Solving 2D equations using color, a story of winding numbers and composition

There’s two things here, the main topic,
and the meta topic. The main topic is a neat algorithm for solving
many two-dimensional equations. That is, equations with two unknown real numbers,
or perhaps those involving a single unknown which is a complex number. So for example, if you want to find the complex
roots of a polynomial, or some of those million dollar zeros of the Riemann zeta function,
this algorithm will do it for you. This method can be super pretty, since there
is a lot of color involved, and the core underlying idea applies to all sorts of math beyond this
algorithm for solving equations, including a bit of topology, which I’ll talk about
afterwards. But what really makes this worth the 20 minutes
or so of your time is that it illustrates a lesson much more generally useful throughout
math, which is to try to define constructs which compose nicely with each other. You’ll see what I mean by that as the story
progresses. To motivate the case with functions that have
2d inputs and 2d outputs, let’s start off simpler, with functions that just take in
a real number and spit out a real number. If you want to know when one function, f(x),
equals another function, g(x), you might think of this as searching for when the graphs of
these functions intersect. Right? That tells you the input where both functions
have the same output. To take a very simple example, say f(x) is
x2, and g(x) is the constant function 2. In other words, you want to find the square
root of 2. Even if you know almost nothing about finding
square roots, you probably see quickly that 12 is less than 2 and 22 is bigger than 2,
and so you realize “Ah, there’s a solution somewhere between 1 and 2”. And then if you wanted to narrow it down further,
you might try squaring the halfway point, 1.5. This comes out at 2.25, a bit too high, so
now you focus on the region between 1 and 1.5. And so on. You can keep computing at the midpoint and
chopping your search space in half. Another way to think about this, which will
make it easier once we get up to higher dimensions, is to instead focus on the equivalent question
of when the difference between the two functions is 0. We found a region of inputs where this was
negative at one end, and positive at the other. And then we split this region in two, narrowing
our attention to a half whose outermost points again produced varying signs. We were able to keep going forever in this
way, taking each region with varying signs on its border and finding a smaller such region
among its halves, knowing that ultimately, we had to be narrowing in on a point where
we would hit exactly zero. In short, solving equations can always be
framed as finding when a certain function is 0. And to do that, use the we use the heuristic
“If f is positive at one point, and f is negative at another point, then we can find
some place in between where it’s zero (…at least, if everything changes smoothly, with
no sudden jumps)”. The amazing thing I want to show you is how
to extend this kind of thinking to two-dimensional equations; to equations between functions
whose inputs and outputs are both two-dimensional. For example, complex numbers are two-dimensional,
and the tool we develop here is perfect for finding solutions to complex equations. Since we’re going to be discussing 2d functions
so much,, let’s take a brief side-step to consider how we illustrate them. Graphing functions with 2d inputs and 2d outputs
would require 4 dimensions, which won’t work so well in our 3d world on our 2d screens,
but we still have a few good options. One is just to look at at both the input and
output space side-by-side. Each point in the input space moves to a particular
point in the output space, and I can show how moving the input point corresponds to
certain movements in the output space. All the functions we consider will be continuous,
in the sense that small changes in input cause small changes in output, without any sudden
jumps. Another option is to think of the arrow from
the origin of the output space to the output point, and attach a miniature version of that
arrow to the input point. This can give us a sense at a glance for where
a given input point goes, or where many different input points go by drawing a full vector field. (Unfortunately, this can also be a bit cluttered;
here, let me make all these arrows the same size, so we can more easily get a sense of
just the direction of the output at each point.) But perhaps the prettiest way to illustrate
2d functions, and the one we’ll use most in this video, is to associate each point
in the output space with a color. Here, we’ve used different “hues” (that
is, where the color falls along a rainbow or color wheel) to correspond to the direction
away from the origin, while we’ve used darkness or brightness to correspond to distance from
the origin. For example, focusing just on this ray of
outputs, all these points are red, but the ones closer to the origin are a little darker
and the ones further away are a little lighter. And focusing just on this ray of outputs,
all the points are green, and again, closer to the origin we get darker and further away
we get lighter. And so on, we’ve just assigned a color to
each direction, all changing continuously. (The darkness and brightness differences here
can be quite subtle, but for this video all we will care about is the directions of outputs,
and not their magnitudes; the hue, not the brightness. The only important thing about brightness
is just that near the origin, which has no particular direction, our colors will fade
to black.) Now that we’ve decided on colors for each
output, we can visualize 2d functions by coloring each point in the input space based on the
color of the point where it lands in the output space. I like to imagine many points from the input
