[00:01] iconic images in all of math but do you know how it's defined you start with some complex number c which will visualize with this movable yellow Dot and then you recursively Define a sequence of complex numbers where the
[00:14] sequence starts with zero and each new value is defined to be the square of the previous Value Plus C so for example on the very first iteration you take 0^2 + the very first iteration you take 0^2 + C meaning Z1 is just C and then for the
[00:29] next iteration you take that number squared plus C meaning Z2 is c^2 + C and in the picture you can see how as I change the choice of C the second value will change in lock step and in general you keep going each new value is the
[00:43] square of the previous Value Plus C creating this infinite sequence in the complex plane which as you can see changes as I change the value of C now for some choices of C the sequence stays bounded but for other choices the terms
[00:57] blow up and go to Infinity if you color all of the values of c that cause this process to stay bounded black and you apply some gradient of colors to the other values where the color depends on how quickly the process blows up to
[01:10] Infinity you get this iconic cardioid with bubbles shape and you can say a little more the main cardioid in the middle corresponds to all of the values of c where this process will approach a single limit point and this big circle
[01:23] choices of C where the process tends to approach a state where it kind of bounces back and forth between two values and then the circles on the top correspond to choices of C where the
[01:35] between three values and in general each part of this image corresponds to some qualitatively distinct behavior of the sequence