Mandelbrot

After Benoit Mandlbrot (1924-2010)

Define a sequence $\{z_n\}$ by $z_{n+1}=z_n^2+c$. Then the Mandelbrot set is defined as the set of $c$'s in the complex plane such that $z_n \not \to \infty$.

Demo Time 🥰

Mandelbrot-simulation from back in the day. (remember to also check out the burning ship fractal)

IFS - Iterated Function Systems

A set of contraction mappings \[ \left\{ f_i: \mathbb R^2 \to \mathbb R ^2 \mid i=1,2,\ldots,N \right \}\] This gives us a fractal that can be thought of as a "fix point": \[ S = \overline {\bigcup_{i=1}^N f_i(S)} \]

Some examples

Many examples here: Paul Bourke (the page is a treasure trove)

Demo time 🥰

Can generate pictures by starting with a point and iterating randomly.

IFS Editor

Abelian Sandpiles

Start with a pile of sand

See Wikipedia and this nice Nautilus article.

In Rust

Show different starting points