[00:00] which is known as Newton's fractal. And aside from being this beautiful, mesmerizing, infinitely detailed mess, what's really cool about the image is how it naturally pops out from something that
[00:14] any of you who've taken a calculus class might have heard of, known as Newton's method. powerful tool for finding approximate solutions to essentially any equation.
[00:26] If you rearrange the equation so that it looks like setting some function of x equal to zero, then visually what it means to solve such an equation is to find a point where the graph of that function crosses the x-axis.
[00:39] Now the way that Newton's method works is to start by making an arbitrary guess. But if you draw a tangent line to the graph at that guess, something much easier to solve explicitly, very often it takes you closer
[00:56] If you wash, rinse, and repeat doing this multiple different times, If you do all the calculus and you work it out, there's a certain formula for how big your step size should be based on this process.
[01:10] formula and you can plug in one value, and what comes out is usually something closer to a true solution of the equation. inputs and real number outputs like we're used to,
[01:26] consider all of the complex number inputs and the corresponding complex number outputs. intersecting the x-axis anymore, but the formula still works.
[01:38] one guess to push it closer to a true solution of the equation. loves complex numbers the way I do, but here's the cool part.
[01:50] Watch what happens if we apply this idea to many many different possible initial guesses. On each iteration, each of those dots is taking a step based on this Newton's method rule. With the example I'm showing, where our function is a degree 5 polynomial,
[02:05] there are five distinct solutions to the equation somewhere in the complex plane. have zeroed in towards one of those solutions. What I'm going to do is color each one of those dots based on which of those five
[02:18] and then we'll kind of roll back the clock to see where each dot came from in the first If you do this at a very fine resolution, treating each pixel of the
[02:33] coloring it based on which route that guess would end up landing on, the pattern that emerges is the image that I showed you at the start. On the full video, I talk much more about why you see a pattern like this,
[02:49] and how it connects to things like the Mandelbrot set. But for now, it's fun to just gawk at the beauty of math.