space hopping over to the output space, which is basically one big color wheel now, getting
painted, and then hopping back to where they came from. This gives a way, just by looking at the input
space, to understand roughly where the function takes each point. For example, this stripe of pink points on
the left tells us that all those points get mapped to the pink direction, the lower left
of the output space. Those three points which are black with lots
of colors around them are the ones that go to zero. Alright, so just like the 1d case, solving
an equation of two-dimensional equations can always be reframed as asking when a certain
function equals 0. So that’s our challenge right now: Create
an algorithm that finds which input points a given 2d function takes to 0. Now, you might point out that if you’re
looking at a color map like this, by seeing these black dots you know where the zeros
of the function are…so, does that count? Well, keep in mind that to create this diagram
we’ve had the computer compute the function at all these pixels on the plane. But part of our goal here is to find a more
efficient algorithm that only requires computing the function on as few points as possible;
only having limited view of the colors of the plane, so to speak. Also, from a more theoretical standpoint,
it’d be nice to have a general construct which gives us conditions for whether or not
a zero even exists inside a given region. Now, in 1-dimension, the main insight was
that if a continuous function is positive at one point, and negative at another, then
somewhere in between it must be 0. How do you extend to two dimensions? You need some analog of talking about signs. Well, one way to think about what signs are
is as directions; positive means pointing right along a number line, and negative means
pointing left along that number line. Two-dimensional quantities also have directions,
although for them, the options are wider: they can point anywhere along a whole circle
of possibilities. So in the same way that for 1d functions,
we were asking whether a given function was positive or negative on the boundary of a
range, which is just two points; for 2d functions we will want to look at the boundary of a
region, which will be some loop, and ask about the direction of the function’s output along
that boundary. For example, we see that on this loop around
this zero, the output goes through every possible direction, all the colors of the rainbow;
red, yellow, green, blue, back to red, and everything in between along the way. But on this loop over here, with no zeros
inside, the output doesn’t go through every color; it only goes through some orangish
colors, but never, say, green or blue. This is promising; it looks a lot like how
things worked in 1d. Maybe, in the same way that if a 1d function
takes both possible signs on the boundary of a 1d region, there’s a zero somewhere
between, we might hypothesize that if a 2d function hits outputs of all possible directions,
all possible colors, on the boundary of a 2d region, somewhere inside that region it
must go to zero. Here, take a moment to really think about
if this should be true, and if so, why. If we start by thinking about a tiny loop
around some input point, we know, since everything is continuous, that our function takes it
to some tiny loop near the corresponding output. But look: most tiny loops of outputs barely
vary in color. If you pick any output point other than zero,
and draw a sufficiently tight loop near it, the loop’s colors will all be about the
same as the color of that point. A tight loop here will be all blue-ish; a
tight loop here will be all yellow-ish, etc. You certainly won’t get every color of the
rainbow. The only point you can tighten loops around
while still going through all the colors of the rainbow is the colorless origin: zero
itself. So it is indeed the case that if you have
loops going through every color of the rainbow, tightening and tightening and narrowing in
on a point, then that point must in fact be a zero. …And so we can set up our 2d equation solver
just like our 1d equation solver: when we find a large region whose border goes through
every color, split it in two, and look at the colors on the boundary on each half. In the example shown here, the border of the
left half doesn’t go through all colors; there are no points that map to the orange
and yellow directions, so we grey this area out as a way of saying we don’t want to
search it further. The right half does go through all colors,
spending a lot of time in the green direction, then passing yellow-orange-red, as well as
blue-violet. Remember, that means points of this boundary
get mapped to outputs of all possible directions, so we’ll explore it further, subdividing
again and checking the boundary colors of each subregion. The boundary of that top right is all green,
so we’ll stop searching there, but the bottom is colorful enough to deserve a subdivision. And just continue like this! Check which subregion has a boundary covering
all colors, meaning points of that boundary get mapped to all possible directions, and
keep chopping those subregions in half like we did in the 1d case. …Except…wait a minute…what happened
here? Neither of those last subdivisions on the
bottom right passes through all colors…so our algorithm stopped without having found
a zero… So being wrong is a regular part of doing
math. We had this hypothesis that led us to a proposed
algorithm, and clearly we were mistaken somewhere. Being good at math is not about being right
the first time, it’s about having the resilience to carefully look back and understand our
mistakes, and how to fix them. The problem here was that we had a region
whose border went through every color, but when we split it down the middle, neither
