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.
Cantor Set
Iterative construction of the Cantor set — at each step, the middle third of every interval is removed. The animation labels each interval with its exact fractional endpoints, shrinking the font as the intervals multiply. Five iterations produce 32 segments, each of length $1/243$.
Cobweb Plot
A cobweb diagram for the logistic map:
$$x_{n+1} = 3.8\, x_n(1 - x_n)$$Starting from $x_0 = 0.2$, the iteration bounces between $f(x)$ and $y = x$ for 100 steps. At $r = 3.8$ the system is chaotic — the trajectory never settles into a periodic orbit, instead visiting a dense subset of $[0, 1]$.
Hilbert Curve
Progressive construction of the Hilbert space-filling curve from order 1 through 8. Each order is a recursive transformation that visits $4^n$ points in a $2^n \times 2^n$ grid while preserving locality — nearby points on the curve tend to be nearby in 2D space. The order-8 curve passes through 65,536 points.