- I'm Jordan Ellenberg, mathematician.
Let's answer some questions from the internet.
This is Geometry Support.
[upbeat music]
@s39gsy asked, "Who tf created geometry?"
Nobody created geometry. Geometry was always there.
It's just part of the way we interact
with the physical world.
The person who first codified it
and formalized it was somebody named Euclid
who lived in North Africa around 2,000 years ago,
and we also know that a lot of what he wrote down
was the work of a lot of other people
that he was collecting and putting in written form.
But this idea of geometry as the set of formal rules
we use to carefully put together demonstrations of facts
about angles, triangles, circles, et cetera,
that's when it sort of stops being purely intuitive
and starts being something we can put in a book.
@alien_searcher asks, "New shapes are just discovered?"
Yes, absolutely.
New shapes are just discovered all the time.
One of the big misconceptions about math that people have
is that math is finished.
People who are geometers are typically thinking
about crazy stuff that's going on in high dimensions
with all kinds of crazy curvature,
but four-dimensional shapes are in some sense
just as real as three-dimensional shapes.
We just have to kind of train our minds
to be able to perceive what shapes in those dimensions
like a hypercube or a tesseract would look like.
@inkbotkowalski asked,
"Wait, wait--a tesseract is a real thing?"
Definitely, yes.
A tesseract is another name
for what's usually called in math a hypercube.
MCU did not create the idea of the tesseract
being in popular science fiction.
That really comes in in Madeleine L'Engle's book
"A Wrinkle in Time."
All right, here you have a square, a two-dimensional figure,
and here you have its three-dimensional counterpart, a cube.
A cube you can think of as two squares,
the top square and the bottom square,
and then you sort of connect them together.
If the cube is the three-dimensional figure
and the square is the two-dimensional figure,
what would be the four-dimensional figure?
I guess the hypercube would have to be something
that was two cubes joined together
and it would have to have twice as many corners
as the cube does or 16.
And now I've gotta connect each corner of the little cube
to the corresponding corner of the big cube.
This is our picture of the hypercube.
And now you may say like, are there really four dimensions
or is that just an invention?
Well, you know what, when we do regular geometry,
we're working in a perfectly flat plane.
Does that exist in the real world?
Like probably not as a physical object.
The two-dimensional plane or three-dimensional space
are just as much of an abstraction
as four-dimensional space.
Okay, @ClaudioJacobo asks,
"If algebra is the study of structure, what is geometry?"
Algebra is the logical and symbolic, right?
It's that side of your brain. Geometry is different.
Geometry is physical, geometry is primal,
and like doing mathematics makes use of this tension
between the algebraic side of our mind
and the geometric side.
@3omega2 asks, "How can I use the Pythagorean Theorem
to solve my problems in life?"
Look, I'm gonna be honest with you, I can't quite imagine
what problem you might have in your life
that would be solved by the Pythagorean Theorem.
The problem that the Pythagorean theorem solves
is the following one.
If for some reason I have some distance I wanna traverse,
and if I know how far west you have to go to get there,
and then how far north,
and I happen to know these two distances,
then the Pythagorean Theorem allows you
to compute this diagonal distance, which we call C,
but we can also write it as the square root
of A squared plus B squared.
Is this the problem you're facing in your everyday life?
If it is, you're in luck,
the Pythagorean Theorem is here for you.
But in most cases, it is not.
@TMSayan asks, "What is special
about Pringle's hyperbolic paraboloid geometry?"
The Pringle is a wonderful geometric form.
What's special about it is this point right here
at the center of the Pringle,
if I move from left to right, I can't help but go up,
so it seems like I'm at the bottom of the Pringle.
But if I move from front to back,
I can't help but go down from the center,
So it's somehow simultaneously at the top.
It's a peak and a valley at the same time.
And this special kind of point,
which is called a saddle point in math,
is what gives the Pringle
its particularly charming geometry.
@DrFunkySpoon asks,
"Sucker MCs maintain cool under pressure,
but who [beeps] with geometry like MC Escher?"
What a good question.
MC Escher, beloved artist of all mathy people,
Escher was famous for studying
and using in his art what are called tessellations,
ways of taking a flat plane and covering it with copies.