subregion’s border went through every color. We had no options for where to keep searching
next, breaking our zero-finder. In 1d, this sort of thing never happened:
any time you had an interval whose endpoints had different signs, when you split it, you
knew you were guaranteed to get some sub-interval whose endpoints still had different-signs. Put another way, any time you have two intervals
whose endpoints don’t change sign, when you combine them, you get a bigger interval
whose endpoints don’t change sign. But in 2d, you can take two regions whose
borders don’t contain every color, and combine them into a region whose border DOES contain
every color. And in just this way, our proposed zero-finder
can break down. In fact, you can have a big loop whose border
goes through every color, without there being any zeros inside We weren’t wrong in our claims about tiny
loops; when we said that forever-narrowing loops that go through every color had to be
narrowing in on a zero. But what made a mess for us is that, the “Does
my border go through every color or not?” property doesn’t combine in a nice, predictable
way when you combine regions. But don’t worry! It turns out, we can modify this slightly,
to a more sophisticated property that DOES combine nicely, giving us what we want. Instead of simply asking whether we find every
color at some point along a loop, let’s keep track more carefully of how those colors
change as as we walk along the loop. Let me show what you I mean with some examples. I’ll keep a little color wheel here in the
corner to help us keep track. When colors along a path of inputs move through
the rainbow from red to yellow, or yellow to green, green to blue, or blue to red, the
output is swinging clockwise. On the other hand, when the colors move the
other way through the rainbow, from blue to green, green to yellow, yellow to red, or
red to blue, the output is swinging counterclockwise. So walking along this short path here, the
colors wind a fifth of the way clockwise through the color wheel. And walking along this path here, the colors
wind another fifth of the way clockwise through the color wheel. Of course, this means that if we go through
both paths, one after another, the colors wind a total of two-fifths of a full turn
clockwise. The total amount of winding just adds up;
this is the kind of straightforward combining that will be useful to us. When I say “total amount of winding”,
I want you to imagine an old-fashioned odometer that ticks forward as the arrow spins clockwise,
but ticks backward as the arrow spins counterclockwise. So counterclockwise winding counts as negative
clockwise winding. The output may turn and turn a lot, but if
some of that turning is in opposite directions, it cancels out. For example, if you move forward along a path,
and then move backward along that same path, the total winding number will end up being
zero: the backwards movement will literally “re-wind” through all the previously seen
colors, reversing all the previous winding and returning the odometer to where it started. We can look at winding along loops, too. For example, if we walk around this entire
loop clockwise, the outputs we come across wind around a total of three full turns clockwise:
the colors swung through the rainbow, in ROYGBIV order, from red to red again… and then again…
and again. In the jargon mathematicians use, we say that
along this loop, the total “winding number” is three. In the case of this loop, the winding number
was three. For other loops it could be any other whole
number. Large ones, if the output swings around many
times as the input walks along a loop. Smaller ones, if the output only swings around
once or twice. Or the winding number could even be a negative
integer, if the output loops around counterclockwise as we walk clockwise around the loop. But along any loop, it will be a whole number,
because by the time we return to where we started, we have to get back to the same output
we started with. (Along paths that don’t end back where they
start, we can get fractions of turns, as we saw earlier, but whenever you combine these
into a loop, the total will be a whole number) Incidentally, if a path actually contains
a point where the output is precisely zero, then we can’t define the winding number
along it, since the output has no particular direction anymore. This won’t be a problem for us, though. Our whole goal is to find zeros, so if this
ever comes up, we’ve just lucked out early. Alright, so winding numbers add up nicely
when you combine paths into bigger paths, but what we really want is for winding numbers
around the borders of regions to add up nicely when you combine regions into bigger regions. Do we have this property? Well, take a look! The winding number as we go clockwise around
this region is the sum of the winding numbers from these paths. And the winding as we go clockwise around
this region is the sum of the winding numbers from these paths. When we combine those two regions into this
bigger region, most of those paths become part of the clockwise border of the bigger
region. And as for the parts that don’t? They cancel out; one is just the other in
reverse, the “re-winding” of the other, as we saw before. So winding numbers along borders of regions
add up just the way we want them to! (As a side note, this reasoning about borders
adding up this way comes up a lot in mathematics, and is often called “Stokes’ Theorem”;
those of you who have studied multivariable calculus may recognize it from that context…) And now, finally, with winding numbers in
hand, we can get back to our equation solving goals. The problem with the region we saw earlier
is that even though its border passed through all colors, the winding number was 0. The outputs wound around halfway, through
yellow to red, then started going counterclockwise back through blue and hitting red from the
