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Math is sometimes a real tease it seduces us with the beauty off reasoning geometrically in two and three dimensions
Where there's this really nice back-and-forth between pairs or triplets of numbers and spatial stuff that our visual Cortex is good at processing
for example if you think about a circle with radius one centered at the origin
You are in effect conceptualizing every possible pair of numbers x and y that satisfy a certain numerical
property that x² + y² is 1 and
The usefulness here is that a lot of facts that look opaque in a purely
analytic context become quite clear geometrically and vice-versa
Honestly this channel has been the direct beneficiary of this back and forth
Since it offers such a rich library of that special category of cleverness that involves connecting two
Seemingly disparate ideas, and I don't just mean the general back and forth between pairs or triplets of numbers and spatial reasoning
I mean this specific one between sums of squares and circles and spheres
It's at the heart of the video I made showing
how pi is connected to number theory in primes and the one showing how to visualize all possible pythagorean triples.
It also underlies the video on the Borsuk-Ulam theorem being used to solve
What was basically a counting puzzle by using topological facts about spheres?
There is no doubt that the ability to frame analytic facts geometrically is very useful for math
But it's all a tease - because when you start asking questions about quadruplets or quintuplets or 100 tuples of numbers
It's frustrating. The constraints on our physical space
Seem to have constrained our intuitions about geometry and we lose this back and forth
I mean it is completely reasonable to imagine that there are problems out there that would have clever and
Illuminating solutions if only we knew how to conceptualize, say, lists of 10 numbers as individual points in some space
for mathematicians or computer scientists or physicists
problems that are framed in terms of lists of numbers - lists of more than three numbers - are a regular part of the job.
And the standard approach to actually doing math in higher dimensions is to use two and three dimensions for analogy
but to fundamentally reason about things just analytically.
Somewhat analogous to a pilot relying primarily on instruments and not sight while flying through the clouds
now what I want to offer here is a hybrid between the purely geometric and the purely analytic views. A method for making the
analytic reasoning a little more visual in a way that generalizes to arbitrarily high dimensions and
to drive home the value of a tactic like this
I want to share with you a very famous example
where analogies with two and three dimensions
cannot help, because of something extremely counterintuitive that only happens in higher dimensions.
The hope though, is that what I show you here helps to make that phenomenon more intuitive
The focus throughout will be on higher-dimensional spheres.
For example, when we talk about a four-dimensional sphere
Say, with radius 1, centered at the origin.
What that actually is is the set of all quadruplets of numbers
x, y, z, w, where the sum of the squares of these numbers is 1.
What I have pictured here now is multiple
three-dimensional slices of a 4D sphere
projected back into three dimensions
but it's confusing and
even if you do wrap your head around it,
it just pushes the question back, to how you would think
about a 5- or a 6- or a 7-dimensional sphere
and more importantly squinting your eyes to understand a projection like this
is not very reflective of what doing math with a 4D sphere actually entails.
Instead, the basic idea here will be to get very literal about it and to think about four separate numbers
I like to picture for vertical number lines with sliders to represent each number.
Each configuration of these sliders is a point in 4D space: A quadruplet of numbers.
And what it means to be on a 4D unit sphere centered at the origin,
is that the sum of the squares of these four values is 1.
Our goal is to understand which movements of these sliders correspond to movements on the sphere.
To do that, it helps if we knock things down to two dimensions where we can actually see the circle.
So ask yourself,
What's a nice way to think about this relation that x²+y² is 1?
Well, I like to think of the value of x² as the real estate belonging to x,
and likewise the value of y² is the real estate belonging to y,
and that they have a total of 1 unit of real estate to share between them.
So moving around on the circle
Corresponds to a constant exchange of real estate between the two variables
Part of the reason I choose this term is that it lets us make a very useful
Analogy: That real estate is cheap near zero, and more expensive away from zero.
To see this, consider starting off in a position where x=1 and y is zero.
