Chaos

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 Clifford attractor is a 2D strange attractor defined by the iterative map: $$ \begin{aligned} x_{n+1} &= \sin(a \cdot y_n) + c \cdot \cos(a \cdot x_n) \\ y_{n+1} &= \sin(b \cdot x_n) + d \cdot \cos(b \cdot y_n) \end{aligned} $$With parameters $a = 1.8$, $b = -1.9$, $c = 1.0$, $d = 1.5$, the system traces out intricate fractal structures. The image above was generated from 300 million iterations, rendered as a density histogram with logarithmic scaling to reveal the fine detail in regions where the trajectory lingers. ...