It was something he learned actually in part
from hanging out at the Alhambra,
this incredible palace from Islamic Spain.
When you go to the Alhambra,
you see these incredibly intricate,
but also very repetitive figures,
which by repetition across the entire wall,
it becomes very complicated and rich.
And that's the feature of a tessellation.
"Who [beeps] with geometry like MC Escher?"
The answer is the unnamed architects
of the Alhambra in Granada, Spain.
@Raspberry_pi asks, "How many holes are there in a straw?"
Fortunately, I always bring a straw with me wherever I go.
How many holes are there in it?
There are the one holers who feel that well look,
there's like one hole, it goes all the way through.
Like what more is there to say?
And there are the two holers whose view is
there's a hole at the top of the straw
and there's a hole at the bottom of the straw.
For the people who think there's two holes,
I would say imagine this straw if you can
getting shorter and shorter.
Like imagine I sort of cut it and it was half as long
and I cut it again until it's so short
that it's actually like shorter than the distance around.
A little bit like this.
Does this have one hole in it or two?
How many holes does a bagel have in it?
That's basically the same shape as this.
If you say a bagel has two holes, I think we all agree
that would be like a very weird thing to say about a bagel.
So now I'm talking to you, triumphant one holers.
If you think this straw is one hole, let's say I take it
and I pinch the bottom like this.
How many holes are there in it now?
There's just like the one hole at the top.
I mean, you could fill this with water, right?
It's basically a bottle.
How many holes are there in the water bottle?
Just the one at the top that you drink out of, right?
But if it has one hole now and I poked a hole in the bottom
and I opened up the bottom, how many holes would it have?
It's gotta have two, right?
I think the way to think about the straw
is that, yeah, there's two holes,
but one of them is the negative of the other.
Top hole plus bottom hole equals zero.
That sounds like an insane thing to say.
Both the one holers and two holers are right
in a way as long as they're willing to learn
about the arithmetic of holes.
@LiberatedSoul_ asks, "The golden ratio in art? Photography?
Is that something to do with perfect composition?"
Yes, the golden ratio is very popular.
It's a number, a kind of unassuming number.
It's about 1.618.
And there have always been people
who felt that this particular number
had some kind of mystical properties.
Why that number?
Well, one way of describing it is that
if I have a rectangle whose length and width
are in that proportion, a so-called golden rectangle,
it has a special property,
which is that if I cut the rectangle
to make one part of it a square,
what's left is again a golden rectangle.
No other kind of rectangle has that property.
Some people would say like you can find it in nature,
like, for instance, I have here the shell
of some kind of invertebrate like a well.
In here, we could find the golden ratio.
They say you can find it in a pine cone.
I mean, I think its mystical significance
has been much overrated.
So I don't wanna sound too salty about this,
but I think you shouldn't look to it
to improve your stock portfolio, help you lose weight,
or help you find the prettiest rectangle.
@zohaibrafiq83 asks, "Why are honeycombs hexagons?"
One thing I can tell you is that
when the bees build the honeycombs, they're not hexagons.
They actually build them round
and then something forces them into that hexagonal shape.
So there's a lot of controversy about this.
For instance, why hexagons are not a grid
of squares or triangles?
And there are people who will say,
"Well, there's an efficiency argument.
Maybe this is the way to give the honeycomb
structural integrity using the least amount of material."
I'm not sure that's completely convincing,
but that's the least one theory that people have.
@BibbitE asks, "How are there
so many different types of triangles?"
This actually speaks to kind of a deep division in math,
the so-called three-body problems,
one of the hardest problems in mathematics.
With two points,
you're making a line segment that looks like this
and there's not a lot of variety among line segments.
They're all basically the same.
Three points, totally different story.
Triangles come in an infinite variety of variations.
I mean, you could have one
that's like very narrow like this.
You could have one that's nice and symmetric,
our friend, the equilateral triangle like that.
You could have a right triangle with a nice right angle.
I could just keep on drawing triangles in this little board
and each one would look different from all the others.
And that is the difference
between two and three problems involving two points.
Simple problems involving three points,
already a completely infinite variety.
