[MUSIC PLAYING] If you had very stretchy pants, could you turn them inside out without taking your feet off the ground?

[MUSIC PLAYING] 9 00:00:19,720 --> 00:00:24,279 Typology is a branch of math which is kind of like geometry.

It studies shapes and spaces, but ones that are arbitrarily stretchy and bendy.

It's sometimes called rubber sheet geometry.

For example, a sphere and a cube are topologically the same thing since you can just kind of mold one into the other.

But a sphere and a donut are fundamentally different things since the donut has a hole and the sphere doesn't.

Two objects are the same if you can mold one into the other by stretching and moving without putting any new holes in it, tearing it, or pushing together any preexisting holes-- gluing it.

This idea about infinitely pliable things gets weird.

The classic example is that a coffee cup and a donut are the same topologically since they both have one hole.

You can morph into the donut.

Any of these crazy straws are also topologically equivalent to a donut since you can first straighten out the crazy straw, then compress it to be a lot shorter, and, finally, you can fatten it into a donut.

Here's one to stretch your imagination a little more.

This is the shell of a donut, what's known as a torus, with a hole in it.

The inside is red and the outside is blue.

Can you move this shape around, stretching and bending without tearing, so that it looks like this, with the red outside and the blue inside?

Pause here to think about it.

There's many ways to think about this, but here's one.

The first step involves expanding the hole in the torus until it's mostly just hole.

You just have two circles left-- the big one that goes around the center of the donut and the little one that goes around the outside.

Notice that the big ring is red on the outside and the little ring is blue on the outside.

So we need to swap the rings, rotate it, and then make the big one little and the little one big.

Now, we take the little ring and stretch it back until it's just a torus with a hole.

In some sense, the punctured torus is made up of two circles, one going around the center hole and one going around the outside ring.

When we flip it inside out, the two circles switch roles.

Here's another counter-intuitive example of two shapes that are topologically equivalent.

This ring, with loops at either end that are linked together, can be morphed into this figure eight.

Pause here if you want to think about how.

So how do deform it?

First, inflate the main loop bigger and bigger until it looks like a ball with two intertwined loops.

Then move the loop attachment along the surface of the ball, disentangling the two loops.

Finally, slide them off the side and shrink the ball body until you get a figure eight both of these examples of topological equivalences and many more are featured in the great book, The Heart of Mathematics.

The figure eight surface you end up with clearly has two holes in it, so in the language of typology we say that it has genus 2.

But it wasn't so obvious in the original drawing that it has genus 2.

The coffee cup and the donut both have genus 1.

Genus is one way topologists classify some shapes.

Another way topologists classify things are in terms of the number of distinct loops you can draw around the surface of a shape.

If we draw these two loops around the donut, I can morph them together.

So they're not really essentially any different.

They're not distinct loops.

But if I draw these two loops I can't morph one into the other.

One of them is stuck around the central hole and the other is stuck around the outside circle.

These are distinct loops.

There's another way I can make distinct loops.

This loop goes around the outside circle once, and this loop goes around twice before it connects back to itself.

Try as you might, you'll never be able to morph one of these into the other.

So you can form distinct loops by going around the central hole as many times as you want, and then around the outside circle as many times as you want.

The description of this collection of loops is called the fundamental group of a shape.

Notice that any loop you draw on the surface of a sphere can be morphed into a simple O and then contracted to a point.

So all loops on a sphere are the same, which is a little boring.

Here's a fun brainteaser related to loops.

You have a picture frame with a loop of string fixed on the back to hang it from and two nails.

Here's the normal way you would probably hang it-- the two nails structurally reinforce each other, but as long as your wall is strong enough they're not both necessary.

Can you think of another way to hang the picture using both nails so that if you pulled either nail out of the wall then the painting would fall down?

Here's the obligatory pause and think break.

If you loop the string around the nails like this, the picture will fall if you remove either nail.

Without the right nail, the string will slide like this.

And without a left nail, the string will slide like this.

There's an article linked in the description that shows you how to hang a picture using n nails so that it falls anytime you remove k of them for any numbers n and k. Math operates on all parts of a spectrum, from considering shapes that are infinitely flexible and stretchy like the ones we've been looking at to completely solid and rigid like you probably learned about in a geometry class.

