/sci/ - Science & Math

[Anonymous No.9654204]

how the fuck do fractals work

[Anonymous No.9654221]

>>9654204
Magnets

[Anonymous No.9654225]

>>9654204
complex numbers
[math] \displaystyle
\\ z_1 = x_1+y_1i \; \; \; \; \; z_2 = x_2+y_2i
\\ z_1^* = x_1-y_1i \; \; \; \; \; z_2^* = x_2-y_2i
\\ | z_1 | = \sqrt{x_1^2+y_1^2} \; \; \; \; \; | z_2 | = \sqrt{x_2^2+y_2^2}
\\ z_1+z_2 = x_1+x_2 + (y_1+y_2)i
\\ \left | z_1+z_2 \right |^2 = \left ( \sqrt{(x_1+x_2)^2+(y_1+y_2)^2} \right ) ^2 = (x_1+x_2)^2 +(y_1+y_2)^2
\\ z_1z_2^* = x_1x_2 -x_1y_2i +y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_2y_1-x_1y_2)i
\\ z_1^*z_2 = x_1x_2 +x_1y_2i -y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_1y_2-x_2y_1)i
\\ z_1z_2^* + z_1^*z_2 = 2(x_1x_2+y_1y_2) = \text{2Re}(z_1z_2^*) = \text{2Re}(z_1^*z_2)
\\ |z_1|^2+|z_2|^2 + z_1z_2^* + z_1^*z_2 = x_1^2+y_1^2 + x_2^2 + y_2^2 + 2(x_1x_2+y_1y_2)
\\ = (x_1^2 + 2x_1x_2 + x_2^2) + (y_1^2 + 2y_1y_2 + y_2^2) = (x_1+x_2)^2 +(y_1+y_2)^2
\\
[/math]

[Anonymous No.9654234]

>>9654225
how do you get different colors from that
how can a computer calculate this shit in real time

[Anonymous No.9654341]

>>9654234
Not the dude that replied to you, but I think the program marks points on a grid of pixels and changes the hue of any already marked pixel.

[Anonymous No.9654506]

>>9654234
https://www.wikihow.com/Plot-the-Mandelbrot-Set-By-Hand

[Anonymous No.9654520]

>>9654506
holy shit a wikihow article thats unironically great

[Anonymous No.9654940]

>>9654234
When you feed a point/imaginary number on the complex plane into the mandelbrot function you'll get a new point that you can feed into the same function and you do this over and over again generating a new point each time. Some spots on the imaginary plane will generate points that gets further and further away from the original point while another spot will generate points that seem to spiral around the original point.
The spot whose generated points did not get exponentially big is part of the mandelbrot set while the spot whose points did is not but we can give that spot a specific colour depending on how many iterations it took for one of the points to go outside a specific limit. The limit is usually set to 2 because inside that region is where the mandelbrot set is located.

So we can make a program that tests all of the points inside a given region of the complex plane and the gap between each point is determined by the pixel resolution of your window. So we test each point, one by one to see if they belong to the mandelbrot set or not by feeding each point into a loop that loops by a set amount and after each iteration inside that loop you can test if the newly generated point is still inside the limit of 2. If it is you continue looping until the loop stops because it reached the maximum number of iterations that you've given it. If one of the points did go outside the limit then you also stop the loop and then check how many times you had to run the loop to make it go outside the limit and then you give it a specific colour based on that. The points that belong to mandelbrot set are usually coloured black.

[Anonymous No.9654955]

>>9654234
>>9654940
One other thing. Just because a point made it through all of the iterations inside the loop does not necessary mean that the point is part of the mandelbrot set. Look at point B for example. It escaped the limit after 8 iterations but what if we had set the loop limit to 7 or lower? The program would think that it's part of the mandelbrot set and colour it black. So the resolution of the mandelbrot set is determined by how many times you let the loop run. The attached .gif is an example of this.

[Anonymous No.9655006]

No idea.
Don't even know how the fuck do branches even work yet.

[Anonymous No.9655078]

[The Lord No.9655091]

>>9654204
>how the fuck do fractals work
Equation 4.270 shows one way fractals could work. You can represent the matrices as tensors, and then increase the tensor rank to create fractal matrix networks on the first and fifth diagonal positions of a 5D metric in de Sitter or Anti de Sitter space. This has the effect of giving 4D spacetime and added fractal dimension.

[Anonymous No.9655112]

If fractals are self repeating patterns then fractals can only exist if they are nested in a fractal. Since fractals exist does that mean that the universe is a fractal??

[Anonymous No.9655581]

>>9654204
https://en.wikipedia.org/wiki/L-system

[Anonymous No.9655587]

It's just a repeating pattern.

[Anonymous No.9655590]

>>9655112
>If fractals are self repeating patterns then fractals can only exist if they are nested in a fractal.
Wrong

[Anonymous No.9655598]

>>9654204
>perform instructions
>repeat ad infinitum
>???
>fractal

[Anonymous No.9655631]

>>9655091
ty mr Copypaste

[Anonymous No.9655885]

>>9654234
Just look at some source code faggot

You could even play with yourself and modify the resultant image

[Anonymous No.9656318]

>>9654204
You could answer this question for yourself in 30 seconds of cursory google searching and 15 minutes of reading, hell you could even spend an hour or two plotting by hand. All of this could be done for yourself. People who refuse to self help in the information age need to kill themselves.

[Anonymous No.9656322]

>>9654204
anything that has an infinite amount of detail is a fractal
even a coastline is considered a fractal

[Anonymous No.9656355]

>>9656322
technically not

[Anonymous No.9656402]

>>9655112
No.
Fractals are mathematical idealizations.
The real world only APPROXIMATES them.
A computer can show fractal detail continuing at a level below the size of atoms, below the size of protons, below the Planck Length.

Rocks and clouds and leaves can be modeled with fractals but don't take it to mean they ARE fractals.

[Anonymous No.9656416]

>>9656402
yeah but the sierpinski triangle is a fractal too and it doesn't have to have an infinitesimal amount of detail
I think fractal just means recursively self-similar

[Anonymous No.9656420]

>>9656416
Fractal dimensions are used to put a measurement on rough surfaces
Everything in the world is a fractal

[Anonymous No.9656536]

>>9656318
Dude, don't be so negative. It's a good thread for a science board.

[Anonymous No.9657607]

>>9656536
A good thread would have been op reading about fractals, postulating an interesting question, and then posting it. I would even have accepted "yo anons lets have an indepth discussion on fractals under a broad space" or some such.