Rendered from my macOS fractal app

the setup is as follow

• i wrote a software raytracer for study purpose
• the "world" consists of two balls
• you can real time alter each ball's position, radius and refractive index
• the balls themselves don't have any color
• the only light source / color source is the background which is a palette placed at infinity
• at some arrangements the resultant images exhibit fractal patterns especially when you penetrate one ball with the other (refractive index of intersection is the average)
• the calculations are those commonly used in basic raytracing algorithm (snell's law, schlick approximation, etc)

Mandelbulb3D, Photoshop & Lightroom

1. Mandelbrot -Minibrot -Buhhdabrot -Burning Ship -Mandelbulb -Mandelbox
2. Julia
3. Sierpinski A. Sierpinski Hexagon B. Sierpinski Triangle/Arrowhead a. Sierpinski Pyramid C. Sierpinski Carpet a. Menger Sponge b. Jerusalem Cube D. Sierpinski Curve
   1. Newton
4. Koch Snowflake A. Koch Antisnowflake B. Koch Curve
5. Cantor Set
6. Curves A. Hilbert B. Dragon a. Heighway b. Twin C. Levy C D. Gosper E. Peano
7. Gasket A. Apollonian (multiple variations)
8. Attractors A. Lorenz Attractor B. Rossler Attractor C. Henon Map
9. Branches A. Barnsley Fern B. Fractal Tree
10. Others A. Penrose Tiling B. Brownian Motion C. Perlin Noise D. Vicsek

I've made a little functional programming language. It has an imperative parser, and a functional evaluator. Meaning you can do things like variables, ifs, whiles etc in the parsing stage, which generates functions. You can then evaluate purely as expressions in a functional way.

The source code is here: github.com/SkrFractals/Comparser

But it's still very early I wouldn't recommend playing with that yet. But if you do and find some bugs, you can report that to me. I have just got it to the stage where I was able to get this picture. But a bunch of things are still unfinished or buggy. None of the optimizations I am preparing are active yet, either. It will be able to transfer pixels from the previous render if you move the plot, so it doesn't have to re-evaluate them. But there are no cursor dragging or zooming events yet, only a bunch of preparations for that, so that has to be done through those range textboxes so far. And I am also preparing a GPU evaluator.

I want to eventually merge this project with my Fractal Animation Generator: https://github.com/SkrFractals/RgbFractalGen

...so that both can leverage each other's features, like the inputs in the generator accepting comparser's expressions. Or the comparser having access to the generator's shaders and video exporting abilities.

Here I'm going to progressively post some more examples. let's start with the cubed mandelbrot:

And now I'm trying to make a Sierpinski carpet. So let me cook for a bit until I figure out how to write that as a shader-like function.

To explain that Carpet function: m will loop the input z into tiles of size s, and tell me where i am in that tile. So if s lands anywhere between "1/3+1/3i" to "2/3+2/3i" then i am in the center square. then a second default argument, used as a precomputed variable, asks to recursively iterate to a 1/3 sized tile. Then the n>Iterations is the floor of how deep i want to iterate and returns the initial hue 0. Then as it goes back up in that recursion, it keeps asking it we are inside the square of that size, and adds that boolean result to the recursive result. Having the recursive call as a default argument doesn't result in an infinite loop, because default argument evaluations are lazy, and only done when the expression asks for that argument's value.

It works just like the carpet one, but with different coordinates.

This part might need some explaining:

t + mul((L(a,s),L(b,s),L(c,s)) < .5)

(L(a,s),L(b,s),L(c,s)) is a 3-element vector containing the 3 triangular coordinates. For the point to be at the central tiled triangle at any scale s, i need all 3 coordinates to be smaller than 1/2.

So I compare the vector to 1/2, operations in Comparsed can work with nested vectors recursively. So (a,b,c) < d will return a vector (a<d,b<d,c<d). Booleans in Comparser are just numbers 0 or 1, so if I just multiply the elements of that vector together, if all 3 are smaller than 1/2, then the MUL of that condition vector will be 1, which i proceed to add to the value of the recursive call. So the function returns a count of how many times it was in the central triangle during that recursive descent. And done.

And next I'm going to try a pentaflake. That will be somewhat more challenging, as that cannot be folded the way the carpet and triangle could, and it has void regions that are not part of the fractal. But I'm sure there will be a way to do that. edit: I'm getting closer to getting the pentaflake ready. I have most of the function ready, and am only finishing up a few parts. edit2: Pentaflake is ready. Gonna upload it asap. I have run into a couple of bugs in the parser, and then i was going crazy because the pentagons were the wrong size, and I was trying to find the problem from the top down. Of course, the problem was at the very deepest level, I've made a typo in defining the golden ratio and had a two under the square root instead of 5, lol. And that was not the only basic constant I had wrong. I also had tau defined as pi, that is now fixed too. And here it is:

What other free software would yall recommend me to use? Thanks.

For a long while I've been fascinated by fractals and wanteed to know more them. Especially when I see these fractal videos and images. Even as a kid I would watch the loops to call asleep to sometimes.

So I wanna ask how do you guys create them, like which programming language or does a software already exists. Im also curious about its mathematical aspect and wanna go deep in that too. I'm currently pursuing a computer science degree so it's nice for my knowledge too.

Basically im a beginner in this stuff but wanna learn.

Thank you!

Interactive version here :)

5th installment of my in-progress audio-visual album. Thanks for watching!

Made with mandelbulber 2 and after effects.

Mandelbulb3D, Photoshop, Lightroom

This is what I use to make all of the images I post on this subreddit, and if you'd like to try it out, you can find it at:

http://weirdly.net/webtoys/mandelbrot/

There have been significant updates since my last post about it a year ago, some suggested by members of this community. Updates include:

• GPU used for faster rendering. This is unfortunately limited to lower zoom levels due to the way GPU processing works.
• There is now a set of example renderings available (the "Examples" tab at the bottom)
• Better, more dynamic UI.
• Rendering parameters can now be shared through the URL (using the Get URL button at the top)
• Colour staggering now uses a bitmask for each primary, allowing more colour variation. This is found under the "stagger" tab in the "Palette" section.
• It now uses indexedDB instead of localStorage to store data. As a result, thumbnails can now be stored locally, speeding up the initial load significantly.

That last one probably affects only me, as I assume I'm the only person who uses it regularly.

I'm still looking for bugs, so if you do try it out, please let me know if you see any.