@tzack16 asks, "What is the random walk theory
and what does it mean for investors?"
Imagine a person with no sense of purpose.
Every day they wake up
and they walk a mile in one direction or another.
You could track that person's motion
over a long period of time.
That purposeless, mindless, unpredictable process?
A lot of people think the stock market
basically works pretty much like that.
This is something that was worked out
actually a really long time ago,
around 1900 by Louise Bachelier.
He was studying bond prices, trying to understand
what are the forces that govern these prices.
And he had this sort of incredible insight, which is to say,
what if those prices just every day
they might happen to go up
or they might happen to go down purely by random chance?
And what he found is that if you model prices that way,
it looks exactly like the prices in real life.
@vikrampunt asks, "Can you believe
you can take the circumference of any circle
and divide it by its diameter
and you will always get exactly pi?"
Yeah, I totally believe that.
And in fact, I would say I relish it
because it's one of the things that makes circle circles.
There's really only one kind of circle.
It could be small or it could be big,
but this one is just a scaled up version of this one.
Whatever the diameter of this circle is,
if this guy has a diameter too,
if this diameter is seven times as big as this one,
then also this circumference,
that's the total distance around the circle,
is seven times the size of this one.
So in particular, the ratio between the circumference
and the diameter is the same in both cases
and that constant ratio, pi, it's about 3.1415.
I don't care so much what pi is
to 10 decimal places or 20 decimal places.
Mathematically, what's important
is that there is such a thing as pi,
that there is a constant that governs all circles
no matter how big or how small.
@taskeenhansa asks, "What is the worst section in maths
and why is it Euclidean geometry?"
Okay, that stings a little bit.
Geometry is the cilantro of math.
Everybody either loves it or hates it.
It's the only part of math
where you're asked to prove that something is true
rather than just getting the answer to a question.
Euclidean geometry is geometry of the plane.
There's lots of other geometries, non-Euclidean geometries.
You guys probably know the fact
that the sum of the angles of a triangle
is supposed to be 180 degrees,
and in Euclid world that's true.
But on a curved surface like a sphere, that's totally wrong.
All right, my lines are not as straight as they might be,
but if you look at this kind of bulgy triangle
and it's three angles, their sum is gonna be around 270,
like way bigger than 180.
And that's a fundamentally non-Euclidean phenomenon
that can only happen in a curved space.
We now know thanks to Einstein
that space actually is curved.
When he revolutionized physics
in the beginning of the 20th century,
the miracle is that non-Euclidean geometry
was already there for him to use.
The mathematicians had already understood
how curved space could work well in time for Einstein
to realize that the world we actually live in is like that.
@wissencmk asks, "Inception, is it really a thesis
on manifold and geometry and four-dimensional space?"
Inception is a little bit more
like what we call in geometry a fractal,
which has the property that it's self-similar,
that if you zoom in on it,
you see a smaller replica of the whole thing.
The more you zoom in, the more detail you see.
And that seems to me
the sort of spirit of the movie Inception.
So I think I'm gonna call that a fractal movie.
@thinkBIGkids asks, "Is there any better way
to teach transformational geometry
than original Nintendo Tetris?"
I spent way too much time playing Tetris in college,
so I've thought about it a lot, tried to make excuses
for why that was actually productive use of my time.
If you take a modern geometry class,
it's not just about angles and circles and shapes.
They also talk about transformations.
They say what happens if you take this shape and reflect it
or take this shape and rotate it?
Tetris teaches you that skill.
Imagine this little dude like marching down the screen.
You have to very quickly mentally figure out
what it's gonna look like rotated
and which version of it
is gonna fit into a space where you need it.
And so I think you can think of Tetris
as like a very, very efficient
and somewhat stressful training device
for exactly that mental rotation skill
that we're now trying to teach kids in geometry.
@meris_crabtree has a joke for me.
"A Mobius strip walks into a bar, sobbing.
The bartender asks, 'What's wrong, buddy?'
The Mobius strip replies, 'Where do I even begin?'"
You'd think in my profession
you'd think I would've heard all the math jokes there are,
but every once in a while I hear a new one.
