An Introduction To The Mandelbrot Set

You guys may have heard this, in your math textbook. They reall-

what? You haven't? Oh, ok then. I guess I should explain it to you.

WARNING: A LOT OF MATH INCOMING; IF YOU'D LIKE TO SKIP THE MATH, SKIP TO THE VIDEO

The Mandelbrot Set

The Mandelbrot Set was deemed the most "Complicated structure in math" by Guinness World Records. Let me tell you about how it works.

First, lets talk about imaginary numbers.

you all know that the square root of -1 is impossible. Nothing squared equals a negative. However, a certain symbol is used for this scenario. i is equal to the square root of negative one.

Now, you can use this for many applications, like equations, quadratics, and the complex plane. The one we're going to be talking about is the complex plane.

if i is the square root of negative one, you can multiply it with the square root of 4 to equal this;

the square root of negative one times the square root of 4 is equal to the square root of negative 4. This can then be further simplified into the square root of one times 2, which is just 2i.

Now, we call the products of a real constant and the imaginary unit pure complex numbers. Wait!
What are complex numbers? Complex numbers are numbers in the form "a+bi". So basically, 4 is a complex number. It is 4+0i. Complex numbers is the set containing both real and imaginary numbers.

Now, This is where it gets interesting.
You should all know the coordinate plane. The complex plane is just like that, except the y-axis is imaginary, and the x-axis is real. It is really easy to graph on it. To graph 4+0i, move right 4, but move none up. To graph 5-4i, move right 5 and go down 4.

Now it's time to generate the Mandelbrot Set.

The formula to generate the Mandelbrot Set is Zn = Zn-1^2+C. Due to the formatting of the forums, you probably didn't understand that, but just search the formula online.

Now, for any complex number as C, we plug it in. Notice this formula is a recursive sequence formula, which means you test C for any increasing value of n.

Now say, the complex number -1.5+i.
If you plug it into the sequence, you may notice Zn gets increasingly larger, never looping. This means it ISN'T in the set.

For every complex number that isn't in the set, you will color it one color. Say, blue.

Now, think about the complex number -1.
If you plug it into the sequence, you may notice that for n=0, Zn is 0. For n=1, Zn is -1. For n=2, Zn is 0. See the pattern? It loops! Any complex number that loops is part of the set! Lets color this point black.

Now. What if you did that for every single complex number, and color-coded each one on how fast it grew?

Get ready for your mind to be blown.

It would look like this;



Strange shape, huh? Now; the coolest thing; the patterns get so complex, that somewhere in the set, you will find the exact copy of this, except a little bit smaller!

Now, the quest is on, to determine how far we can zoom in the this complex structure.

 

i thought that was 1 :thonk:

 

c

crap

 

I was with you until the math. So basicly the mandelbrot set is recursive?

 

welp, looks like I found out that there is no meaning to life because of whatever the fuck I just read.
time to go tell everyone to just forget about life and to just jump off this stupid flat earth we live on.
(flat earthers are proof that there are people without brains)

 

isnt that called a fucking fractal
(sorry for my incompetent little brain, i may not have read the entire thing)

 

it's a type of fractal

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yep

 

Mind = blown