There's also a bunch of stuff in between with various degrees of flexibility and rigidity required of the shape.

It seems likely that the reason these topological puzzles are so tricky is that our intuition about the world is pretty rigid.

Real materials are limited in how much they can be deformed, and these topological puzzles live outside of those restrictions in the unlimited world of mathematics.

Finally, back to the pants.

Let your pants be topological.

They're super stretchy and bendy, but you still can't rip them or sew them back together.

Now, can you turn them inside out without lifting your feet off the ground?

Amazingly, you can.

But it requires some clown pants-- like these.

Two fantastic Cornell mathematicians, Tara Holm and Lila Greco, developed and sewed these pants just for demonstrations like this.

Notice that they have the same shape as regular pants, just stretched out.

Turning the pans inside out is basically a matter of crawling first through one leg like this and then crawling back through the other leg like this, which is only medium hard to do without falling over.

So me wearing the pants this way, with the red side facing out, and me wearing the pants the other way, with the yellow side facing out, are topologically indistinguishable, just like the coffee cup and the donut.

Here's your challenge problem for the day.

Come up with two shapes that are topologically equivalent so you can bend, stretch, or distort one into the other without tearing or gluing anything.

Email us a drawing of your shapes at pbsinfiniteseries@gmail.com.

Use the subject typology challenge and provide an explanation of how to distort one into the other.

See you next time.

Thanks to Squarespace for supporting this episode.

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Squarespace provides an all-in-one platform with templates that allow you to easily set up a web site.

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So if you want to build a web site full of topological illusions, start your free trial with Squarespace at squarespace.com/infiniteseries and enter the offer code infinite to get 10% off your first purchase.

My wonderful friend Vanessa Hill, the host of Brain Craft, has just released her exciting new documentary about genetic engineering called Mutant Menu.

She investigates the scientific, interviewing the world's leading experts on what items will be on the menu of mutations in the near future, and the societal, asking folks how they feel about modifying their genes.

Fright or delight?

You can watch it right here on YouTube right now.

Just click on the link in the description.

Hello.

On our video about how to build infinite bridge, you asked a lot of great questions about infinite series, so I figured I'd answer two of them.

Here's the first one.

Louis JX asked, when the series diverges doesn't that mean it doesn't converge?

True.

So when we say a series converges, we mean that the sum approaches a number.

But when it diverges, it just doesn't do that.

So it can either diverge by becoming infinitely large, which is what happens if we add 1 plus 1 plus 1 plus 1 plus 1 plus 1 together, it grows up to infinity, or it can diverge by just bouncing around.

So the sum negative 1 plus positive 1 plus negative 1 plus positive 1 just bounces around and it never settles on the value.

So that's another way something can diverge.

I didn't address it in the video because it wasn't relevant, but it's a good point.

Also Erdminnelchen asked, if the harmonic series is infinite and 1 plus 1/2 plus 1/4 plus 1/8 and so in is finite, is there an infinite sum between those sums when is on the edge of being infinite and finite?

I don't know if there is one good answer to this question, but what it make me think of was the geometric series.

So if you take a number smaller than 1, let's call it r, and you add r to 0 plus r to 1 plus r to 2 plus r to 3 and so on, you get 1 over is 1 minus r. This is true if are smaller than the 1.

If r is bigger than 1, you get infinity.

So the closer r gets to 1, the bigger your sum will be.

So if you pick r to be really, really close 1, just a little bit smaller than 1, your sum will be really, really, really big.

So that's somehow on the border between being finite and infinite.

All right.

For our challenge winners for who can build the coolest bridge, I couldn't pick just one.

So we have two challenge winners this week.

The first one is Alan Chang.

It thought it was pretty funny that he used these yellow math graduate textbooks.

They're a staple of every math department and they just cover the walls.

There's lots and lots of yellow books in math departments.

So I thought they were pretty funny.

And also they were being attacked Pokemon.

Our second winner is chalkface, who's submitted on Twitter.

First, chalkface is a really funny Twitter handle.

Second, there's a cute baby in the picture.

And third, and my favorite part of this picture, is that there's some subliminal messaging happening.

Can you find the word logarithm in that picture?

It's pretty sneaky.

All right.

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