So a Mobius strip is a geometric figure
with a rather unusual quality
that's not visible to the naked eye,
which is that it only has one side.
I'm gonna Mark a little spot with an X.
And now I'm gonna take my finger, put it on the X,
and I'm gonna start moving my way around the band.
Watch me very closely. I'm not switching sides.
I'm moving, I'm moving.
My finger is staying on the band.
And look where I am.
I'm sort of in the same spot, but I'm on the other side.
Somewhat miraculously,
what appear to be two different sides of the band
are actually connected.
@Rebecca57219 asks, "Anyone currently in a position
where you use Pascal's triangle?"
I definitely use Pascal's triangle
and the numbers in it all the time.
Here, I have one with me.
There's these numbers written in the form of a triangle,
and the rule, if you wanted to make one of these yourself,
is just that each number
is the sum of the two numbers above it.
So right, see how the six is the sum of three and three,
and then if I didn't know what went in here,
I could look above it and see a four and a six.
Oh, those add up to 10, so I have to put a 10 there.
So the cool thing is that these numbers
actually mean something.
Actually, they mean a lot of different things,
but one of my favorite things that they mean
is they record the likelihood of various outcomes
in a random scenario like flipping coins.
So how do you turn these numbers into probabilities?
Well, if you were to add up all six of these numbers,
you would get 32.
So you should really think of these numbers as fractions,
like one out of 32, 5 out of 32, 10 out of 32.
Those fractions are probabilities.
If I flip a coin five times,
there's six things that can happen.
I can get zero heads, one head, two heads,
three heads, four heads.
Okay, well I ran out of fingers, or five heads,
that's a sixth possibility.
And those correspond exactly to these six numbers
in the fifth row of Pascal's triangle.
If you did an experiment
and you flipped five coins thousands and thousands of times,
the proportion of those times
that you would get two heads out of five
would converge to 10 out of 32.
@harpua71burner asks,
"Why does the shape of a district matter?"
And I'm gonna assume that the question here
is about congressional districts.
The reason is that if you see one with a very strange shape,
that is an indication that someone has designed
that district for a political purpose.
I'm sorry to say
that like rather advanced mathematical techniques
are used in order to effectively explore
that geometric space to find the most partisan advantage
that you can squeeze out of a map.
Legislators choosing their voters
instead of the voters choosing their legislators.
So that's why we care.
@pw1111, okay, I don't know how many ones there are,
there's a lot of ones.
"Why do GPS systems need to use geometry
based on the sphere in order to work?"
What GPS essentially does is there's a bunch of satellites
which are in positions that we know.
They can tell you what is your distance
when you're somewhere on the Earth
from each one of those satellites.
And knowing those numbers
is actually enough to specify your exact location.
Let's say I know I'm exactly 5,342 kilometers
from a given satellite.
The set of all points that are at exactly that distance
from the satellite is a sphere
whose center is that satellite.
That's what the definition of a sphere is.
It's the set of all points
at a fixed distance from a given center.
If I have two satellites,
I'm at the intersection of two spheres.
Once you have four or more of those spheres,
they're never gonna have more than one point in common.
That's exactly the geometry that underlies GPS.
@Quantum_Stat asked,
"What can the geometry of deep learning networks tell us
about their inner workings?"
I'm gonna tell you the strategy that it uses.
It's basically a very intensive form of trial and error.
We make sort of some modest change to our behaviors
and sort of see if it gives us better results.
And if it does, we keep doing that thing.
I think of that as a kind of exploration of a space.
Geometry in the modern sense is any context
in which we can talk about things being near and far.
We know what it means for two people
to be near each other geographically.
Similarly, the space of all strategies
for recognizing a face, those have geometries too.
There are some strategies that are near each other
and some that are far away.
Any context in which we can talk about near and far,
whether that's the surface of the Earth
or a social network or your family
where you can talk about close relatives or far relatives,
I know I'm kind of sounding like I'm just saying
geometry is everything, but I'm gonna be honest,
that is kind of what I think.
Okay, so those are all the questions we have time for today.
I hope my answers made some sense
or messed with your mind a little, or best of all,
maybe did some combination of those two things.
Thanks for watching Geometry Support.