Fractals

A collection of mathematical animations built with Manim. Each one visualizes a concept from fractal geometry or dynamical systems. Box-Counting Dimension Estimates the fractal dimension of a binary image by overlaying grids at progressively finer scales and counting how many cells contain structure. The box-counting dimension is the slope of the log-log regression: $$d = \lim_{r \to 0} \frac{\log N(r)}{\log(1/r)}$$where $N(r)$ is the number of boxes of size $r$ that intersect the fractal. The animation sweeps through grid sizes from 4 to 128, plotting each $(log(1/r),\ \log N(r))$ point in real time and fitting the line at the end. ...

The Julia set for a complex number $c$ is the boundary between points that escape to infinity and those that remain bounded under iteration of: $$z_{n+1} = z_n^2 + c$$Each value of $c$ produces a different fractal. Connected Julia sets correspond to points inside the Mandelbrot set; disconnected “dust” fractals come from points outside. Controls: Click the Mandelbrot set (left) to choose $c$. Drag to pan, scroll to zoom the Julia set. ...