Here.

Come on in.

If you keep going, you will eventually emerge out my other end.

And for this reason, it has been said that the human body is like a doughnut.

Yeah.

Yeah.

You are just a bunch of meat packed around a central hole.

You're sorry.

Humans have more than one whole, right?

I mean, we've got nostrils and ears.

The whole sweetpea out of the holes were born from nibbles.

Our pores there are subatomic gaps between the molecules were made out of.

Where?

Don't we have holes?

Well, that is the rub literally.

If you cut a clove of garlic in it and then rub that fresh raw end on your foot about an hour later, you will start to taste garlic in your mouth.

That's because the molecules that give garlic it's taste are small enough and have just the right properties to permeate skin cells in your foot, enter your bloodstream and reach your mouth.

But you are even holier than that.

Every second day and night, about 60 a 1,000,000,000 neutrinos from the sun passed through.

Just your thumbnail so clearly at small enough scales.

How many holes does a human have becomes a meaningless question.

Ultimately, the human body isn't a solid thing that could even have holes.

It's just a loose constellation of atoms and molecules.

But if we accept a minimum whole size, the answer becomes pretty interesting.

And a good choice for this minimum is about 22 60 million THS of a meter, about the width of a human hair.

A magic spaceship 60 microns wide could fly into your pores like they were giant holes that craters.

But it couldn't continue on through the vasculature at the other end or diffuse through cells or slip between molecules.

And that is significant.

It highlights the fact that not all holes are equal.

A 60 micron wide ship or string could be threaded into your mouth and come out somewhere else, but it couldn't do that by entering a poor or hair follicle.

This makes the G I tract what engineers call Ah through, whereas pours Yuri throws nipples, ears, hair follicles, birth canals and the Sinuses are blind holes.

They could be entered, but eventually dead in usually at narrow capital Aires permeable only buy things smaller than a single blood so, and the determination to not be stopped.

The eyeball can be squeezed under, but you'll eventually be stopped by the Khan junk Taiba.

The Sinuses are nice big rooms in our schools, but the only way out is the same Ostia you came in through.

As for the ear, well, the ear is a blast to go inside.

But if you're 60 microns wide, the air tight ear drum will block further passage.

It's a blind hole now, altogether, counting all of your pores and hair follicles.

You've got millions of blind holes all over your body.

But are they actually holds?

That's a real humdinger because you know what a hole is, what what a whole really is.

It's a word a colloquial, fuzzy, imprecise Lex seem that refers to a host of disparate, utterly un reconcilable things that eludes a single precise mathematical definition.

In fact, holes might not even exist.

I mean, think about it.

If I eat Ah, whole doughnut, have I eaten the whole like?

Is the hole inside me or could I eat a doughnut without eating its hole?

Could I go to a store and buy Swiss cheese?

But leave the holes at the store?

Clearly, Holes are at best ontological e parasitic.

Their existence depends upon the existence of something else that they can inhabit or be a disturbance in.

Of course, the philosophy of holes rarely matters.

In your day to day life, you can call something a hole and context will do its work.

And people will know what you're talking about.

But take a look at this.

Does this have a hole in it?

Well, yeah, right.

Obviously.

Right here, there's a hole.

I can put my hand in it.

It can store things.

It's got a hole.

But now imagine that I could mould it like it was made out of clay and I molded it down into the shape of a drinking glass.

You could see how that could happen.

Right?

Well, does a drinking glass have a hole in it?

If this does, then this should too, right?

I mean, I didn't pinch the whole shut or glue anything together.

I sure Fine.

I mean, I could accept that a drinking glass technically has a hole in it.

But now imagine that I took this glass and I molded it out, and I widened its opening until I had a shape like this.

A bowl now.

Does a bowl have a hole in it?

Now?

We're really stretching the use of the word hole.

I mean, if someone said they're bull had a hole in it, I would think that it had a hole somewhere else and it was leaking.

But sure, let's call this a hole.

It's not very prototypical one, but I think you see where I'm going with this.

If I then molded the bowl and flattened its sides all the way out until I had a plate, a shape like this, well, does a plate have a hole in it?

Not really so if a plate doesn't have a hole in it.

But this shape did, and I continuously molded from here to the glass to the bowl to the plate.

And I never glued anything shut where the whole go.

Clearly, blind holes are pretty unique.

They can be removed without closing or pinching anything.

Shut.

Compare that to the through hole of a doughnut.

There is no way to remove a doughnuts through hole or add a new through whole without gluing stuff together, squishing things together that used to not be together or ripping pieces apart, poking a hole through and breaking it.

That is extremely significant, but let's go back to the body before we get Ah, head of ourselves.

The mouth is an entrance to both blind and through holes.

A 60 micron wide traveler could enter it, meander down the esophagus and keep going until they were well dumped out but turned down the trachea and they would dead end in the lungs.

Now the area of the throat behind the mouth is called the Pharynx.

It's a pretty chill place, except not really.

It's actually quite warm.

It plays a role in warming and moistening and filtering the air that we breathe before it enters the lungs, including the air we inhale through our nostrils.

Now each nostril leads into a separate nasal vestibule.

That's the tunnel that you could explore when you pick your nose.

Eventually, those tunnels meet and sniffed.

Air enters the nasal cavity.

A hollow, air filled room in your face.

Protruding from the walls of the nasal cavity, are mucus e Finns called the nasal Conkey, or terminates that warm and moist in the air that passes around them.

From there, the air flows via the pharynx down the trachea so your nostrils and your mouth are connected.

A string could go into your mouth or nose and come out your butt.

