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hey Vsauce Michael 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 you are
just a bunch of meat packed around a
central Hulk or are you humans have more
than one hole right I mean we've got
nostrils and ears the whole sweetpea out
of the holes were born from nipples our
pores yes 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 half 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 its 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 billion
neutrinos from the Sun pass 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 can even have holes
it's just a loose constellation of atoms
and molecules but if we accept a minimum
hole size the answer becomes pretty
interesting and a good choice for this
minimum is about twenty to sixty
millionths 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 like 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 pore or hair
follicle
this makes the GI tract what engineers
call a through-hole where as pores
urethras nipples ears hair follicles
birth canals and the sinuses are blind
holes they can be entered
but eventually dead end usually at
narrow capillaries permeable only by
things smaller than a single blood cell
and the determination to not be stopped
the eyeball can be squeezed under but
you'll eventually be stopped by the
conjunctiva the sinuses are nice big
rooms and our skulls 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 airtight
eardrum 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 holes
that's a real humdinger because you know
what a hole is what what a hole really
is it's a word a colloquial fuzzy
imprecise lexeme that refers to a host
of disparate utterly unreconciled things
that eludes a single precise
mathematical definition in fact holes
might not even exist I mean think about
it if I eat a 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
ontologically 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 mold 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 -
right I mean I didn't pinch the hole
shut or glue anything together all right
sure fine I mean I can 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 their ball 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 a very prototypical
one but I think you see where I'm going
with this if I didn't 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'd the
hole 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
Donuts through hole or add a new through
hole 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 ahead
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 turn 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 can 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 mucusy fins called
the nasal Concha or turbinates 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
eardrum blocks the way if it didn't
there would be clear passage from the
outside into the middle ear and then
down the eustachian tube into the nasal
cavity via an opening about here the
eustachian tube controls air pressure in
the middle ear behind the eardrum and is
normally collapsed shut but if the
outside pressure is dramatically
different than the air pressure in the
middle ear swallowing and yawning can
get 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 orifices that lead from the
outside into this place your nasal
cavity and they are the lacrimal punctum
there is one near each of your eyelids
they're tiny openings about a third of a
millimeter wide into which tears
the fluid constantly moistening and
protecting your eyeball drain once
inside the lacrimal punctum ASA lacrimal
ducts tear ducts into your nasal cavity
which is why when you're making a lot of
tear fluid and have to blow your nose
that's not snot that's mainly tears the
point is a 60 micron wide string could
be pushed into any of your four lacrimal
punctum threaded through your tear ducts
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 hole that Forks
is it two that combined 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 two
essentially identical pieces of material
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now they are no longer identical or are
they geometrically sure their shapes are
now different but what didn't change
about them well that is what topology
studies topology is concerned with the
properties that persist so long as
something isn't ripped apart the famous
joke that a topologist 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 or sewing
required topologists call these gentle
continuous transformations
homeomorphisms 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 are homeomorphic 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 now blind holes
can be erased through a homeomorphism as
such topologists don't even really
consider them they're just geometric
disturbances topological holes on the
other hand cannot be massage it away and
unlike a blind hole where what qualifies
and 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 homeomorphic with but
first let's play around with some
topological puzzles here is a to hold
donut 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 a third 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 shape such that one
wire is in front and I choose one of the
other wires to be the middle of the
doughnut 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 donut well
then the rod is seen to be passing
through two holes likewise if you
continuously deform our original to hold
donut into the three tube thing and pick
this tube to be the new middle tada 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 tada
freedom alright let's define
homeomorphism 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 homeomorphism so long as you
glue everything back together the way it
was in the end more precisely a
homeomorphism is a bijective 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 by
jek ssin means that it must be a special
kind of function where there is a
one-to-one correspondence between points
in 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
homeomorphism if parts are scooched over
everything else must flow with the
scootch 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
now 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 preimage the
input point that only 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 by continuity
means that a function must be continuous
in both directions okay now that we can
homey amorphous let's start using it to
count holes
remember this shape it wasn't
immediately obvious earlier how to count
its holes but it is easy if we can use a
homeomorphism to turn it into something
with an easy to count number of holes
which we can both of these shapes are
homeomorphic they both have two holes to
see why
simply drag in mold and flow the opening
of one of these shapes holes into the
tunnel of the other and there we go
since we didn't cut or glue the number
of holes hasn't changed so this thing
just like this thing always had two
holes
okay what about a straw well short
informally it can often make sense
depending on the context to
differentiate between two openings in a
straw the one you put in your drink and
the one you put in your mouth but that
does not mean it has two holes it only
has one a straw is homeomorphic to a
torus both openings are part of the same
single hole but this process is a
homeomorphism and thus does not create
any new holes you can also see that
openings aren't holes by stretching one
of the straws openings until it becomes
the outer part of a doughnut there
really was only ever just one hole
but enough about straws let's get back
to the body we found eight external
openings orifices interconnected by
tunnels but openings aren't holes
they're parts of holes and we can count
those holes as it turns out at a scale
of 60 microns the human body has seven
through holes the human body is not a
doughnut it is a seven hold doughnut
this shape can be molded and stretched
into you first we choose a hole to be
the GI tract the mouth anus tunnel now
into this we roll half of the other
orifices okay now we've got something
that looks pretty dang human seven holes
with eight external orifices that meet
in a common space the nasal cavity if we
squish all the matter in towards the
tunnels will notice that our seven hold
torus is topologically equivalent to
four pairs of pants sewn together at the
waists your body isn't a doughnut it's a
bodysuit for a spider okay now to finish
let's make two of the legs the nostrils
and four of them the tear ducts one the
mouth and inflate the material into the
form of a head now let's inflate the
boundary of the final tube into the
shape of a body with its opening in the
rear and we've done it
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the human body is a seven hold doughnut
or is it for every piercing you have
that's one more hole in your body
well two more if the piercing goes
through a through hole like I don't know
if you had a thin piercing into your
face that went through a tear duct and
came back out or if something like a
bullet pierced into your chest through
your esophagus and came out the other
side that would mathematically count as
two new holes and there's more
some people have supernumerary lacrimal
punctum on their eyes each additional
punctum they have over four total adds
an additional hole to the standard seven
and remember the sinuses and the Ostia
connecting them into the nasal cavity
well they're just blind holes
depressions but as many as half of us
may have at least one accessory ostium
an extra hole connecting a sinus to the
nasal cavity well now we're talking
about a through hole you can enter one
opening and exit via another now these
may not be external orifices but for
every accessory Ostia you have that's
another topological hole you need to add
to your bodies total the thing is though
most of us have no idea how many
accessory Ostia we have unless you've
had serious sinus problems or have had
extensive scans of your nasal region
that have been studied from multiple
angles so to answer this videos question
the human body has millions of blind
holes like at least five million and at
Birth seven through holes it would be
better if there was a clear answer that
applied to all of us for our entire
lives or if finding out how many you had
right now was easier but you'll have to
speelunk in your sinuses to know for
sure and that's beautiful isn't it we
can rigorously define the properties of
holes in all sorts of dimensions and we
can study that temperatures at the
bottom of craters on Pluto but few of us
will ever truly know
the whole truth of our own bodies and as
always thanks for watching
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you