Meaning x has all of the real estate to itself
which in our usual geometric picture means we're on the right most point of the circle.
If you move x down just a bit to 0.9 the value of x²
changes to 0.81, so it has in effect given up 0.19 units of real estate
But for y² to increase by that same amount
Y has to move an entire 0.44 units away from zero more than four times the amount that x moved
in other words, X changed a little to give up expensive real estate,
So that y could move a lot and gain the same value of cheap real estate.
In terms of the usual circle drawing this corresponds to the steep slope near the right side.
A small nudge in x allows for a very big change to y
Moving forward, let's add some tick marks to these lines to indicate
what 0.05 units of real estate looks like at each point
that is how much would x have to change so that the value of x² changes by 0.05
As you walk around the circle the trade-off in value between x² and y²
gives this piston looking dance motion, where the sliders are moving more slowly away from 0
Because real estate is more expensive in those regions.
There are just more tick marks to cover per unit distance
Also, a nice side effect of the term real estate is that it aligns naturally with the fact that it comes in units of distance squared.
So the square root of the total real estate among all coordinates gives us the distance from the origin.
For a unit sphere in 3 dimensions the set of all triplets x,y,z, where the sum of their squares is 1
All we have to do is add a third slider for z
But these three sliders still only have the one unit of real estate to share between them
To get a feel for this imagine holding x in place at 0.5,
where it occupies 0.25 units of real Estate
What this means, is that y and z can move around
in the same piston dance motion we saw before
as they trade off the remaining 0.75 units of real Estate.
In terms of our typical way of visualizing a sphere,
this corresponds to slicing the sphere along the plane where x is 0.5
and looking at the circle formed by all of the choices for Y and Z on that sphere.
As you increase the value of x the amount of real estate left over for Y and Z is smaller
And this more constrained piston dance is what it feels like for the circular slice to be smaller
eventually once x reaches the value 1
there's no real estate left over, so you reach this singularity point where y and z are both forced to be 0
The feeling here is a bit like being a bug on the surface of the sphere
You are unable to see the whole sphere all at once instead
You're just sitting on a single point, and you have some sense for what local movements are allowed
In 4 dimensions and higher we lose the crutch of the global view that a spatial visual offers
But the fundamental rules of this real estate exchange remain the same
If you fix one slider in place and watch the other three trade off
This is basically what it means to take a slice of the 4D sphere to get a small 3D sphere in
Much the same way that fixing one of the sliders for the three dimensional case
Give us a circular slice when the remaining two were free to vary
Now watching these sliders move about and thinking about the real estate exchange is pretty fun
but it runs the risk of being aimless unless we have an actual high dimensional puzzle to sink our teeth into
so let's set aside the sliders for just a moment and bring in a very classic example of something that seems reasonable and
even dull in two and three dimensions
But which is totally out of whack in higher dimensions.
To start, take a 2x2 box centered at the origin
Its corners around the vertices (1, 1), (1, -1), (-1, 1) and (-1, -1).
Draw four circles each with radius one centered
At these four vertices so each one is tangent to two of its neighbors
Now I want you to think of the circle centered at the origin which is just large enough to be touching those corner circles
Tangent to each one of them
What we want to do for this setup and for its analogies in higher dimensions is find the radius of that inner circle
Here in two dimensions
We can use the pythagorean theorem to see that the distance from the origin
To the corner of the box is the square root of two which is around 1.414
Then you can subtract off this portion here, the radius of the corner circle, which by definition is 1
And that means the radius of the inner circle is square root of 2 minus 1 or about 0.414
No surprises here, that seems pretty reasonable.
Now do something analogous in three dimensions:
draw a 2 x 2 x 2 cube.