The nasal cavity is quite the hub.

I mean, your ear holes would almost lead into it, but the ear drum blocks the way.

If it didn't, there would be clear passage from the outside into the middle ear and then down the use station tube into the nasal cavity via an opening about here.

The you station tube controls air pressure in the middle here behind the ear drum, and it's normally collapsed shut.

But if the outside pressure is dramatically different than the air pressure in the middle ear, swallowing and yawning can gape it open, equalizing the pressure.

That's what happens when you pop your ears.

It's cool, but it's not a through hole, and that's what we're looking for.

And as it turns out, there are four more.

Four more orifice is that lead from the outside into this place, your nasal cavity and they are the lack Rimmel punk tha there is one near each of your eyelids.

There tiny openings, about 1/3 of a millimeter wide into which tears the fluid, constantly moistening and protecting your eyeball drain.

Once inside the lateral puncture, tears flow through nasal Akram el ducts tear ducts into your nasal cavity, which is why when you're making a lot of tearful stiffly, I have to blow your nose.

That's not snot.

That's mainly tears.

That the point is a 60 micron wide string could be pushed into any of your four lack rebel Ponca threaded through your tear ducks into your nasal cavity into the pharynx and then pushed all the way out your butt.

Pretty cool.

That gives us eight external openings that don't dead end.

But how many through holes is that?

I mean, how many holes does a straw have?

This clearly has two holes, but how many does this have?

Is it one whole that forks?

Is it to that combine?

Gosh, maybe it's three.

Well, what about this?

How many holes does this thing have?

Or this topology can help us answer every single one of those questions Here I have to essentially identical pieces of material.

Now they are no longer identical.

Where are they geometrically sure, their shapes are now different, but what didn't change about well.

That is what apology studies Topology is concerned with the properties that persist so long as something isn't ripped apart.

The famous joke that eight apologised doesn't know the difference between a doughnut and a coffee cup is based on the fact that a coffee cup can be gently continuously molded into a doughnut by simply stretching and squashing.

No cutting, gluing, ripping our sewing required apologists call these gentle, continuous transformations, Homie.

Oh, morph ISMs and the cutting and ripping and gluing that they disallow are exactly the kinds of actions required to make new holes or remove old ones.

So since a coffee cup and a doughnut, our homey amore FIC, they must have the same number of through holes, and they do one.

We can now more precisely describe the difference we saw earlier between blind holes and through holes and understand why we are separately counting them.

Not blind holes can be erased through ah, home your dwarfism.

As such apologists don't even really consider them.

They're just geometric disturbances.

Top A logical holes, on the other hand, cannot be massaged away.

And unlike a blind hole where what qualifies in what doesn't is a matter of opinion.

The number of through holes, a surface like your body and three dimensions has can be clearly defined if we are having a hard time counting through holes.

All we need to do is find something with an easy to count arrangement of through holes that it is homie amore Fick with.

But first, let's play around with some top.

A logical puzzles Here is a to hold doughnut with an infinitely long, unbreakable unmovable rod through one of its holes, without cutting or separating any part of the shape.

Can you figure out how to manipulate it such that the rod goes through both holes?

Pause the video.

If you want to think about it, remember this shape.

It looks like it might have three holes, right.

It's got a hole there, a second hole there and 1/3 hole here.

But if I flatten it, you can see that it only has two holes.

It has one there, and one there.

If the rod is threaded through the shapes such that one wire is in front and I choose one of the other wires to be the middle of the donut for example, this one.

Then the rod passes through just one of the two holes.

But if I choose the wire in the front to be the middle of the doughnut will, then the rod is seen to be passing through two holes.

Likewise, if you continuously deform our original two whole doughnut into the three tube thing and pick this tube to be the new middle, the rod is now going through two holes.

No cutting or gluing required one more puzzle without cutting or breaking.

Can you unlock this?

Shapes intertwined loops?

Well, pause.

If you want to figure it out yourself, here's a solution.

Simply inflate the bulb of the shape until you can skate a leg of each loop around until they're untangled.

And Tara Freedom All right, Let's define home.

You morph is, um, a little better.

We said it was a rubber sheet or clay like molding procedure with no cutting or breaking or gluing.

And that's a good introduction.

But honestly, you can cut all you want during a home you morph ism so long as you glue everything back together the way it was in the end.

More, more precisely, a home dwarfism is a bye jek tive, and by continuous function it's a function because it is a list of ordered pairs where each point starts is paired with where it goes, requiring that it be a bye Jek Xin means that it must be a special kind of function where there is a 1 to 1 correspondence between points and one object and in the other, no two points can map to the same location, and no point can get magically turned into multiple new points.

Basically, material cannot be added or subtracted by continuous means that any cuts made must be later mended perfectly, with points going back amongst the same neighboring points they had before in a home you morph ism.

If parts are scorched over, everything else must flow with the scooch, as if all the points are kind of sticky.

There is no smooth sliding along abrupt fissures.

The precise test for whether a function is by continuous is pretty cool.

At first, I consider a point in one arrangement.

Now where the function takes that point is its image.

Okay, Now I choose some neighborhood around the image with a radius larger than zero, and I consider all of the points within it.

If the function is continuous in this direction, I should be able to find a neighborhood around the pre image, the input point that Onley contains points that map inside the images neighborhood.

In this case, I can.

But in this case we've got an original point and where it went but given a neighborhood around where it went, every neighborhood around the original, no matter how small, will always contain some stuff that didn't make it over.

Which means points got separated but not put back.

So the function is not continuous bye.

Continuity means that a function must be continuous in both directions.