Fractals etc
Fredrik Meyer (19/10-2023)
Something beautiful
in the intersection of math and programming
Also motivating with visual feedback and deep problems.
What are fractals?
- Self similarity
- Detailed at all scales
- Tendencies to chaos
No precise definition covering all use cases exist.
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.
Abelian Sandpiles
Start with a pile of sand
See Wikipedia and this nice Nautilus article.
In Rust
Show different starting points
Conclusion
- Fun
- Learning (some algorithms are slow)
- Math is beautiful
Takk for alt!
My last fagdag-talk (in a while!?) 🥺 It has been really fun to get to know you all, and I really hope our paths will continue crossing.