other direction, then wound clockwise again in such a way that the total winding netted
out to be zero. But if you find any loop with a nonzero winding
number and split it in half, at least one of the halves is guaranteed to have a nonzero
winding number, since things add up nicely. And in this way, we can keep going, narrowing
in further and further on a point… As we narrow in on that point, we’ll be
doing so using tiny loops with nonzero winding number, which must be tiny loops which go
through every color, and therefore, as we discussed before, that point must be a zero. And that’s it! We have now created our 2d equation solver,
and this time, I promise, there are no bugs. “Winding numbers” are precisely the tool
we needed to make this work. We can now solve equations “Where does f(x)
=g(x)?” in 2d just by considering how the difference between f and g winds around. Whenever we have a loop whose winding number
isn’t zero, we can run this algorithm on it, and are guaranteed to find a solution
somewhere within it. And what’s more, just as in 1d, our equation
solver is remarkably efficient; we keep narrowing in to half the size of our region each round,
thus quickly narrowing in on our zeros, and all the while, we only have to check the value
of the function along points of these loops, rather than checking it on all the many, many
points inside. So in some sense the overall “work” we
do is proportional only to our search space’s perimeter, not its area. Which is amazing! It is weirdly mesmerizing to watch this in
action; just giving it some function and letting search for zeros. Like I said before, complex numbers are 2d,
so we can apply this algorithm to equations between functions from complex numbers to
complex numbers. For example, here’s our algorithm finding
all the zeros of the function x5 – x – 1 over the complex numbers, starting by considering
a very large region around the origin. Each time you find a loop with nonzero winding
number, you split it in half, and figure out the winding numbers of the two smaller loops. Either one or both of the smaller loops will
have nonzero winding number; when you see this, you know there is a zero to be found
in that loop, and so you keep going in the same way in that smaller search space. As for loops whose winding number is zero,
you don’t explore those further inside. (We also stop exploring a region when we stumble
directly across a zero point inside it, right on the nose, as happened once on the right
half here; these rare occurrences interfere with our ability to compute winding numbers,
but hey, we get a zero!). And letting our equation solver continue in
this same way, it eventually converges on lots of zeros of this polynomial. Incidentally, it’s no coincidence that the
overall winding number came out to 5 in this example. With complex numbers, the operation xn directly
corresponds to winding around n times as we loop around the origin, and for large enough
inputs, every term in a polynomial other than the leading term becomes insignificant in
comparison. So any complex polynomial of degree n has
a winding number of n around a large enough loop. In this way, our winding number technology
actually guarantees us that every complex polynomial has a zero; mathematicians call
this the Fundamental Theorem of Algebra. Having an algorithm for finding numerical
solutions to equations like this is extremely practical, but I should say that we’ve left
out a few details on how you’d implement it. For example, to know how frequently you should
sample points, you’d want to know how quickly the direction of the output changes, we’ve
left some more details in the description. But the fundamental theorem of algebra is
a good example of how these winding numbers are also quite useful on a theoretical level,
guaranteeing the existence of a zero for a broad class of functions under suitable conditions. We’ll see a few more amazing applications
of this in a follow-up video, including correcting a mistake from an old 3blue1brown video! (Which one? Rewatch all of our videos, everything we’ve
ever made, and see if you can spot the error first!) The primary author of this video is one of
the newest 3blue1brown team members, Sridhar Ramesh, and with the last video on the Basel
problem you’ve already seen the work of the other new addition Ben Hambrecht. I did a little Q&A session with both of them,
which you can find on the Patreon page, and where you can have fun laughing at how bad
we are with live action filming and lighting. Sridhar and Ben are both incredibly smart
and talented, and additions like these will be crucial for covering all the topics I’d
like to with this channel. The fact is, a lot of time and care goes into
each of these videos. There’s taking the time to explore which
topics have are most likely to deepen people’s relationship with math; and even once you
have that, any of you who write know how many iterations can go into putting together clear
and engaging storyline for a given topic. Obviously creating the visuals to best clarify
an idea in math takes serious time, and quite often involves writing the more general code
for fundamentally different types of visuals. What’s neat is that unlike similar products,
such as movies or college courses, we’re able to offer these stories and lessons for
free. What makes that possible is that a little
less than half a percent of subscribers to this channel have decided this is content
worth paying for, in the form of Patreon pledges, which is amazing! So thank you. And if you’re not in a position to support,
just, don’t worry about it, that’s exactly why the content is free. Just sit back enjoy, and if you really want
to help out, share it with others.

Leave a Reply

Your email address will not be published. Required fields are marked *