Whose corners have vertices (1, 1, 1), (1, 1, -1), on and on and on.
and then we're gonna take eight different spheres each of which has a radius one and
Center them on these vertices so that each one is tangent to three of its neighbors
Now again think about the sphere centered at the origin which is just large enough to be barely touching those eight corner spheres
As before, we can start by thinking about the distance from the origin to the corner of the box
say the corner at (1, 1, 1)
By the way
I guess I still haven't yet explicitly said that the way distances work in higher dimensions is
Always to add up the squares of the components in each direction and take the square root
If you've never seen why this follows from the pythagorean theorem just in the two-dimensional case
it's actually a really fun puzzle to think about and I've left the relevant image up on the screen for any of you who want
want to pause and ponder on it.
Anyway, in our case the distance between the origin and the corner (1, 1, 1) is the
Square root of 1²+1²+1² or square root of 3, which is about 1.73
So the radius of that inner sphere is going to be this quantity
Minus the radius of a corner sphere, which by definition is 1.
And again, 0.73 seems like a reasonable radius for that inner sphere.
But what happens to that inner radius as you increase dimensions?
Obviously the reason I bring this up. Is that something surprising will happen,
and some of you might see where this is going.
But actually don't want it to feel like a surprise -
as fun as it is to wow people with counterintuitive facts in math
The goal here is genuine understanding, not shock
For higher dimensions we'll be using sliders to get a gut feel for what's going on
But since it's kind of a different way of viewing things it helps to get a running start by looking back at how to analyze
the two and three-dimensional cases in the context of sliders
First things first,
how do you think about a circle centered at a corner, like (1, -1)?
Well previously for a circle centered at the origin
the amount of real estate belonging to both x and y was dependent on their distance from the number 0
And it's the same basic idea here as you move around the center
It's just that the real estate might be dependent on the distance between each coordinate and some other number
so for this circle centered at (1, -1)
the amount of real estate belonging to x is the square of its distance from 1
Likewise the real estate belonging to y is the square of its distance from negative 1
Other than that the looking feel with this piston dance trade-off is completely the same
for simplicity we'll only focus on one of these circles the one centered at (1, 1)
Now ask yourself
What does it mean to find a circle centered at the origin?
Large enough to be tangent to this guy when we're thinking just in terms of sliders
Well, notice how this point of tangency happens when the x and y coordinates are both the same
Or phrased differently, at the point of this corner circle closest to the origin
The real estate is shared evenly between x and y
This will be important for later. So let's really dig in and think about why it's true
Imagine perturbing that point slightly.
Maybe moving x a little closer to 0 which means y would have to move a little away from 0
The change in x would have to be a little smaller than the change in y
Since the real estate it gains by moving farther away from 1 is more expensive than the real estate that y
loses by getting closer to 1
But from the perspective of the origin point (0, 0) that trade-off is reversed.
The resulting change to x² is
smaller than the resulting change to y² since when real estate is measured with respect to (0, 0)
That move of y towards 1 is the more expensive one
What this means is that any slight perturbation
Away from this point where real estate is shared evenly results in an increasing distance from the origin
The reason we care
Is that this point is tangent to the inner circle
So we can also think about it as being a point of the inner circle
And this will be very useful for higher dimensions. It gives us a reference point to understanding the radius of that inner circle
Specifically you can ask how much real estate is shared between x and y at this point
when real estatemeasurements are done with respect to the origin (0, 0)
For example down here in two dimensions
both x and y dip below 0.5 in this configuration
So the total value x²+y² is going to be less than 0.5²+0.5²
Comparing to this halfway point is really going to come in handy for wrapping our mind around what happens in higher dimensions
Taking things one step at a time, let's bump it up to three dimensions
Consider the corner sphere with radius 1 centered at (1, 1, 1).
The point on that sphere that's closest to the origin
Corresponds to the configuration of sliders where x, y and z are all reaching down toward 0 and equal to each other
Again, they all have to go a little beyond that halfway point because the position 0.5
only accounts for 0.5² (or 0.25) units of real estate
so with all three coordinates getting 1/3 of a unit of real estate they need to be farther out.
