
another half twist?
Turns out that's not a Möbius loop. It's two-sided. Give it yet another half twist. It becomes a Möbius loop.
Probably the most famous Möbius loop has three half twists to it
It's the recycle sign - and you can see there's 1, 2, 3 half twists to it.
That's a Möbius loop, just like
that's a Möbius loop. To be a Möbius loop
you have to be able to walk on all one side
without crossing an edge. Yep. Ok, got that one
Oh, my fingers are blue from having tried to put blue
dye into a Klein bottle. Ignore that, they'll wash off in the next two or three weeks.
So, I can make a three twist
Möbius loop. Can I do the same thing with a Klein bottle? What if I take a Klein bottle,
and add another Klein bottle to it?
Oh, I can go over the glass shop, get a scissors cut that there, cut that there, get another one and build - this.
Is this a Klein bottle? It's two Klein bottles welded together. Oh, let's get a -
pen. Okay, let's start over here and write the letter A,
and the letter B.
Well, let's start at A, going along here.
Oh, I can't keep going. I'm on the inside of this part. Still on the inside, still on the inside, still on the inside.
Well, come out over here and I come back to A.
So, I can go from A, all the way back to A but I never get to the letter B
Meaning that this side here,
the A side, is different from the B side.
B and A are on different sides. - (Brady: So, Cliff, someone watching might be thinking)
(oh, no. I can just run my finger directly from A to B, but - )
You're, you're cheating if you go like this and go bonk like that. No! We're talking about, we're talking about a
mathematical thing like this. Where, where, where -
my glass blower had actually cut a hole here, so you can go, ziip - zip through it go around the inside.
Nah, this is something - this is a figment of our - figment of our three-dimensional universe. Ignore that one.
(Brady: That self-intersection doesn't count?)
That self-intersection doesn't count. From the outside here, Bzz! Now you're on the *bzz*
inside of this
Bzzz! Now you're on the outside of this and you're back there. So this
turns out to have two sides, and a hole there. This is homeomorphic to a torus.
Okay, got that. What if I add another one to this? I'll go from here to here,
Here's three
Klein bottles arranged. Well, let's try the same experiment.
I'll write A over here,
B,
And C.
Okay, triple Klein bottle go from A, *Bzzz*
Over here. Now, I'm on the inside of B
...still on the inside...
I come out here,
Now I'm on the outside of C.
Okay,
outside of C, go back here. Outside, outside, outside, outside. Oh look, I end up on the inside of A. Aah!
You go around the inside of A. I'm now right here, so, without
lifting the pen from the surface of the glass, I've gone from the
outside - from the outside over here over to the inside. This is one sided. Let's keep going.
I'm still on the inside of this one - I need - a millimeter away from here is where I'm trying...I see,
Still on the inside, still on the inside, I come around here,
I've hit the outside, I've hit B! Hot Diggity! Ah, cool! Now, I come around,
And I'm on the outside, *glook* through there,
now I'm on the inside of this, and I've reached the other side of the C surface. I stand that other side,
come out here, I am back to where I started.
I've covered the entire surface. An ant could walk on the entire surface of this,
just, you know, ignoring the,
little places where, you can't get through.
So this is a Klein bottle. It's one-sided. There's no edge. At no point is there sharp edge that the
caterpillar crawling the surface ever got cut on. (Brady: You could, )
(paint the whole thing by pouring paint over one side and the - ) - That's right, if I, if I hired a painter they could paint the
whole thing with one side, and one colour - piece of cake works like gangbusters!
Showed this to a friend of mine:
Lucas Clarke, glass blower. He thought about it for a while and he said:
"Why, you're trying to tell me that any
even number of Klein bottles linked together as a torus, any odd number -
1, 3, 5 -
Is a Klein bottle?" And I said "Yeah, that makes sense"
He makes
a Klein bottle, of
17 individual Klein bottle units. Brady check this out,
check this out. Each of these each little unit is a Klein bottle.
And, if you count them,
you'll count
17 Klein bottles!
But wait, wait.
To show
how nutty
Lucas is - each Klein bottle, in this linked set of 17 Klein bottles,
twists, just enough,
so that, it's a
Möbius loop of
Klein bottles.
[hushed] He actually made a mistake, but I'll ignore it. Don't tell anyone that. He sort of flipped one around - but
This is a Möbius ring' of Klein bottles! Just as you could say that a Klein bottle is
the connected sum of two Möbius loops;
here is [laughs]
the connected sum of 17 Klein bottles, as a Möbius ring!
Oh my God! [gasps] Oh, my, God.
[gasps] Lucas!
You're amazing!
(Brady: Was it hard to make?)
Cliff: Was it hard to make?! Oh why, why, why knock on the door at the glass shop?
I won't say that everybody runs and hides,
but,
multiple Klein bottle
manufacturing, such as this, or
3 X turtle Klein bottles, or
suppose instead of making three Klein bottles by cutting these together and hang them together like this,
suppose I removed this section, and put a smaller Klein bottle right there. So I,
cut this, and
build it in there and that over here I go -
take this part out and cut that out and put an even smaller Klein bottle in. So I have three Klein bottles, which are
internally linked, rather than externally linked. Well, in that case I'll end up with something...
that looks like...
This. Okay, this is homeomorphic
to
this Klein bottle. Both of which are homeomorphic to an ordinary, regular,
Klein bottle.
Like this guy here.
This is three internally linked Klein bottles.
So, this hole here is
map - well is really the same as this hole here. And
that leads outward through this tube,
coming back to the next one in,
goes to here,
comes up and around out here.
That tube goes to...
another Klein bottle, which is completely linked up here, and comes out this way
to the outside,
which comes back to here.
In other words, here's a triple Klein bottle which has the same properties as;
this triple Klein bottle,
which in turn, which in turn, has the same properties as
this guy,
which also have the same properties as
this Klein bottle. All of which is to say,
if you're looking for
zero volume, one sided,
R_3 immersed - not embedded -
Klein bottles, uh,
there's plenty to explore with glass.
Brady, if you're going to show Felix Klein behind a Klein bottle, there's Möbius, framed by a Möbius loop.
(Brady: You've already broken it once haven't you?)
Yeah, I've, broken it once and
Adams and George; George Chittenden, over at Adams and Chittenden glass, laughed at me and said "I'll fix it for you, just this once"
He patched it up just, so I'm not gonna - I treat it with delicacy
Enjoy, it's a
the idea of making a
17 unit Klein bottle into a Möbius loop. Oh
Lucas Clarke over here, Lucas is a complete nut
Lucas, this is astonishing! [gasps] You're brilliant!
(I'm gonna have to take a picture and send this to my friend now so she knows what the big - okay)