Constructing Fractals -- Iterated Function Systems

Introduction

Last week, after having written my first blog post, I got interested in creating a more general framework which supports more general Iterated Function Systems (IFSs).

In general, an IFS can described as a finite set of contractive functions

f_j: X \to X,

where $X$ is a compact metric space, and the associated fractal $S$ for a generating subset $S_0$ is obtained iteratively through:

\begin{align*} S_{i+1} &= \bigcup_{j} f_j(S_i),\\ S &= \limsup_{i \to \infty} S_i. \end{align*}

This is My implementation is available on my Github, on the manim-projects repository. Here are some project examples:

A visualization of iterations of the Barnsley Fern, linearly interpolated with a linear rate function:

And a visualization of the Vicsek fractal:

Orientability of surfaces

The Sierpiński Triangle