And again, since this is a point where the corner sphere is tangent to the inner sphere,
it's also a point of the inner sphere.
so with reference to the origin (0, 0, 0)
Think about the amount of real estate shared between x, y, and z in this position corresponding to the tangent point
It's definitely less than 0.75
Since all three of these are smaller than 0.5
so each one has less than 0.25 units of real estate and
Again, we sit back and feel comfortable with this result, right?
The inner sphere is smaller than the corner spheres
But things get interesting when we move up into four dimensions
Our 2x2x2x2 box is going to have 16 vertices
at (1, 1, 1, 1), (1, 1, 1, -1) and so on,
with all possible binary combinations of 1 and -1
What this means, is that there are 16 units spheres centered at these corners,
each one tangent to four of its neighbours
As before, we'll just be focusing on one of them,
the one centered at (1,1,1,1).
the point of the sphere closest to the origin
Corresponds to the configuration of sliders where all four coordinates reach exactly halfway between 1 and 0.
And that's because when one of the coordinates is 0.5 units away from 1
it has 0.25 units of real estate with respect to the point 1.
We do the same trick as before
Thinking of this now as a point of the inner sphere and measuring things with respect to the origin
But you can already see what's cool about four dimensions:
as you switch to thinking of real estate with respect to (0, 0, 0, 0)
It's still the case that each of these four coordinates has 0.25 units of real estate
Making for a total of 1 shared between the four coordinates
In other words that inner sphere is precisely the same size as the corner spheres
This matches with what you see numerically by the way
where you can compute the distance between the origin and the corner (1,1,1,1) is the square root of four,
And then when you subtract off the radius of one of the corners spheres what you get is 1.
But there's something much more satisfying about seeing it, rather than just computing it in particular
Here's a cool aspect of the fact that that inner sphere has radius one
move things around so that all of the real estate goes to the coordinate x and
You'll end up at the point (1,0,0,0).
this point is actually touching the 2x2x2x2 box.
And when you're stuck thinking in the two or three dimensional cases,
this fact, that the inner sphere has radius one,
The same size as the corner spheres and that it touches the box
Well it just seems too big
But it's important to realize this is fundamentally a four dimensional phenomenon, and you just can't cram it down into smaller dimensions
But things get weirder. Let's knock it up to five dimensions.
In this case, we have quite a few corner spheres:
32 in total.
But again, for simplicity, we'll only be thinking about the one centered at (1, 1, 1, 1, 1)
Think about the point of the sphere closest to the origin where all five coordinates are equally splitting the one unit of shared real-estate
This time, each coordinate is a little higher than 0.5
if they reach down to 0.5
Each one would have .25 units of real estate giving a total of 1.25 which is too much?
But, the tables are turned when you view this as a point on the inner sphere
Because with respect to the origin this configuration has much more than one unit of real estate
Not only is every coordinate more than .5 units away from zero
But the larger number of dimensions means that there's more total real estate when you add it all up
specifically you can compute that the radius of that inner sphere is about 1.24
The intuitive feel for what that means,
is that the sliders can roam over more territory than what just a single unit of real estate would allow
One fun way to see what this means is to adjust everything so that all of the real estate goes to just one coordinate
Because this coordinate can reach beyond 1, what you are seeing is that this five dimensional inner sphere
Actually pokes outside the box
But to really get a feel for how strange things become as a last example. I want to jump up into ten dimensions
remember, all this means is that points have ten coordinates for a sphere with radius one a
Single unit of real estate must be shared among all ten of those coordinates
as always the point of this corner sphere closest to the origin is the one where all ten coordinates split the real estate evenly and
here you can really see just how far away this feels from the origin or
phrased differently, that inner sphere is allowed to have a very large amount of real estate.
In fact, you can compute that the radius of the inner sphere is about 2.16
and viewed from this perspective,
where you have tenfold the mentions to share that real estate
Doesn't it actually feel somewhat reasonable that the inner sphere should have a radius more than twice as big as all those corner spheres?
