README.md

Multibrot & Julia Sets Visualization

This project is a visualizer for Multibrot and Julia Sets. It allows to create and visualize complex fractals using various coloring techniques. The renderer is highly customizable. The project is written in TypeScript and uses WebGL for rendering, with GLSL shaders for fractal generation and coloring.

Try the web demo.

🔎 About

✤ Multibrot Set

The Multibrot Set $\mathcal{M}$ is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations.

$$f_c : \mathbb{C} \longrightarrow \mathbb{C}$$

in the form:

$$f_c(z) = z^p + c$$

For each complex parameter $c$ we consider the behavior of the succession $(0, f_c(0), f_c(f_c(0)), \dots )$ obtained by iterating $f_c(z)$ from the point $z = 0$ this can either diverge to infinity or be limited. The Multibrot set is defined as the set of points $c$ such that the corresponding subsequence is limited. The previous succession can also be written as:

$$\begin{cases}z_0 = 0 \\ z_{n+1} = z^p + c\end{cases}$$

if we indicate with $f^n_c(z)$ the $n$-th iteration of $f_c(z)$, $(f_c \circ f_c \circ \dots \circ f_c)(z)$ self composed $n$ times, the Multibrot Set is:

$$\mathcal{M} = \{\ c \in \mathbb{C} : \sup_{n \in \mathbb{N}}\ |f^n_c(z)| < \infty \}\ $$

The case of $p = 2$ is the classic Mandelbrot set from which the name is derived.

To visualize a Multibrot Set, one typically starts with a grid of complex numbers in the complex plane. For each complex number $c$ on the grid, we generate the sequence by iteratively applying the function $f_c$ starting from $z = 0$. Each point $c$ on the grid is then colored according to the behavior of its corresponding sequence. The point is colored according to the number of iterations it took for the sequence to exceed a certain threshold. This process results in a fractal image that represents the Multibrot Set.

✷ Julia Set

The Julia Set $\mathcal{J}$ is a subset of the complex plane that, like the Multibrot Set, is associated with the behavior of complex dynamical systems. However, while the Multibrot Set is defined by the behavior of a sequence of complex numbers under iteration by a function, the Julia Set is defined by the behavior of a single complex number under iteration by the same function.

For a given complex number $c$, the Julia Set is defined as the set of all complex numbers $z$ such that the sequence $(z, f_c(z), f_c(f_c(z)), \dots)$ does not tend towards infinity. In other words, the Julia Set for a given $c$ is the set of all $z$ for which the sequence remains bounded under iteration by the function $f_c(z) = z^p + c$. We have to fix a value $B &gt; 0$ such that $B^2 - B \ge |c|$, then the filled Julia set for this system is the subset of the complex plane given by

$$\mathcal{J} = \{\ z \in \mathbb{C} : \forall n \in \mathbb{N}, |f^n_c(z)| \le B\}\ $$

Each Julia Set is associated with a unique point in the Multibrot Set, and the shape of the Julia Set can provide insights into the behavior of the complex dynamical system associated with its corresponding point in the Multibrot Set. Values of the parameter $c$ for which Julia's set is connected are those that are part of Multibrot. For values of $c$ outside the Multibrot Set the Julia Set is a Cantor Space (it's omeomorfic to the Cantor Set).

Visualizing the Julia Set involves a process similar to the Multibrot Set, but with a slight variation. We begin with a point $c$ from the Multibrot Set, then for each complex number $z$ we iterate the function $f_c(z)$, with each point colored according to the number of iterations it took for the sequence to exceed a certain threshold.

🟢 Getting started

✔️ Prerequisites

You need to have Node.js and npm installed on your machine. You can download Node.js here and npm is included in the installation.

🛠 Installation

First, clone the repository to your local machine:

git clone https://github.com/xhst/multibrot-set.git

then, navigate to the project directory and install the dependencies with:

npm install

▶️ Build

To build the application, run:

npm run build

Or, to have a localhost webpack server, run:

npm run serve

🖼️ Screenshots

Mandelbrot and Julia sets, point $(0.2815, -0.0085i)$, with grayscale coloring

Mandelbrot and Julia sets, point $(-0.245, -0.655i)$, with linear coloring

Mandelbrot and Julia sets, point $(-0.8, -0.155i)$, with normalized iteration count (6 bail-out radius) coloring

10 iterations Mandelbrot

100 iterations Mandelbrot

1000 iterations Mandelbrot