Mandelbrot Voyage 2

A fully interactive GPU-accelerated customizable fractal zoom program aimed at creating artistic and high quality images & videos.

Mandelbrot set is a set defined in the complex plane, and consists of all complex numbers which satisfy $|Z_n| < 2$ for all $n$ under iteration of $Z_{n+1}=Z_n^2+c$ where $c$ is the complex number in question and $Z_0=0$.

Points inside the set are colored black, while points outside the set are colored based on $n$ (how many iterations it took for the point to diverge).

Features

• Fully customizable equation in GLSL syntax
• Perturbation theory for zooming further than you'll have the patience for
• 11 built-in unique fractals
• Smooth coloring
• Normal mapping for shadows
• TAA (temporal anti-aliasing) and SSAA (super sampling anti-aliasing)
• Customizable color palette with up to 16 colors
• Hold right-click to see the orbit, the corresponding Julia set, and hear the sound waves of the orbit for any point
• Zoom video creation to uncompressed AVI

output.mp4

Zoom video demonstrating perturbation theory using one central reference orbit, 70K iterations, taking 3 minutes to render

2024-05-22.00-45-10.mp4

Custom equations

Users can write custom equations to draw fractals in the complex plane. The following functions are available in addition to the standard functions in GLSL:

Custom functions reference

Double-precision transcendental functions

Function	Definition
double atan2(double, double)	$\tan^{-1}(x/y)$
double dsin(double)	$\sin(x)$
double dcos(double)	$\cos(x)$
double dlog(double)	$\ln(x)$
double dexp(double)	$e^x$
double dpow(double, double)	$x^y$

Complex-defined double-precision functions

Function	Definition
dvec2 cexp(dvec2)	$e^z $
dvec2 cconj(dvec2)	$\bar{z} $
double carg(dvec2)	$\arg{(z)}$
dvec2 cmultiply(dvec2, dvec2)	$z\cdot w$
dvec2 cdivide(dvec2, dvec2)	${z}/{w} $
dvec2 clog(dvec2)	$\ln{(z)}$
dvec2 cpow(dvec2, float)	$z^x, x \in \mathbb{R}$
dvec2 csin(dvec2)	$\sin(z)$
dvec2 ccos(dvec2)	$\cos(z)$

Furthermore, the following variables are also exposed to the user:

Local variables

Name	Description
dvec2 c	Corresponding point in the complex plane of the current pixel
dvec2 z	$Z_n$
dvec2 prevz	$Z_{n-1}$
int i	Number of iterations so far
dvec2 xsq	$\Re^2(Z_n)$, for optimization purposes
dvec2 ysq	$\Im^2(Z_n)$, for optimization purposes
float power	Uniform variable of type float, adjustable from the UI
int max_iters	Maximum number of iterations before point is considered inside the set
double zoom	Length of a single pixel in screen space in the complex plane
dvec2 center	Center point of the window in the complex plane
dvec2 mouseCoord	Point in the complex plane that the mouse cursor is on
dvec2 initialz	Initial value of $Z_n$

The first input (the main equation, must evaluate to a dvec2) is the new value of $Z_{n+1}$ after each iteration. The second input (bailout condition, must evaluate to a bool) is the condition which, when true, the current pixel will be considered inside the set. The third input (initial Z, must evaluate to a dvec2) is $Z_0$.

User-controlled variables can also be defined, which can then be used in the equation and adjusted in real time using the sliders below. "Power" is a default slider that cannot be deleted and corresponds to the power variable above.

Note

Due to limitations in GLSL, zoom is limited to $10^4$ for any non-integer or bigger than 4 power. See Limitations for more information. Click "Round" to round the power to the nearest integer to zoom further.

Building

Dependencies

• Git
• CMake version >= 3.21

Clone the repository with --recurse-submodules, then go into the directory

git clone --recurse-submodules https://github.com/Yilmaz4/MV2.git
cd MV2

Windows

Install MSYS2 to C:\msys64, and from a MSYS2 UCRT64 terminal, run

pacman -Syu mingw-w64-ucrt-x86_64-gcc mingw-w64-x86_64-ninja

Then add C:\msys64\mingw64\bin to PATH.

On Linux

Arch Linux

sudo pacman -S base-devel cmake ninja

Debian/Ubuntu

sudo apt install build-essential cmake ninja-build

Fedora

sudo dnf install gcc-c++ cmake ninja-build pkg-config

openSUSE

sudo zypper in -t pattern devel-basis
sudo zypper install cmake ninja

Finally, generate the build files with CMake and build

cmake -S . -B build -G Ninja
cmake --build build

You can then find the binary in the bin directory

Limitations

• Any custom equation utilizing dvec2 cpow(dvec2, float) where the second argument $\not\in {2, 3, 4}$ will be limited to single-precision floating point, therefore limiting amount of zoom to $10^4$.
• Maximum zoom for all fractals other than the Mandelbrot set is $10^{14}$ due to finite precision. Perturbation can be enabled for the Mandelbrot set to zoom further.

Known issues

• Shader linkage takes very long on Intel iGPUs with Mesa drivers on Linux, causing the program to open only after several minutes, I have no idea why
• Enabling perturbation causes "glitches" to appear such as same-color blobs or noise. This can be minimized with glitch detection algorithms and more reference orbits as described here.
• The zoom videos do not play in VLC or Windows Media Player, however they do in MPV.

Contributing

Contributions are highly welcome, it could be anything from a typo correction to a completely new feature, feel free to create a pull request or raise an issue!