hi I'm going to talk about the mandelbrot set the mandelbrot set is a fractal discovered by benois B mandelbrot it is generated with this recursive formula on the complex plane what does that mean let's start from the basics every number has a square root right wrong -1 has no real square root because a negative number squared is always positive its square root must be imaginary the square root of1 is an imaginary number we will call I any number involving I is known as a complex number the neat thing about I is that when you square
it it equals 1 this can make some interesting things in mathematics possible this is the complex plane instead of an X and Y AIS we have a real and imaginary axis cool for example this complex number would be here on the plane 1 and 1/2 units left and one unit up Mega cool oh but it gets cooler remember our recursive formula from from before the C here represents a complex number let's use our good old friend -1.5 + i z Sub 0 = 0 Z sub 1 = 0^ 2 + C which is our number
Z sub 2 = that squar plus our number that's 0.25 - 2 I Zu sub 3 = that squar plus our number which equals 54375 + 2 I when we do this for a while we can see that the number keeps growing and growing it's therefore not in the mandal BR set so we'll color it oh a shade of blue let's try the equation with the number -1 Z sub 1 = 0^ 2 - 1 Z sub 2 = -12 - 1 Z sub 3 = 0^ 2 - 1 Z sub 4 = -1 2
- hey wait a minute we're going in a loop that means -1 is a part of the mandle BR said we'll color that point black now what if we did this with every point in the complex plane color coding each point based on how fast the number grows if the number goes in a loop we color it black doing that we get this shape crazy the edges are kind of rough though let's zoom in and see what this thing really looks [Music] like wow look at that structure check it out we zoom in here and
find some fivefold Symmetry and next to it Sevenfold symmetry then ninefold symmetry then 11 then 13 and if we go way far down each arm Curves in on itself making an incredible intricate structure within the mandal BR set we can find a near perfect copy of the entire thing and look here's another but it's a little crooked no one designed this mathematical creature the order structure and Beauty we find is simply embedded in reality in this image I see God we live in a world of which beauty is an inherent Inseparable part think for a
second if no one created this image how could it be so beautiful how could it be that way it just is sometimes there is no reason [Music] why