To get a sense for just how big this inner sphere is,
look back in two dimensions and imagine a 4x4 box
bounding all four circles from the outside
Or go to three dimensions, and imagine a 4x4x4 box
bounding all of those corner spheres from the outside
Way up here in ten dimensions, that "inner sphere" is actually large enough to poke
Outside of that outer bounding box since it has a diameter bigger than four
I know that seems crazy
But you have to realize that the face of the box is always two units away from the origin
no matter how high the dimension is,
and fundamentally, it's because it only involves moving along a single axis
But the point (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) which Determines the inner spheres radius is
Actually really far away from the center all the way up here in 10 dimensions
and it's because all 10 of those dimensions add a full unit of real estate for that point.
And of course as you keep upping the dimensions that inner sphere just keeps growing without bound
Not only is it poking outside of these boxes, but the proportion of the inner sphere lying inside the box
Decreases exponentially towards zero as the dimension keeps increasing
So taking a step back,
one of the things I like about
Using this slider method for teaching
is that when I shared it with a few friends
The way they started to talk about higher dimensions became a little less
Metaphysical and started to sound more like how you would hear a mathematician talk about the topic
Rather than skeptically asking whether or not 10 dimensional space is a real thing
Recognizing that it's exactly as real as numbers are
People would actually probe at what other properties high dimensional spheres have and what other shapes feel like in terms of sliders
This box situation is just one in a number of things that feel very crazy about higher dimensional spheres
And it's really fun to think about these others in the context of sliders and real estate
It's obviously limited
I mean
You're a bug on the surface of these objects only getting a feel for one point at a time and for the rules of movement
Also, Geometry can be quite nice when its coordinate free and this is the opposite of that
But it does give a foothold into thinking about high dimensional shapes a little more concretely
Now you could say that viewing things with sliders is no different from thinking about things purely analytically
I mean, it's honestly a little more than representing each coordinate literally. It's kind of the most obvious thing you might do
but this small move makes it much more possible to play with the thought of a high dimensional point and
Even little things like thinking about the squares of coordinates as real estate
Can shed light on some seemingly strange aspects of high dimensions like just how far away the corner of a box is from its center
If anything the fact that it's such a direct representation of a purely analytic description is
Exactly, what makes it such a faithful reflection of what genuinely doing math in higher dimensions entails
we're still flying in the clouds trusting the instruments of analytic reasoning, but this is a
Redesign of those instruments; one which better takes advantage of the fact that such a large portion of our brains goes towards image processing.
I mean, just because you can't visualize something doesn't mean you can't still think about it visually
Before you go, I've got a couple announcements for you guys, including a Q&A follow-up that I owe you. First off,
I want to tell you about who helped to support this video
this is just such a good example of strong alignment between sponsor and content creator.
A lot of people ask me for advice on where and how to learn more math and more than anything I cannot emphasize enough
How important it is to
Prioritize active problem-solving with things like lectures and readings serving as a supplement to that main goal
And there's an irony in me saying that since the videos I create are a form of passive education
But my real hope with these is that they inspire people to go and actively play with math is one of the best places on the internet to do that
I'm a member of their subscription service, and I think anyone watching this video would enjoy their course on the joy of problem solving or
aptly enough outside of the box geometry
Also their complex algebra course is really good
But there's just many many good things to choose from both in and out of math
the user experience is one of going through a very thoughtfully curated list of puzzles and questions and
Honestly, it's just really fun
If you follow the link that lets them know that you came from here
Also, I'm starting a podcast. It's with two very smart guys each named ben where we talk about education and various other things
Some of you may remember that last video I elicited Q&A questions and the first episode includes my responses to many of them
For the version of it up on YouTube I actually had some fun and made the background video just a point wandering aimlessly
Around on a 10 dimensional sphere for an hour, and it's weirdly captivating to watch
so if you want to learn stuff go check out brilliant if you want more passive consumption check out the podcast and
As always if you want to help out the channel consider sharing the video or supporting on Patreon