Michael Spivak's "A Comprehensive Introduction to Differential Geometry" - he really means topology, I suspect, but...
It says: "A classical theorem of topology states....blah blah blah. To which of these 'standard' surfaces are the following homeomorphic?"
A hole in a hole in a hole. If I solve this, what is it?
Well, I head over to the glass shop and, I said, "I can make that."
So I made a hole in a hole in a hole.
There's a hole... that... goes through a hole, that's in a hole.
In the book it's in paper, here, it's glass.
How do I simplify this to something that answer Spivak's question?
So, let me take the inner hole and stretch it a little bit.
[Glass sounds]
Now, one of these holes is untouched,
but what used to be a ring around this is now stretched out - it's almost an ellipse.
Well, that's okay in topology, because I can stretch things.
Haven't changed much. Let's go a little bit further - I'm going to keep pulling that this way.
*grunts* Until this piece of plastic I'm just touching right now, the end of my finger.
It's coming all the way up. I'm going to stretch this, oh, about 2 or 3 cm further this way until it starts to pop out the other end.
So now what had been a ring is stretched out and I'm now touching that place
Let's keep going. Now, this part of the hole
and this part of the hole - I'm going to move these, I'm going to push the glass
until it's almost wishbone shaped.
This will pull this way, I'll pull this one over here.
This is the original beast,
however, instead of this coming over like this, that piece of glass now is continuous across
>> We haven't broken any rules yet?
We've broken no topological rules. In fact, we've just assumed that this is strictly... flexible glass.
And, at 1200 degrees fahrenheit, it *is* flexible glass.
Now that we have this
I'm going to pull this *brrrp*
Around here. And this *keeeywp* over to there.
We're going to straighten it out
So the wishbone, here, has
become a t-bone.
Haha. A t-bone and this is straight through.
It's still the same 'beast' we started with,
but we just moved the holes around, we moved things around.
Is it still a hole, in a hole, in a hole?
Well it's still the original thing,
that's been topologically reshaped.
The rules are, we can't cross things, we can't tear anything, but we can move these around, and that's what we've been doing.
Let's instead of going straight across, lets go boink, boink.
Let's make this that's something a little more curvy.
From here,
over to here.
Notice, we still have,
This hole is the same as this hole
(Is it ok if I'm pointing with my nose?)
So, these are still topologically the same, the intersection of the 'T' is now the intersection of two curves.
And you can see what we going to do next.
The next step I'm going to do is to pull this tube over here and pull this tube over here.
Remember here's one of those holes.
Now, we've separated those holes so that's is now three tubes.
One tube here, this tube, this tube.
Let's go back to what's the source of that.
And you can see we've simply pulled those tubes along.
Nothing was torn, haven't added, haven't subtracted from it.
BRADY: It does seem a bit like it's torn, but no?
Has it been torn? All I'm doing is pulling this hole along here,
and this hole further along here.
So I'm just pulling that outward,
and in so doing,
forcing this tube to the side, this tube to the side.
I now have a tube, a tube, and a hole!
Given this, let's straighten the tube.
So I've got this tube, I'm gonna bring this hole around here,
this hole around here, pull them really straight.
Like this. And now those two curved tubes are two straight tubes,
And, I have another straight tube here.
Ok, now it's a piece of cake!
All I have to do is pull these two to be parallel to this one,
Or, alternatively, rotate this tube to be parallel with these two.
So, let's try it!
Here is all three of those tubes parallel.
So all three holes are now parallel.
And! [CHUCKLES]
Now this is getting close to a solution!
I'll take this in one hand, this in another,
and I'm gonna squash it! I'm gonna push it down this way.
So instead of a sphere,
I'm going to compress it down till it's more like a pancake.
I'll get something that's pancake-shaped,
but the three holes are still there.
There's also a tiny little hole there.
When you make hand-blown glass,
it's necessary to have a little hole to let out hot air --- ignore that guy.
These are the main three, and oh!
Doesn't take much to say this is a three-holed torus.
This is a torus with one, two, three holes in it.
Which, is a nice way of answering Spivak's question.
How would I manipulate this into a solution?
A three hole torus will make any topologist in the universe happy! in the universe happy!
But, let's go back and try something else.
Let's go back one step. Remember we had this guy? And...
I said, we're going to squash this down to here.
Instead, let's pull this tube...
over towards the sides. So right now it's near the middle,
let's pull it over towards the sides.
In fact, let's pull it almost to the edge, and we find that,
It would be possible to go through it.
If we kept pulling it, we could pop it out!
From here...
We can make a sphere with three handles.
I am jumping over several steps, but think of this:
Right here is a handle, I could put a chain right through here,
pick it up, I put a string right through here,
pull this outward.
This, is the same as this.
Notice that hole there, is the same as...
This hole, a second hole, and a third hole.
Here's a three-handled sphere,
a sphere with one, two, three handles.
This three-handled sphere, is homeomorphic to this three-holed sphere,
and that's homeomorphic to this three-holed torus!
If I have this three-handled sphere,
I can go to my glass shop, heat it up,
put my hand right there, push down right here, pshh,
and push down really hard,
and, from here, I can transform this,
into a three-handled coffee mug.
This is a three-handled mug,
a three-handled drinking glass,
that's homeomorphic to, and has the same properties as this three-holed torus,
a three-holed donut,
and that is...
the solution to the question: What does this map onto.
BRADY: thanks for Audible.com for supporting this video.
If you haven't discovered audio books yet as a great way to catch up on excellent writing,
well, you've been missing out.
After numerous recommendations from friends, I've finally been listening to Ready Player One.
I actually became a bit obsessed with it on a recent holiday,
every spare few minutes I had I was listening to a bit more of the book.
I don't want to give anything away about it,
because I really enjoyed going into it cold, knowing nothing.
I want you to have the same if that's possible, but I do recommend it highly.
But whatever kind of writing you're into, you can rest assured,
Audible is going to have something you'll like.
And when you do sign up, you get your first month and your first books for free.
Do that on audible.com/numberphile, so they know that you come from this channel,
and if you do download a book that say you don't like, Audible will let you exchange it for another.
I've actually done the exchange myself, and it was really easy,
but I don't think you want to exchange Ready Player One.
It's a really good recommendation for your first audio-book,
really well-read by Wil Wheaton.
Give that book a try, give Audible a try, and thanks to them for supporting Numberphile yet again.
CLIFF: the changes that have happened from one to the next to the next,
have not required moving things through one another.